A high-precision calibration method for a mechanical arm based on multi-station measurement

Abstract

This invention discloses a high-precision calibration method for a robotic arm based on multi-station measurement, comprising: 1. Installing a 3D measuring device at the j-th station and installing a measuring rod at the end of the robotic arm; 2. Driving the robotic arm to bring the ball head at the end of the measuring rod into the measurement area, at which point the robotic arm is in the i-th robotic arm posture, and measuring the coordinate x of the ball center relative to the coordinate system of the 3D measuring device, calculating the offset of the measuring rod caused by gravity and / or measurement contact force, and compensating it into the ball center coordinate x to obtain the compensated ball center coordinate x'; 3. Adjusting the posture of the robotic arm and repeating step 2; 4. Installing the 3D measuring device at other stations and repeating steps 2 and 3; 5. Converting the compensated ball center coordinate x' to the same coordinate system to obtain the ball center coordinate x”; 6. Identifying the true kinematic parameters of the robotic arm based on the joint angle q and the ball center coordinate x” obtained from multiple measurements.

Description

Technical Field

[0001] This invention relates to the field of robotics, and in particular to a high-precision calibration method for robotic arms based on multi-station measurement. Background Technology

[0002] The repeatability of robotic arms is generally very high, but their absolute positioning accuracy is usually low due to the influence of machining and assembly errors, deformation of the rods, etc. To improve the absolute positioning accuracy of robotic arms, kinematic parameter calibration is generally required. This requires the use of high-precision measuring equipment and appropriate parameter identification methods to identify the accurate parameters of the robotic arm model.

[0003] Taking CN113084798A as an example, a robot calibration device based on multi-station measurement includes a calibrated device, a calibration device, and a multi-station template. The multi-station template provides multiple calibration measurement positions to expand the calibration measurement range of the calibration device for the calibrated device. The positional relationship between the calibrated device and the calibration device is either that the calibrated device is connected to the robot arm while the calibration device is connected to the multi-station template, or that the calibration device is connected to the robot arm while the calibrated device is connected to the multi-station template. This calibration device can quickly calibrate robot arms at any time in the industrial field. It is simple and convenient to operate and has high calibration accuracy. By using the multi-station template to achieve measurement at multiple stations, a larger measurement range can be obtained. At the same time, the relative positions between each station are known information, which can be used for the identification of robot arm kinematic parameters, thereby improving calibration accuracy.

[0004] However, during the calibration process, the measuring rod is subjected to gravity and / or measurement contact force. This gravity and / or measurement contact force can cause the measuring rod to deviate, and the amount of deviation caused by gravity and / or measurement contact force can affect the high-precision calibration of the robotic arm. Currently, there is no mature and reliable calibration method to address this issue. Summary of the Invention

[0005] The purpose of this invention is to provide solutions to existing technical deficiencies and unmet technical requirements.

[0006] To achieve the above objectives, the present invention provides the following technical solution:

[0007] A high-precision calibration method for a robotic arm based on multi-station measurement includes the following steps:

[0008] Step 1: Install the 3D measuring device on the j-th station of the multi-station base, and install the measuring rod at the end of the robotic arm;

[0009] Step 2: Drive the ball head at the end of the measuring rod of the robotic arm into the measurement area of ​​the 3D measuring device. At this time, the robotic arm is in the i-th robotic arm posture. Record the joint angle q of the robotic arm at this time, and measure the coordinate x of the ball center relative to the coordinate system of the 3D measuring device through the 3D measuring device. Based on the posture angle of the measuring rod at this time, calculate the offset of the measuring rod caused by gravity and / or measurement contact force, and compensate it into the ball center coordinate x to obtain the compensated ball center coordinate x'. Eliminate the measurement error caused by the deformation of the measuring rod.

[0010] Step 3: Adjust the posture of the robotic arm and repeat Step 2 to obtain data for multiple measurement configurations;

[0011] Step 4: Install the 3D measuring device on other workstations of the multi-station base, and repeat Step 2 and Step 3;

[0012] Step 5: Based on the relative pose relationship between different workstations on the multi-station base, convert the compensated sphere center coordinates x' to the same coordinate system to obtain the sphere center coordinates x” after unifying the coordinate system;

[0013] Step 6: Based on the joint angle q and ball center coordinate x” of the robotic arm obtained from multiple measurements, identify the true kinematic parameters of the robotic arm and complete the calibration.

[0014] Preferably, in step two, before each data recording, the center of the ball at the end of the measuring rod is driven by the robotic arm to reach the same point. This point is a fixed point relative to the 3D measuring device. Under the displacement feedback of the three displacement sensors of the 3D measuring device, the robotic arm gradually adjusts the angle of each joint of the robotic arm so that the center of the ball at the end of the measuring rod reaches the fixed point.

[0015] Preferably, when the 3D measuring device is installed at the j-th workstation and the robotic arm is in the i-th robotic arm posture, the coordinate x of the center of the ball of the measuring rod is denoted as... The compensated coordinates of the sphere's center, x', are denoted as The compensated coordinates of the sphere's center are calculated using the following formula:

[0016]

[0017] in It is a 3×3 rotation matrix between the robotic arm flange coordinate system {T} and the measurement coordinate system {D}. Let be the offset vector of the ball center of the measuring rod relative to the robot arm flange coordinate system {T} at the j-th station and the i-th robot arm posture.

[0018] Preferably, for 3D measuring devices using contact measurement, the TCP offset of the measuring rod can be calculated using the following formula:

[0019]

[0020] Alternatively, for 3D measuring devices that use non-contact measurement, the TCP offset of the measuring rod can be calculated using the following formula:

[0021]

[0022] in To measure the TCP offset of the measuring rod caused by the contact force at the j-th station and the i-th robot arm posture, C represents the TCP offset of the measuring rod caused by gravitational acceleration. F It is the contact force of the measuring rod on the measuring device. The compliance matrix, C g It is the acceleration of the measuring rod relative to gravity. The flexibility matrix, Let be the vector value of gravitational acceleration relative to the flange coordinate system {T}. To measure the vector value of the contact force relative to the flange coordinate system {T}.

[0023] As a preferred option, C F It is the contact force of the measuring rod on the measuring device. The compliance matrix, Calculate using the following formula:

[0024]

[0025] in It is the compliance coefficient of the measuring rod in different directions, which can be obtained through theoretical calculation, physical testing, or finite element simulation analysis.

[0026] As a preferred option, C g It is the acceleration of the measuring rod relative to gravity. The flexibility matrix, Calculate using the following formula:

[0027]

[0028] Its internal elements It is the compliance coefficient of the measuring rod in different directions, which can be obtained through theoretical calculation, physical testing, or finite element simulation analysis.

[0029] Preferably, at the j-th workstation and the i-th robotic arm posture, the vector value of the contact force relative to the flange coordinate system {T} is measured and denoted as... Calculate using the following formula:

[0030]

[0031] in, It is a 3×3 rotation matrix between the coordinate system {D} of the 3D measuring device and the flange coordinate system {T} of the robotic arm; It is the rotation matrix between the base coordinate system {O0} and the flange coordinate system {T} of the robotic arm, which is calculated using the current joint angles and nominal kinematic parameters of the robotic arm; It is the rotation matrix between the robot arm base coordinate system {O0} and the 3D measuring device coordinate system {D} under the j-th workstation and the i-th robot arm posture, based on the 6D base parameters obtained during calibration. get, Let be the vector value of the measuring contact force of the 3D measuring device installed at the j-th station on the measuring rod relative to the coordinate system {D} of the 3D measuring device.

[0032] As a preferred option Calculate using the following formula:

[0033]

[0034] Where, χ j The coordinate system {D} of the 3D measuring device is the coordinate system {D} of the j-th workstation. j The 6D vector of relative pose between}, R(χ j F is a 3×3 rotation matrix derived from the attitude angles in the 6D vector. D When a 3D measuring device is installed at any workstation, the measuring contact force of the 3D measuring device on the measuring rod is relative to the workstation coordinate system {D}. j The vector value of}.

[0035] Preferably, when the 3D measuring device uses contact measurement, if the center of the ball at the end of the measuring rod is located at a fixed point relative to the 3D measuring device, the measuring contact force F of the probes of the three displacement sensors of the 3D measuring device on the ball is constant. Even if the 3D measuring device is installed at different workstations, the measuring contact force F relative to the coordinate system {D} of that workstation remains constant. j The vector value F of} D It is also constant, the vector value F D The force can be obtained by measuring the probe thrust of the three displacement sensors separately using a force gauge.

[0036] Preferably, the vector value of gravitational acceleration relative to the flange coordinate system {T} is g. T The calculation is performed using the following formula:

[0037]

[0038] Where g 0It is the vector value of gravitational acceleration relative to the coordinate system {O0} of the robotic arm base; It is the rotation matrix between the robot arm's base coordinate system {O0} and the flange coordinate system {T}, calculated using the robot arm's current joint angles and nominal kinematic parameters; since the Z-axis of the robot arm's base coordinate system {O0} is the vertical direction, therefore g 0 =g=[0, 0, 9.81] T (Unit: m / s) 2 ).

[0039] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0040] 1. When the 3D measuring device adopts contact measurement, the offset of the measuring rod caused by gravity and measurement contact force is calculated and compensated into the ball center coordinate x, so as to obtain the compensated ball center coordinate x'. The measurement error caused by the deformation of the measuring rod is eliminated, the measurement accuracy is improved, and the calibration accuracy of the robotic arm is ultimately improved.

[0041] Second, when the 3D measuring device adopts non-contact measurement, the offset of the measuring rod caused by gravity is calculated and compensated into the ball center coordinate x, so as to obtain the compensated ball center coordinate x'. The measurement error caused by the deformation of the measuring rod is eliminated, which improves the measurement accuracy and ultimately improves the calibration accuracy of the robotic arm. Attached Figure Description

[0042] Figure 1 This is a schematic diagram illustrating the bending deformation principle of the measuring rod of the present invention.

[0043] Figure 2 This is a schematic diagram of the structure of the robotic arm of the present invention;

[0044] Figure 3 This is a state diagram of the measuring rod ball head of the 3D measuring device of the present invention;

[0045] Figure 4 This is an enlarged view of the measurement area structure of the 3D measurement device of the present invention. Detailed Implementation

[0046] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0047] like Figure 1 and Figure 2The measuring rod 13 is mounted on the robotic arm 11, and a ball head 12 is mounted on the front end of the measuring rod 13. The 3D measuring device 8 is mounted on the base, which has multiple components.

[0048] Or such as Figure 3 and Figure 4 The base adopts a multi-station base 2, which has several stations. The 3D measuring device 8 can be installed in different stations. The 3D measuring device 8 has a measuring area 9 for the ball head 12 to be placed in. Three displacement sensors are provided in the measuring area 9. The probe 14 of the displacement sensor exerts a certain thrust on the ball head 12.

[0049] A high-precision calibration method for a robotic arm based on multi-station measurement includes the following steps:

[0050] Step 1: Install the 3D measuring device 8 on the j-th station of the multi-station base 2, and install the measuring rod 13 at the end of the robotic arm 11;

[0051] Step 2: Drive the ball head 12 at the end of the measuring rod 13 with the robotic arm 11 into the measurement area 9 of the 3D measuring device 8. Before each data recording, drive the center of the ball head 12 at the end of the measuring rod 13 with the robotic arm 11 to the same point. This point is a fixed point relative to the 3D measuring device 8. Under the displacement feedback of the three displacement sensors of the 3D measuring device 8, the robotic arm 11 gradually adjusts the angle of each joint of the robotic arm 11 so that the center of the ball head 12 at the end of the measuring rod 13 reaches the fixed point. At this time, the robotic arm 11 is in the i-th posture of the robotic arm 11. Record the joint angle q of the robotic arm 11 at this time, and measure the coordinate x of the center of the ball head 12 relative to the coordinate system of the 3D measuring device 8 through the 3D measuring device 8. Based on the posture angle of the measuring rod 13 at this time, calculate the offset of the measuring rod 13 caused by gravity and / or measurement contact force, and compensate it into the ball center coordinate x to obtain the compensated ball center coordinate x'. Eliminate the measurement error caused by the deformation of the measuring rod 13.

[0052] Step 3: Adjust the posture of robotic arm 11, repeat step 2, and obtain data for multiple measurement configurations;

[0053] Step 4: Install the 3D measuring device 8 on other stations of the multi-station base 2, and repeat steps 2 and 3.

[0054] Step 5: Based on the relative pose relationship between different stations on the multi-station base 2, convert the compensated sphere center coordinates x' to the same coordinate system to obtain the sphere center coordinates x” after unifying the coordinate system;

[0055] Step 6: Based on the joint angle q and ball center coordinate x” of the robotic arm 11 obtained from multiple measurements, identify the true kinematic parameters of the robotic arm 11 and complete the calibration.

[0056] The 3D measuring device 8 can be used for contact or non-contact measurement.

[0057] For the case where the 3D measuring device 8 uses contact measurement: when the center of the ball head 12 at the end of the measuring rod 13 is located at a fixed point relative to the 3D measuring device 8, the measuring contact force F of the probes 14 of the three displacement sensors of the 3D measuring device 8 on the ball head 12 is constant. Even if the 3D measuring device 8 is installed at different work positions, the measuring contact force F relative to the coordinate system {D} of that work position remains constant. j The vector value F of} D It is also constant, the vector value F D The thrust of the probe 14 of the three displacement sensors can be measured using a force gauge. in These are the thrusts of the displacement sensors in the X, Y, and Z directions, respectively, and the force gauge can be a spring force gauge.

[0058] Then the vector value of the measuring contact force of the 3D measuring device 8 installed at the j-th workstation relative to the coordinate system {D} of the 3D measuring device 8 is... Calculate using the following formula:

[0059]

[0060] Where, χ j The coordinate system {D} of the 3D measuring device is the coordinate system {D} of the j-th workstation. j The 6D vector of relative pose between}, R(χ j F is a 3×3 rotation matrix derived from the attitude angles in the 6D vector. D When the 3D measuring device 8 is installed at any workstation, the measuring contact force of the 3D measuring device 8 on the measuring rod 13 relative to the workstation coordinate system {D} is... j The vector value of}.

[0061] In the j-th workstation and the i-th robotic arm's 11-position, Transformed into the flange coordinate system {T} at the end of the robotic arm 11, the vector value of the measured contact force relative to the flange coordinate system {T} is denoted as... Calculate using the following formula:

[0062]

[0063] in, It is a 3×3 rotation matrix between the coordinate system {D} of the 3D measuring device 8 and the flange coordinate system {T} of the robotic arm 11; It is the rotation matrix between the base coordinate system {O0} and the flange coordinate system {T} of the robotic arm 11, which is calculated using the current joint angles and nominal kinematic parameters of the robotic arm 11; It is the rotation matrix between the robot arm 11 base coordinate system {O0} and the 3D measuring device 8 coordinate system {D} under the j-th workstation and the i-th robot arm 11 posture, based on the 6D base parameters obtained during calibration. get.

[0064] Similarly, the vector value of gravitational acceleration relative to the flange coordinate system {T} is g. T A similar method can be used, and the calculation can be performed using the following formula:

[0065]

[0066] Where g 0 It is the vector value of gravitational acceleration relative to the coordinate system {O0} of the base of the robotic arm 11; It is the rotation matrix between the base coordinate system {O0} and the flange coordinate system {T} of the robotic arm 11, calculated using the current joint angles and nominal kinematic parameters of the robotic arm 11; since the Z-axis of the base coordinate system {O0} of the robotic arm 11 is the vertical direction, therefore g 0 =g=[0, 0, 9.81] T (Unit: m / s) 2 ).

[0067] Next, a deformation model of the measuring rod is established. Based on the knowledge of mechanics of materials, it can be determined that the contact force being measured... Caused TCP offset Contact force for measurement The relationship is linear; the TCP offset can be predicted simply by obtaining the compliance matrix and the measured contact force. This includes the TCP offset of the measuring rod caused by the measured contact force at the j-th station and the i-th robotic arm posture. Calculate using the following formula:

[0068]

[0069] Where C F It is the contact force of the measuring rod on the measuring device. The compliance matrix, its internal... It is the compliance coefficient of the measuring rod in different directions, which can be obtained through theoretical calculation, physical testing, or finite element simulation analysis.

[0070] acceleration due to gravity Caused TCP offset With gravitational acceleration It also shows a linear relationship.

[0071] Calculate using the following formula:

[0072]

[0073] Where C g It is the acceleration of the measuring rod relative to gravity. The flexibility matrix, whose internal elements It is the compliance coefficient of the measuring rod in different directions, which can be obtained through theoretical calculation, physical testing, or finite element simulation analysis.

[0074] Finally, and Add them together to get the final overall TCP offset.

[0075]

[0076] Therefore, given the known vector values ​​of the contact force and gravitational acceleration in the flange coordinate system {T}... With g T In this case, only the compliance matrix C needs to be obtained. F With C g This allows us to predict the overall TCP offset.

[0077] When the 3D measuring device 8 is installed at the j-th workstation and the robotic arm 11 is in the i-th robotic arm 11 posture, the coordinate x of the center of the ball of the measuring rod is denoted as... The compensated coordinates of the sphere's center, x', are denoted as The compensated coordinates of the sphere's center are calculated using the following formula:

[0078]

[0079] in It is a 3×3 rotation matrix between the flange coordinate system {T} and the measurement coordinate system {D} of the robotic arm 11, which can be calculated based on the forward kinematics formula of the robotic arm 11.

[0080] Alternatively, the 3D measuring device 8 may employ non-contact measurement: since no contact force is measured. Caused TCP offset The TCP offset of the measuring rod can be calculated using the following formula:

[0081]

[0082] When the 3D measuring device 8 is installed at the j-th workstation and the robotic arm 11 is in the i-th robotic arm 11 posture, the coordinate x of the center of the ball of the measuring rod is denoted as... The compensated coordinates of the sphere's center, x', are denoted as The compensated coordinates of the sphere's center are calculated using the following formula:

[0083]

[0084] in It is a 3×3 rotation matrix between the flange coordinate system {T} and the measurement coordinate system {D} of the robotic arm 11, which can be calculated based on the forward kinematics formula of the robotic arm 11.

[0085] The foregoing has shown and described the basic principles, main features, and advantages of the present invention. It will be apparent to those skilled in the art that the invention is not limited to the details of the exemplary embodiments described above, and that the invention can be implemented in other specific forms without departing from its spirit or essential characteristics. Therefore, the embodiments should be considered illustrative and non-limiting in all respects, and the scope of the invention is defined by the appended claims rather than the foregoing description. Thus, all variations falling within the meaning and scope of equivalents of the claims are intended to be included within the scope of the invention. No reference numerals in the claims should be construed as limiting the scope of the claims.

[0086] Furthermore, it should be understood that although this specification describes embodiments, not every embodiment contains only one independent technical solution. This narrative style is merely for clarity. Those skilled in the art should consider the specification as a whole, and the technical solutions in each embodiment can also be appropriately combined to form other embodiments that can be understood by those skilled in the art.

Claims

1. A high-precision calibration method for a robotic arm based on multi-station measurement, characterized in that, Includes the following steps: Step 1: Install the 3D measuring device on the j-th station of the multi-station base, and install the measuring rod at the end of the robotic arm; Step 2: Drive the ball head at the end of the measuring rod of the robotic arm into the measurement area of ​​the 3D measuring device. At this time, the robotic arm is in the i-th robotic arm posture. Record the joint angle q of the robotic arm at this time, and measure the coordinate x of the ball center relative to the coordinate system of the 3D measuring device through the 3D measuring device. Based on the posture angle of the measuring rod at this time, calculate the offset of the measuring rod caused by gravity and / or measurement contact force, and compensate it into the ball center coordinate x to obtain the compensated ball center coordinate x'. Eliminate the measurement error caused by the deformation of the measuring rod. Step 3: Adjust the posture of the robotic arm and repeat Step 2 to obtain data for multiple measurement configurations; Step 4: Install the 3D measuring device on other workstations of the multi-station base, and repeat Step 2 and Step 3; Step 5: Based on the relative pose relationship between different workstations on the multi-station base, convert the compensated sphere center coordinates x' to the same coordinate system to obtain the sphere center coordinates x” after unifying the coordinate system; Step 6: Based on the joint angles q and ball center coordinates x” of the robotic arm obtained from multiple measurements, identify the true kinematic parameters of the robotic arm and complete the calibration; When the 3D measuring device is installed at the j-th workstation and the robotic arm is in the i-th robotic arm posture, the coordinate x of the center of the ball of the measuring rod is denoted as... The compensated coordinates of the sphere's center, x', are denoted as The compensated coordinates of the sphere's center are calculated using the following formula: in It is a 3×3 rotation matrix between the robotic arm flange coordinate system {T} and the measurement coordinate system {D}. Let be the offset vector of the ball center of the measuring rod relative to the robot arm flange coordinate system {T} at the j-th station and the i-th robot arm posture; For 3D measuring devices that use contact measurement, the TCP offset of the measuring rod is calculated using the following formula: Alternatively, for 3D measuring devices that employ non-contact measurement, the TCP offset of the measuring rod is calculated using the following formula: in To measure the TCP offset of the measuring rod caused by the contact force at the j-th station and the i-th robot arm posture, C represents the TCP offset of the measuring rod caused by gravitational acceleration. F It is the contact force of the measuring rod on the measuring device. The compliance matrix, C g It is the acceleration of the measuring rod relative to gravity. The compliance matrix, Let be the vector value of gravitational acceleration relative to the flange coordinate system {T}. To measure the vector value of the contact force relative to the flange coordinate system {T}; Where C F It is the contact force of the measuring rod on the measuring device. The flexibility matrix, Calculate using the following formula: in It is the compliance coefficient of the measuring rod in different directions, which is obtained through theoretical calculation, physical testing, or finite element simulation analysis; Where C g It is the acceleration of the measuring rod relative to gravity. The compliance matrix, Calculate using the following formula: Its internal elements It is the compliance coefficient of the measuring rod in different directions, which is obtained through theoretical calculation, physical testing, or finite element simulation analysis; At the j-th workstation and the i-th robotic arm posture, the vector value of the contact force relative to the flange coordinate system {T} is denoted as... Calculate using the following formula: in, It is a 3×3 rotation matrix between the coordinate system {D} of the 3D measuring device and the flange coordinate system {T} of the robotic arm; It is the rotation matrix between the base coordinate system {O0} and the flange coordinate system {T} of the robotic arm, which is calculated using the current joint angles and nominal kinematic parameters of the robotic arm; It is the rotation matrix between the robot arm base coordinate system {O0} and the 3D measuring device coordinate system {D} under the j-th workstation and the i-th robot arm posture, based on the 6D base parameters obtained during calibration. get, Let be the vector value of the measuring contact force of the 3D measuring device installed at the j-th station on the measuring rod relative to the coordinate system {D} of the 3D measuring device; Calculate using the following formula: Where, χ j The coordinate system {D} of the 3D measuring device is the coordinate system {D} of the j-th workstation. j The 6D vector of relative pose between}, R(χ j F is a 3×3 rotation matrix derived from the attitude angles in the 6D vector. D When a 3D measuring device is installed at any workstation, the measuring contact force of the 3D measuring device on the measuring rod is relative to the workstation coordinate system {D}. j The vector value of}; The vector value of gravitational acceleration relative to the flange coordinate system {T} T The calculation is performed using the following formula: Where g 0 It is the vector value of gravitational acceleration relative to the coordinate system {O0} of the robotic arm base; It is the rotation matrix between the robot arm's base coordinate system {O0} and the flange coordinate system {T}, calculated using the robot arm's current joint angles and nominal kinematic parameters; since the Z-axis of the robot arm's base coordinate system {O0} is the vertical direction, therefore g 0 =g=[0,0,9.81] T .

2. The high-precision calibration method for a robotic arm based on multi-station measurement according to claim 1, characterized in that, In step two, before each data recording, the center of the ball at the end of the measuring rod is driven by the robotic arm to reach the same point. This point is a fixed point relative to the 3D measuring device. Under the displacement feedback of the three displacement sensors of the 3D measuring device, the robotic arm gradually adjusts the angle of each joint of the robotic arm so that the center of the ball at the end of the measuring rod reaches the fixed point.

3. The high-precision calibration method for a robotic arm based on multi-station measurement according to claim 1, characterized in that, When a 3D measuring device uses contact measurement, if the center of the ball at the end of the measuring rod is located at a fixed point relative to the 3D measuring device, the contact force F between the probes of the three displacement sensors of the 3D measuring device and the ball is constant. This applies even if the 3D measuring device is installed at different workstations, the contact force F relative to the coordinate system {D} of that workstation remains constant. j The vector value F of} D It is also constant, the vector value F D The force was obtained by measuring the probe thrust of the three displacement sensors using a force gauge.

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