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Theoretical research on multi-coordinate transformation modeling and its application in computer-aided manufacturing of five-axis laser machining system

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Abstract

Multi-axis laser processing has significant advantages and broad application prospects. Multi-coordinate spatial positioning is the key to realize its technical application. In this paper, the mathematical modeling of multi-axis positioning was carried out for a typical five-axis laser processing equipment. The space vector of the rotation axis was calibrated. The construction principle and calculation method of the workpiece machining coordinate system (WCS) were expounded. On the basis of solving the rotation and translation matrix, the geometrical information including machining position and posture in the CAD coordinate system was converted into the machine coordinate system (MCS) through coordinate transformation, which lays a technical foundation for multi-coordinate laser processing. At the same time, the verification of machining application, such as multi-axis positioning processing of rings on planar features, and spatial helical positioning machining on cylindrical surfaces were carried out, which proved the feasibility and effect of the proposed method. The modeling method proposed in this work is scalable and can be applied to different types and structures of equipment in the computer-integrated manufacturing. The research results of this work have important theoretical and technical value for the high-performance application of five-axis laser processing equipment.

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Abbreviations

(x 10  y 10  z 10  C0  A0):

The position coordinate of the C point (the point marked as PC0)

(x 20  y 20  z 20  C1 A0):

The corresponding position coordinate of point C when turning C angle to C1 in a counterclockwise from top to bottom (the point marked as PC1)

(x 30 y 30  z 30  C2 A0):

The current position coordinate of point C when turning C angle to C2 in a counterclockwise from top to bottom (the point marked as PC2)

(deltaX1 deltaY1 deltaZ1):

The coordinate increment of point PC1 relative to point PC0 in the three directions

(deltaX2 deltaY2 deltaZ2):

The coordinate increment of point PC2 relative to point PC1 in the three directions

(x 1  y 1  z 1):

Treat (x1 y1 z1) as the zero point of the relative incremental coordinates system

(x 2  y 2  z 2):

Treat (x2 y2 z2) as the incremental coordinates of PC1 relative to the zero point PC0

(x 3  y 3  z 3):

Treat (x3 y3 z3) as the incremental coordinates of PC2 relative to the zero point PC1

(xcc ycc zcc):

The rotating center coordinate corresponding to the coordinates of three positions of the calibration point CP at the initial position (the point marked as Pcenter0)

ab, ac, xm, and ym :

The intermediate quantity in the calculation process

(axc ayc azc):

The rotation vector for the calibration of C axis

(axcc aycc azcc):

The unitized rotation vector for the calibration of C axis

(x10a y10a z10a C0 A0):

The position coordinate of the A point (the point marked as PA0)

(x20a y20a z20a C0 A1):

The corresponding position coordinate of point A when turning A angle to A1 in a counterclockwise from top to bottom (the point marked as PA1)

(x30a y30a z30a C0 A2):

The current position coordinate of point A when turning A angle to A2 in a counterclockwise from top to bottom (the point marked as PA2)

(xcaa ycaa zcaa):

The coordinate data of the center of the rotary axis corresponding to the characteristic point A at the initial position (the point marked as Pcenter0A)

(deltaX 1a deltaY 1a deltaZ 1a):

The coordinate increment of point PA1 relative to point PA0 in the three directions

(deltaX 2a deltaY 2a deltaZ 2a):

The coordinate increment of point PA2 relative to point PA1 in the three directions

(x 1a  y 1a  z 1a):

Treat (x1a y1a z1a) as the zero point of the relative incremental coordinates system

(x 2a  y 2a  z 2a):

Treat (x2a y2a z2a) as the incremental coordinates of PA1 relative to the zero point PA0

(x 3a  y 3a  z 3a):

Treat (x3a y3a z3a) as the incremental coordinates of PA2 relative to the zero point PA1

aba, aca, xma, and yma :

The intermediate quantity in the calculation process

(axa 1a  aya 1a  aza 1a):

The rotation vector for the calibration of A axis

(axaa ayaa azaa):

The unitized rotation vector for the calibration of A axis

(X mzp Y mzp Z mzp):

Zero point of the machining coordinate system

RTfLs :

The rotation transformation matrix for three-axis laser machining condition

T mzp :

The translate transformation matrix for three-axis laser machining condition

CAD pcv::

He one-dimensional matrix vector formed by processing point data in CAD coordinate system

MCS pcv :

The calculated one-dimensional matrix vector

(x fLpMA1  y fLpMA1  z fLpMA1):

The coordinates of the first feature point on the feature line PfLp1 after the workpiece is clamped in MCS for multi-axis laser machining condition

(x fLpMA2  y fLpMA2  z fLpMA2):

The coordinates of the second feature point on the feature line PfLp2 after the workpiece is clamped in the MCS for multi-axis laser machining condition

(x fspMA1  y fspMA1  z fspMA1):

The coordinates of the first feature point on the feature surface PfSp1 after the workpiece is clamped in MCS for multi-axis laser machining condition

(x fspMA2  y fspMA2  z fspMA2):

The coordinates of the second feature point on the feature surface PfSp2 after the workpiece is clamped in MCS for multi-axis laser machining condition

(x fspMA3  y fspMA3  z fspMA3):

The coordinates of the third feature point on the feature surface PfSp3 after the workpiece is clamped in MCS for multi-axis laser machining condition

(deltaX1 fsp  deltaY1 fsp  deltaZ1 fsp):

The incremental coordinates in three directions between point PfSp2 and point PfSp1

(deltaX2 fsp  deltaY2 fsp  deltaZ2 fsp):

The incremental coordinates in three directions between point PfSp3 and point PfSp1

(axc fsp  ayc fsp  azc fsp):

The calculated vector of Z-axis vector in MCS

(ax fsp ay fsp az fsp) and (i_ax fsp j_ay fsp k_az fsp):

Unit vector of Z-axis vector in MCS

P fLLen :

The square of the length of the vector formed by the two machining feature points along the X axis

MZP :

The machining zero point in MCS

MachZvec :

The calculated unit vector of Z-axis vector in MCS

MachXvec :

The unit vector of X-axis vector in MCS

MachYvec :

The unit vector of Y-axis vector in MCS

RMachSys :

Rotation transformation matrix between the CAD model coordinate system and the machine coordinate system

TMachSys :

Translation transformation matrix between the CAD model coordinate system and the machine coordinate system

RTfLs :

The inverse of the translation matrix RMachSys

T mzp :

The inverse of the translation matrix TMachSys

(x CADp  y CADp  z CADp):

The coordinate values of start point of the vector in CAD coordinate system

(x CADpEnd  y CADpEnd  z CADpEnd):

The coordinate values of end point of the vector in CAD coordinate system

(XXCAD pBegMC S  YYCAD pBegMC S  ZZCAD pBegMCS):

The coordinate values of start point of the vector in MCS coordinate system

(XXCAD pEndMC S  YYCAD pEndMC S  ZZCAD pEndMCS):

The coordinate values of end point of the vector in MCS coordinate system

(i_axMCSp2 j_axMCSp2 k_axMCSp2) and (ax ay az):

The vector in the MCS coordinate system

sinthetaz :

Rotate the current vector \(\overrightarrow {V}\) by the angle of thetaz around the OZ axis, and the vector that is marked as \(\overrightarrow {V}^{\prime }\) can be rotated into the YOZ plane. sinthetaz is the sine of thetaz angle.

costhetaz :

Rotate the current vector \(\overrightarrow {V}\) by the angle of thetaz around the OZ axis, and the vector that is marked as \(\overrightarrow {V}^{\prime }\) can be rotated into the YOZ plane. costhetaz is the cosine of thetaz angle.

sinthetax :

Rotate the current vector \(\overrightarrow {V}^{\prime }\) by the angle of thetax around the OX axis, and the vector that is marked as \({\overrightarrow{V}}^{^{\prime\prime}}\) can be rotated into the XOZ plane. sinthetax is the sine of thetax angle.

costhetax :

Rotate the current vector \(\overrightarrow {V}^{\prime }\) by the angle of thetax around the OX axis, and the vector that is marked as \({\overrightarrow{V}}^{^{\prime\prime}}\) can be rotated into the XOZ plane. costhetax is the cosine of thetax angle.

C :

The rotation angle around C axis

A :

The rotation angle around A axis

Rxthetax_PP :

The rotation transformation matrix around the X axis vector

Rythetay_PP :

The rotation transformation matrix around the Y axis vector

Rzalpha_PP :

The rotation transformation matrix around the C axis vector

T_LT :

The translation matrix that moves the zero point of local coordinate system of the rotation axis C to the absolute zero point of the machine tool

T_PP :

The inverse transformation matrix of the translation matrix that moves the zero point of the rotation axis C local coordinate system to the absolute zero point of the machine tool

\(\alpha_{A}\) :

The rotation angle of the processing point around the A axis vector

Ta1_PP :

The translation matrix that moves the zero point of local coordinate system of the rotation axis A to the absolute zero point of the machine tool

RzthetazA_PP :

The rotation transformation matrix around the Z axis vector

RythetayA_PP :

The rotation transformation matrix around the Y axis vector

RvalphaA_PP :

The rotation transformation matrix around the A axis vector

Ta_PP :

The inverse transformation matrix of the translation matrix that moves the zero point of the rotation axis A in local coordinate system to the absolute zero point of the machine tool

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Funding

This work is funded by the Zhejiang Natural Science Foundation (LY20E050004), National Natural Science Foundation of China (51505468), Zhejiang Province Key R&D Program(2020C01036), and Ningbo Science and Technology Innovation 2025 Major Special Project (2019B10074).

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Authors and Affiliations

  1. Ningbo Institute of Materials Technology and Engineering, Chinese Academy of Sciences, Ningbo, 315201, Zhejiang, China

    Xiaoxiao Chen, Heng Wang & Wenwu Zhang

  2. Zhejiang Provincial Key Laboratory of Aero Engine Extreme Manufacturing Technology, Ningbo, 315201, Zhejiang, China

    Xiaoxiao Chen, Heng Wang & Wenwu Zhang

  3. University of Chinese Academy of Sciences, Beijing, 100049, China

    Xiaoxiao Chen & Wenwu Zhang

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  1. Xiaoxiao Chen
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  2. Heng Wang
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Chen, X., Wang, H. & Zhang, W. Theoretical research on multi-coordinate transformation modeling and its application in computer-aided manufacturing of five-axis laser machining system. Int J Adv Manuf Technol 123, 1037–1058 (2022). https://doi.org/10.1007/s00170-022-10191-6

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  • DOI: https://doi.org/10.1007/s00170-022-10191-6

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