Full-scale measurement of CNC machine tools

Abstract

Geometric error measurement is crucial for error compensation and machining accuracy of CNC machine tools. However, traditional double-ball bar (DBB) method cannot be used for full-scale measurement of geometric errors due to the constant length of bars. In this paper, a full-scale measurement of geometric errors for CNC machine tools with high efficiency is carried out based on the double ball bar with a ball joint (J-DBB) method. Firstly, the principle of J-DBB is introduced. The J-DBB can be used for a full-scale measurement thanks to its freely measuring radius. Then, the equations of geometric errors which include squareness error, straightness error, roll error, pitch error, and yaw error of the axes are theoretically analyzed. In the end, we compared the theoretical trajectories of circular interpolation with the actual ones. The geometric errors between theoretical trajectories and actual ones can be measured and shown by J-DBB method. The results revealed that the J-DBB can be applied to conduct the full-scale measurement of geometric errors in CNC machine tools. Besides, the measurement range of geometric errors is extended from a two-dimensional circle to a three-dimensional sphere with a radius of 0.259L to 2L.

Appendix. Error equation of the J-DBB

For J-DBB, the first ball is fixed in the machine’s table. And the second ball is fixed in the machine tool spindle. In the measuring coordinate system, origin Oof coordinates is located in the center of the first ball (Fig. 3). The xoy plane is parallel to the machine table of CNC machine tools. Coordinate axis z is parallel to the CNC machine tool spindle.

The center of the second ball located in the machine tool spindle is assumed to be point P. The coordinate of point P is (x, y, z). Accordingly, the point P(x, y, z) can be expressed as

$$ \left\{\begin{array}{l}{x}_0=L\cos {\theta}_1+L\cos \left({\theta}_1+{\theta}_2\right)\\ {}x={x}_0\cos {\theta}_3\\ {}y={x}_0\sin {\theta}_3\\ {}z=L\sin {\theta}_1+L\sin \left({\theta}_1+{\theta}_2\right)\end{array}\right.. $$

(18)

where L is the length of bar;

θ₁ is the angle between the first bar and xoy plane;

θ₂ is the angle between the first bar and the second bar;

θ₃ is the angle of the machine tool spindle moved at t second with the velocity v, θ₃ = vRt.

The measuring radiusRcan be expressed as

$$ {R}^2={x}^2+{y}^2+{z}^2 $$

(19)

When the machine tool spindle starts to move, the center of the second ball moves from point P to P^'. At the moment, the coordinates of machine tool spindle in measuring coordinates is located in point P ' (x', y', z'). The actual measuring radius R' can be expressed as

$$ {R}^{\hbox{'}2}={x}^{\hbox{'}2}+{y}^{\hbox{'}2}+{z}^{\hbox{'}2} $$

(20)

There are 38 errors in the five-axis CNC machine tools; therefore, the machine tool spindle cannot move along a standard circle. The measuring radius R will increase or decrease with the motion of the machine tool spindle. The errors of the machine tools can be shown by the variation of the measuring radius R. In addition, it is observed that the measuring radius R will change with the interpolation trajectories of machine tool spindle.

The position errors of the five-axis CNC machine tools at point O and P are assumed to be C₀ and C₁, as shown in Fig. 3. Then,

$$ \left\{\begin{array}{c}{C}_0=\left({C}_{x0},{C}_{y0},{C}_{z0}\right)\\ {}{C}_1=\left({C}_{x1},{C}_{y1},{C}_{z1}\right)\end{array}\right. $$

(21)

The actual coordinate of the first ball in measuring coordinates is O₁(C_x0, C_y0, C_z0). And the actual coordinates of the second ball is P₁(x + C_x1, y + C_y1, z + C_z1).

Due to the errors in CNC machine tools, the actually measuring radius R^' can be expressed as

$$ {R}^{\hbox{'}2}={\left(R+\varDelta R\right)}^2={\left(x+{C}_{x1}-{C}_{x0}\right)}^2+{\left(y+{C}_{y1}-{C}_{y0}\right)}^2+{\left(z+{C}_{z1}-{C}_{z0}\right)}^2 $$

(22)

Simplify Eq. (22) and ignore higher-order terms, the error is

$$ \varDelta R=\frac{1}{R}\left[x\left({C}_{x1}-{C}_{x0}\right)+y\left({C}_{y1}-{C}_{y0}\right)+z\left({C}_{z1}-{C}_{z0}\right)\right] $$

(23)

For convenience, we define an error vector E

$$ E=\left({C}_{x1}-{C}_{x0},{C}_{y1}-{C}_{y0},{C}_{z1}-{C}_{z0}\right)=\left({C}_x,{C}_y,{C}_z\right) $$

(24)

Substituting Eq. (24) into Eq. (23), the error equation of J-DBB is as the following

$$ \varDelta R=\frac{1}{R}\left(x{C}_x+y{C}_y+z{C}_z\right) $$

(25)

where (x, y, z) denotes the coordinates of machine tool spindle in the measuring coordinate system;

(C_x, C_y, C_z) denotes the position errors of machine tool spindle in measuring coordinate system.