After the kinematic model of the multi-axis LEMT was established, the parameters need to be measured to ensure the accuracy of the model. There are seven linkage parameters and six positioning parameters in the kinematic model. The measurement is divided into two parts: linkage parameters measurement and positioning parameters measurement. This paper proposed a rapid measure method of linkage parameters, which was performed once after the machine tool was assembled. As for positioning parameters, this paper proposed a direct measurement method based on structured light scanner, which was carried out every time a workpiece was mounted.
3.1. Measurement of Linkage Parameters
As mentioned above, there are seven unknown linkage parameters in the kinematic model of multi-axis LEMT. This paper proposes a rapid measurement method of linkage parameters taking into account the characteristics of laser processing. This method is implemented by combining the machine tool motion with the laser marking action. The measurement steps of linkage parameters are as follows:
Step 1: Measure the zero point of A-axis and C-axis. Make the laser beam direction parallel to the Z-axis movement direction, and the movement angle of A-axis at this time can be defined as A = 0. Then make the rotation centerline of A-axis parallel to the X-axis movement direction, define the movement angle of C-axis at this time as = 0.
Step 2: Measure the distance from the laser focus to the A-axis rotation centerline (
). As shown in
Figure 5a, adjust the A-axis to
A = 0
, align the laser exit port with the upper surface of workbench, the displacement of Z-axis at this time is
. Prepare a marking board and place it on the workbench and ensure the upper surface is level. Adjust the Z-axis with a fixed step, move the Y-axis slightly at each position, use the laser focus to draw a line on the marking board. After that, use a microscope to observe the thinnest line or the line width is equal to the diameter of laser focus, the displacement of Z-axis at this time is
. The linkage parameter
can be calculated as:
where
h is the thickness of the marking board,
d is the distance from laser exit port to the rotation centerline of A-axis, which is the known quantity of the machine tool.
Step 3: Measure the distance between laser beam and the A-axis rotation centerline (
). As shown in
Figure 5b, place the marking board perpendicular to the workbench, adjust the C
-axis to
= 90
, adjust the A-axis to
A = 90
, move the Y-axis slightly, use the laser focus to draw a line on the marking board defined as
. Then, keep the C
-axis in
= 90
, adjust A-axis to
A = −90
, rotate the C
-axis by 180
, move the X-axis to make the laser focus fall on the marking board, move the Y-axis slightly, draw another line defined as
. After that, use a microscope to observe the distance between
and
, which is the linkage parameter
.
Step 4: Measure the distance from the ACS to C
CS (
) and the staggered distance between the A-axis rotation centerline and the C
-axis rotation centerline (
). As shown in
Figure 6a, place the marking board on the workbench, adjust the A-axis to
A = 0
, adjust the C
-axis to
= 0
, move the Y-axis slightly, use the laser focus to draw a line defined as
, record the position of the X-axis at this time as
, and then move the X-axis slightly, use the laser focus to draw a vertical line defined as
. After that, keep the A-axis in
A = 0
, adjust the C
-axis to
= 180
, move the X-axis by a large margin to make the laser focus fall on the marking plate again, move the Y-axis slightly, use the laser focus to draw a line defined as
, record the position of the X-axis at this time as
, and then move the X-axis slightly, use the laser focus to draw a vertical line defined as
. After that, use a microscope to observe
to
, the distance between
and
can be defined as
, the distance between
and
can be define as
, the linkage parameter
can be calculated as:
The linkage parameter
can be calculated as:
Step 5: Measure the position coordinates of the X-axis, Y-axis, and Z-axis when the origin of the C
CS coincides with the center of the C
-axis (
,
, and
). As shown in
Figure 6b, place the marking board on the workbench, adjust the A-axis to
A = 0
, adjust the C
-axis to
= 90
, make the laser focus approximately fall on the center of the C
-axis, rotate the C
-axis, the laser focus will draw a circle on the marking board. And then adjust the position of X-axis and Y-axis continuously until the circle becomes a point, record the position of the X-axis, Y-axis, Z-axis at this time as
,
,
. The linkage parameters
,
, and
can be calculated as:
Through the above-mentioned steps, the measurement of seven linkage parameters can be completed. This method makes full use of the characteristics of laser processing and can be conveniently applied to multi-axis LEMT, which does not require the complicated and expensive measuring instruments.
3.2. Measurement of Positioning Parameters
There are six unknown positioning parameters in the kinematic model of multi-axis LEMT. This paper proposes a positioning parameter measurement method by combining the motion of the machine tool with the position of the object measured by structured light scanner. The process of positioning parameter measurement is to obtain the translation values and rotation values from BCS to WCS, which is divided into two parts: measure parameters from BCS to MCS and measure parameters from MCS to WCS. The measurement steps of positioning parameters are as follows:
Step 1: Measure the direction vector of X-axis (in MCS). Fix a marking target on the workbench. Make the X-axis carry the marking target and move several steps in the positive direction, as shown in
Figure 7a. During the movement, make the structured light scanner shoot the marking target one time at each position, and record a coordinate (in MCS) of marking target defined as
,
… respectively. These coordinates can be fitted into a straight line using the least square method, the direction vector of X-axis (in MCS) can be expressed as:
Step 2: Measure the direction vector of Y-axis (in MCS), further calculate the rotation matrix from BCS to MCS (
). Make the Y-axis carry the marking target and move several steps in the positive direction, as shown in
Figure 7b. During the movement, make the structured light scanner shoot the marking target one time at each position, and record a coordinate (in MCS) of marking target defined as
,
… respectively. Similarly, the direction vector of Y-axis (in MCS) can be expressed as:
According to the orthogonality of the Cartesian coordinate system, the direction vector of Z-axis (in MCS) is: . Synthesize the above vectors, the rotation matrix from BCS to MCS is: .
Step 3: Measure the direction vector of the rotation centerline of C
-axis (in MCS). Move the C
-axis to a suitable position close to the structured light scanner, and record its position in the CNC system as
. Make the C
-axis carry the marking target and rotate several steps in the positive direction, as shown in
Figure 8. During the movement, make the structured light scanner shoot the marking target one time at each position, and record two coordinates (in MCS) of marking target, which are defined as
,
,
,
…,
respectively.
,
… can be fitted into a circle by the least square method, assume that the coordinate of circle center is
.
,
… can also be fitted into a circle, and the coordinate of circle center is
. Then, the direction vector of the rotation centerline of C
-axis can be expressed as:
Step 4: Measure the coordinate of a standard sphere, further calculate the translation parameters from BCS to MCS (
). Fix a standard sphere on the machine tool turntable, as shown in
Figure 9, make the structured light scanner shoot the surface of the standard sphere, and obtain the 3D point cloud data of the surface, use the least square method to calculate the sphere center coordinate, which is defined as
=
. The sphere center
can be projected to
L and obtain the projection point
. After the action of the rotation matrix
, the coordinate of projection point can be expressed as:
Use height ruler measure the distance between
and the machine tool turntable, the distance is defined as
. The projection point
is on the rotation centerline of the C
-axis. So, the coordinate of the projection point in BCS is
=
0
. Due to the coordinate system conversion relationship is
=
+
, the translation parameters from BCS to MCS can be calculated as:
The above calculations are all carried out with the coordinate of C
-axis center being
. When the C
-axis center moves to any position in the CNC system, assume that the coordinate is
, the translation parameters from BCS to MCS can be calculated as:
The translation matrix from the BCS to MCS can be expressed as:
. Synthesize the rotation matrix
and the translation matrix
, the transformation matrix from BCS to MCS can be calculated as:
Step 5: Measure the coordinate transformation matrix from MCS to WCS (
), further calculate the positioning parameters. Clamp the workpiece to the worktable with the structured light scanner fixed as shown in
Figure 10. Make the structured light scanner shoot the workpiece to obtain the 3D point cloud of the workpiece. Perform IRLS-ADF robust registration between the 3D point cloud and the CAM model of the workpiece, the coordinate transformation matrix from MCS to the WCS can be obtained [
21]. The coordinate transformation matrix from MCS to WCS can be denoted as
, and the transformation matrix from BCS to WCS can be calculated as:
=
×
. Decompose the transformation matrix
, the translation values (
and
z) and the rotation values (
,
and
) from BCS to WCS can be calculated.
Through the above-mentioned steps, the measurement of six positioning parameters can be completed. The measurement of the coordinate transformation matrix from BCS to MCS () needs to be done after the structured light scanner is fixed. Remeasurement will not be needed until the position of structured light scanner is changed. Meanwhile, the measurement of the coordinate transformation matrix from MCS to WCS () is carried out every time a workpiece is mounted.