Volumetric error compensation of machine tool using laser tracer and machining verification


Volumetric error is a primary factor that affects the accuracy of machining tools. The ability to ascertain volumetric error in a rapid and simple manner is of significant importance, and this can remarkably reduce the probability of temperature variation during the experiment. This study proposes a strategy for volumetric error measurement and compensation based on laser tracer for the 3-axis numerical control (NC) machining tool, and verifies the machining accuracy of a concave semi-spheroid test piece. In this study, the calibration methods employed by the laser tracer and the error separation algorithm of the machining tool are initially investigated. Following this, the accuracy of the geometric error measurement is verified using the laser interferometer. The cubic spline interpolation method is then employed to establish the tool path volumetric error model in the 3-axis NC machining tool, and the G-code modification is conducted for volumetric error compensation. Experiment results show that when the error compensation is performed, the improvement in the accuracy as compared with the initial state exceeds 50% not involving machining. To evaluate the accuracy and effectiveness of the proposed method, two machining tests to obtain a concave semi-spheroid test piece with and without volumetric error compensation strategy are studied, and the corresponding accuracies are measured by a high precision coordinate–measuring machine. It is found that the machining accuracy after having performed the error compensation is approximately 43% higher than that obtained on the pre-volumetric error compensation.

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Theoretical measuring point, i = 0,1,2,…, N

Ai' (xi',yi',zi' ):

Calibration of measuring point

bi,ci :

Integration constants

C :

\( {x}_{p1}^2+{y}_{p1}^2+{z}_{p1}^2-{L}_1^2 \)

D :

Coefficient matrix

E i :

Expanded form of geometric error function

h i :

xi − xi−1

j :

Measurement sequence of laser tracer

l 1i :

Measured length values from P1 to ith measuring point

L 1 :

Residual error

M i :

Undermined function expression

Pi(xpi, ypi, zpi):

Location of laser tracer, i = 1,2,…,6

Q i :

21 individual errors at point Ai

R(i) :

xi, Δyi, Δzi]T

S :

Distance between laser tracer and measuring point

S xy :

Perpendicularity error between X- and Y-axis

S xz :

Perpendicularity error between X- and Z-axis

S yz :

Perpendicularity error between Y- and Z-axis


y i−1


y i


Second-order derivation of the fitting function at the knot xi

tx,ty,tz :

Cutting tool length in x/y/z direction, tx = 0 and ty = 0

δ xx :

Positioning error of X-axis

δ xy :

Straightness error of Y-axis in X direction

δ xz :

Straightness error of Z-axis in X direction

δ yx :

Straightness error of X-axis in Y direction

δ yy :

Positioning error of Y-axis

δ yz :

Straightness error of Z-axis in Y direction

δ zx :

Straightness error of X-axis in Z direction

δ zy :

Straightness error of Y-axis in Z direction

δ zz :

Positioning error of Z-axis

∆L j :

Error of measured length between laser tracer and measuring point

∆p 1 :

xp1, Δyp1, Δzp1]


Coordinate error of measuring points

xi :


yi :


∆z i :


ε xx :

Roll error of X-axis

ε xy :

Pitch error of Y-axis

ε xz :

Pitch error of Z-axis

ε yx :

Pitch error of X-axis

ε yy :

Roll error of Y-axis

ε yz :

Yaw error of Z-axis

ε zx :

Yaw error of X-axis

ε zy :

Yaw error of Y-axis

ε zz :

Roll error of Z-axis


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This work was supported by the Shenzhen Science and Technology Project (JCYJ20180306170733170), Natural Science Foundation of Jiangsu Province (BK20190218), and Major National Science and Technology Projects (2018ZX04002001).

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Correspondence to Jun Zha.

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Zha, J., Wang, T., Li, L. et al. Volumetric error compensation of machine tool using laser tracer and machining verification. Int J Adv Manuf Technol 108, 2467–2481 (2020). https://doi.org/10.1007/s00170-020-05556-8

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  • Error measurement
  • Laser tracer
  • Machine tool
  • Volumetric error compensation