Dynamic performance test under complicated motion states for five-axis machine tools based on double ballbar

During five-axis CNC machining, the dynamic tracking error caused by imperfect servo dynamic performance is becoming the major factor affecting the accuracy during high-speed and high-precision manufacturing. The double ballbar (DBB) test is one of the commonly used dynamic performance tests, but existing DBB test methods do not have satisfactory test capability for the requirement of complex freeform surface machining. In this paper, on the basis of summarized of dynamic tracking error's mechanism and characteristics, an improved dynamic performance test based on DBB, which is named the circle-8 test, is proposed. In this test, a provenly efficient RTCP test trajectory is used to rebuild the DBB test process, which includes complex movements of all the five motion axes. To exhibit the improvement of the circle-8 test, this test and the BK3 test of ISO standard, which is considered as a comparison, are conducted in a five-axis machine tool with a tilting rotary table. According to the simulation and experiment results, for common dynamic inaccuracy situations, the circle-8 test always has better sensitivity of dynamic performance test than BK3 test. The above-improved dynamic performance test can be applied to provide effective data for the research of modeling and error reduction of five-axis machine tools.

Keywords: Five-axis CNC machining; Dynamic tracking error; Double ballbar (DBB); Complicated motion states

Five-axis CNC machine tools are very important equipment for the manufacturing industry, especially for complex-contour and high-speed manufacturing. The machining error of the five-axis machine tool can be decoupled into geometric error, thermal error, cutting force error, and dynamic tracking error [[1]]. In the recent years, some researchers have shown that, during the high-speed and high-precision manufacturing process, the dynamic tracking error caused by imperfect dynamic performance may make up a substantial percentage of total error [[2]].

The dynamic tracking error is not inherent in machine tool but caused by the delay of each motion axis control process, which is different from geometric error, thermal error, and cutting force error. There are many research issues about the dynamic tracking error. Most of the existing researches about dynamic tracking error can be concluded into dynamic tracking error reduction and dynamic performance test. As for the research of dynamic tracking error reduction, servo dynamic tuning [[4]–[7]], contour error predictive pre-compensation [[8]–[12]], improvement on control system [[13]–[18]], and tool path calculation or planning [[19]–[25]] are the most popular methods, as listed in Table 1.

Recent research issues on dynamic tracking error reduction methods

For dynamic performance test, the commonly used dynamic performance test can be divided into two types, the standard piece test and non-machining test. The standard piece test means determining the dynamic performance by machining and measuring a standard piece. The cone-frustum test piece created by NAS979 [[26]] and the S-shape test piece [[27]–[29]] are the most typical standard pieces recently. It is worth mentioning that the S-shape test piece has been included in latest ISO 10791-7 standard [[30]], which means the dynamic performance test with five axes' complicated motions has become a growing concern in recent years. Machining and measuring a standard piece can be used to test and evaluate overall performance comprehensively, but the deviation of the test piece surface is affected by other factors during the manufacturing process, such as tool condition, workpiece materials, and so on, and it is still difficult to separate the dynamic tracking error from the whole deviation. Hence, non-machining test, the another one, is a more suitable type of methods for testing dynamic accuracy.

Non-machining test, also called multi-axis kinematic test, which means making machine tool move and measuring the accuracy of the movement by instrument. The typical non-machining tests can be divided into rotation tool center point (RTCP) and double ballbar (DBB) test.

The RTCP test is based on the rotation tool center point function. In the RTCP test, the tool tip is set to remain still, while the rotary axes are set to run and the deviation of the tool tip from the set point can be considered as the dynamic tracking error of the machine tool. Weikert [[31]] first invented an instrument termed R-test for the RTCP test, and then many researchers have done a lot of improvement for the RTCP test. Jywe et al. [[32]], Hong [[33]], and Ding et al. [[34]] did some research about the improvement of the RTCP device, and Zhong et al. [[35]], Jiang et al. [[36]], and Ding et al. [[37]] did some work on the RTCP test trajectory.

Another typical non-machining test is the DBB test. During the test, tool tip is set to move along a circle or arc trajectory and the deviation of tool path is measured to evaluate the dynamic performance. The same advantage of RTCP test and DBB test is high measuring accuracy based on small measuring range, which is suitable for dynamic performance test. However, there are also some different characteristics for the two methods. Because of the motion characteristics during the two tests, the circle or arc test with DBB can offer wider movement ranges of the five motion axes, especially for the linear motion axes, which can result in stronger testing capability. The sensor in DBB can only measure the deviation in the radial direction of the circle or arc test, but the three sensors of RTCP test device can measure the errors in X-, Y-, and Z-directions, which is more suitable for error tracing. Bryan [[38]] developed the DBB device firstly, including a displacement sensor in the bar and two precision balls separately in the tool holder side and table side. Lei et al. [[4], [40]] designed a combo of dynamic accuracy measuring methods and summarized the DBB testing error patterns of servo gain mismatch, backlash, and tracking direction as a diagnosis of the error sources. Flohic et al. [[42]] proposed a model-based servo tuning method based on DBB to reduce tracking error. However, the above DBB tests only include circle or arc test trajectories, which cannot sensitively reflect the dynamic performance of five-axis machine tools when machining complex freeform surfaces.

ISO 10791-6 [[43]] provides a standard DBB test trajectory named BK3 (for different types of machine tools, it is named AK3 or CK3). During the test, the machine tool should be programmed to move along with a cone, angle between the base circle of the programmed cone and the table surface. The DBB shall be set approximately perpendicular to the cone surface. The ISO BK3 test is valid as a dynamic performance test, but previous researches [[35]] have shown that the motion and velocity change of this test are relatively smooth, which cannot fully reflect the dynamic performance of five-axis machine tools when machining complex freeform surfaces. Kato and Tsutsumi et al. [[44]–[47]] did a lot of works about the improvements of the BK3 cone test, including the adjustment of the cone's half apex angle, center point, DBB's sensitive direction, and so on. According to the above adjustment, the cone test will have more powerful dynamic performance test ability than so-called ISO-BK3 test. However, the above-improved cone tests are also limited by the base of the cone test, whose geometric shapes are simple and regular, so it cannot be sure that adjusting cone test is the best method to improve the DBB dynamic test ability.

In this paper, on the basis of summarized of dynamic tracking error's mechanism and characteristics, a novel dynamic performance test based on DBB, which is named circle-8 test, is proposed. In this test, free from popular the cone test form, a provenly efficient RTCP test trajectory is used to rebuild the DBB test process, which includes complex movements of all the five motion axes. To exhibit the improvement of the circle-8 test, this test and the BK3 test of ISO standard, which is considered as a comparison, are conducted in a five-axis machine tool with a tilting rotary table. According to the simulation and experiment results, the conclusion can be drawn that, for common dynamic inaccuracy situations, the circle-8 test always has better sensitivity of dynamic performance test than BK3 test. The above-improved dynamic performance test can be applied to provide effective data for the research of modeling and error reduction of five-axis machine tools.

The mechanism and characteristics of dynamic tracking error can be summarized as Fig. 1. In five-axis machine tools, all of the motion axes, including linear axes and rotary axes, are driven by servo system. The servo control systems of the linear axis and rotary axis are established as displayed in Figs. 2 and 3 respectively, where the physical meaning of symbols in these figures is shown in Tables 2 and 3. For linear motion axes, the relationship between actual axis position xr and ideal position xi can be expressed as Eqs. 1 and 2.

1 $$x_r\left(t\right)=x_i\left(t\right)+{\xi }_{\mathrm{vx}}\cdot x_i\text{'}\left(t\right)+{\xi }_{\mathrm{ax}}\cdot x_i\text{'}\text{'}\left(t\right)+o\left(t\right)$$

Graph

2 $$\left\{\begin{array}{l}{\xi }_{\mathrm{vx}}=-2\pi /\left(K_{\mathrm{pp}}\cdot l\right)\\ {\xi }_{\mathrm{ax}}=-\frac{2\pi }{\left(K_{\mathrm{pp}}\cdot l\right)}\cdot \left[-\frac{2\pi }{K_{\mathrm{pp}}\cdot l}+\frac{C_t}{K}+\frac{T_i}{\mathit{DA}\cdot T_{\mathrm{rg}}\cdot K_{\mathrm{vi}}}\cdot \left(C_b+C_t\cdot \frac{l^2}{4{\pi }^2}\right)\right]\end{array}\right)$$

Graph

Graph: Fig. 1 Mechanism of dynamic tracking error of five-axis machine tools

Graph: Fig. 2 Servo system configuration of each linear axis

Graph: Fig. 3 Servo system configuration of each rotary axis

Physical meaning of symbols in Fig. 2

Physical meaning of symbols in Fig. 3

where o(t) is the Lagrange remainder, which can be treated as the residual of model simplification and ignored. Similarly, for rotary motion axes, the relationship between actual axis position θr and ideal position θi can be expressed as Eqs. 3 and 4.

3 $${\theta }_r\left(t\right)={\theta }_i\left(t\right)+{\xi }_{\mathrm{vA}}\cdot {\theta }_i\text{'}\left(t\right)+{\xi }_{\mathrm{aA}}\cdot {\theta }_i\text{'}\text{'}\left(t\right)+o\left(t\right)$$

Graph

4 $$\left\{\begin{array}{c}{\mathrm{\xi }}_{\mathrm{vA}}=1/K_{\mathrm{pp}}{R}_{\mathrm{g}}{R}_{\mathrm{w}}\\ {\mathrm{\xi }}_{\mathrm{aA}}=\frac{2}{K_{\mathrm{pp}}{R}_{\mathrm{g}}{R}_{\mathrm{w}}}·\left[-\frac{1}{K_{\mathrm{pp}}{R}_{\mathrm{g}}{R}_{\mathrm{w}}}+\frac{C_{\mathrm{w}}}{K_{\mathrm{g}}}+\frac{C_{\mathrm{t}}}{K_{\mathrm{w}}}+\frac{C_{\mathrm{w}}{R}_{\mathrm{w}}^2}{K_{\mathrm{g}}}\right]\\ +\frac{2}{\left(K_{\mathrm{pp}}{R}_{\mathrm{g}}{R}_{\mathrm{w}}\right)}·\left[-\frac{T_i}{\mathit{DA}·T_{\mathrm{rg}}·K_{\mathrm{vi}}}+\left(C_{\mathrm{m}}+C_{\mathrm{t}}{R}_{\mathrm{g}}^2{R}_{\mathrm{w}}^2+C_{\mathrm{w}}{R}_{\mathrm{g}}^2\right)\right]\end{array}\right)$$

Graph

The deviation between actual and ideal output can be defined as tracking error, as shown in Eqs. 5 and 6.

5 $$e_{\mathrm{L}}=x_r\left(t\right)-x_i\left(t\right)={\xi }_{\mathrm{vx}}\cdot x_i\text{'}\left(t\right)+{\xi }_{\mathrm{ax}}\cdot x_i\text{'}\text{'}\left(t\right)+o\left(t\right)$$

Graph

6 $$e_{\mathrm{R}}={\theta }_r\left(t\right)-{\theta }_i\left(t\right)={\xi }_{\mathrm{vA}}\cdot {\theta }_i\text{'}\left(t\right)+{\xi }_{\mathrm{aA}}\cdot {\theta }_i\text{'}\text{'}\left(t\right)+o\left(t\right)$$

Graph

where eL and eR are the tracking errors of linear motion axis and rotary motion axis respectively.

The tracking error of each motion axis will cause the deviation of the machine tool's tool tip. Herein, a five-axis machine tool with a tilting rotary table (B-type) is considered as an example, as shown in Fig. 4. The kinematic chain of the machine tool can be described by multi-body system theory [[48]], as shown in Fig. 5, and the kinematic transform can be formulated as Eqs. 7 and Eq. 8.

7 $$P_{\text{workpiece}}=T_{12}^{-1}{T}_{01}^{-1}{T}_{03}{T}_{34}{T}_{45}{P}_{\text{tool}}$$

Graph

8 $$T_{12}=\left[\begin{array}{ccc}cos\left(C_{\mathit{\text{axes}}}\right)& -sin\left(C_{\mathit{\text{axes}}}\right)& 00\\ sin\left(C_{\mathit{\text{axes}}}\right)& cos\left(C_{\mathit{\text{axes}}}\right)& \begin{array}{cc}0& 0\end{array}\\ \begin{array}{c}0\\ 0\end{array}& \begin{array}{c}0\\ 0\end{array}& \begin{array}{c}\begin{array}{cc}1& 0\end{array}\\ \begin{array}{cc}0& 1\end{array}\end{array}\end{array}\right],T_{01}=\left[\begin{array}{ccc}1& 0& \begin{array}{cc}0& 0\end{array}\\ 0& cos\left(C_{\mathit{\text{axes}}}\right)& \begin{array}{cc}-sin\left(C_{\mathit{\text{axes}}}\right)& 0\end{array}\\ \begin{array}{c}0\\ 0\end{array}& \begin{array}{c}sin\left(C_{\mathit{\text{axes}}}\right)\\ 0\end{array}& \begin{array}{c}\begin{array}{cc}cos\left(C_{\mathit{\text{axes}}}\right)& 0\end{array}\\ \begin{array}{cc}0& 1\end{array}\end{array}\end{array}\right],T_{03}=\left[\begin{array}{ccc}1& 0& \begin{array}{cc}0& 0\end{array}\\ 0& 1& \begin{array}{cc}0& Y_{\mathit{\text{axes}}}\end{array}\\ \begin{array}{c}0\\ 0\end{array}& \begin{array}{c}0\\ 0\end{array}& \begin{array}{c}\begin{array}{cc}1& 0\end{array}\\ \begin{array}{cc}0& 1\end{array}\end{array}\end{array}\right],T_{34}=\left[\begin{array}{ccc}1& 0& \begin{array}{cc}0& X_{\mathit{\text{axes}}}\end{array}\\ 0& 1& \begin{array}{cc}0& 0\end{array}\\ \begin{array}{c}0\\ 0\end{array}& \begin{array}{c}0\\ 0\end{array}& \begin{array}{c}\begin{array}{cc}1& 0\end{array}\\ \begin{array}{cc}0& 1\end{array}\end{array}\end{array}\right],T_{45}=\left[\begin{array}{ccc}1& 0& \begin{array}{cc}0& 0\end{array}\\ 0& 1& \begin{array}{cc}0& 0\end{array}\\ \begin{array}{c}0\\ 0\end{array}& \begin{array}{c}0\\ 0\end{array}& \begin{array}{c}\begin{array}{cc}1& Z_{\mathit{\text{axes}}}\end{array}\\ \begin{array}{cc}0& 1\end{array}\end{array}\end{array}\right]$$

Graph

where (Xaxes, Yaxes, Zaxes, Aaxes, Caxes) is the position of each motion axis, Pworkpiece is the coordinates of the tool center point, and Ptool is the coordinate of the tool center point on tool coordinate system, where Ptool = [0, 0, −L, 1] and L is the tool length. Thus, the relationship between (Xaxes, Yaxes, Zaxes, Aaxes, Caxes) and the tool tip position (Xtool, Ytool, Ztool) can be formulated by Eq. 9.

9 $$\left\{\begin{array}{c}{X}_{\text{tool}}=X_{\text{axes}}·cos{C}_{\text{axes}}+Y_{\text{axes}}·cos{A}_{\text{axes}}·sin{C}_{\text{axes}}+Z_{\text{axes}}·sin{A}_{\text{axes}}·sin{C}_{\text{axes}}-L·sin{A}_{\text{axes}}·sin{C}_{\text{axes}}\\ Y_{\text{tool}}=X_{\text{axes}}·sin{C}_{\text{axes}}+Y_{\text{axes}}·cos{A}_{\text{axes}}·cos{C}_{\text{axes}}+Z_{\text{axes}}·sin{A}_{\text{axes}}·cos{C}_{\text{axes}}-L·sin{A}_{\text{axes}}·cos{C}_{\text{axes}}\\ Z_{\text{tool}}=-Y_{\text{axes}}·sin{A}_{\text{axes}}+Z_{\text{axes}}·cos{A}_{\text{axes}}-L·cos{A}_{\text{axes}}\end{array}\right)$$

Graph

Graph: Fig. 4 Five-axis machine tool with a tilting rotary table

Graph: Fig. 5 Topological structure and coordinates transform of the machine tool in Fig. 4

Thus, the relationship between motion axes' tracking errors (eX, eY, eZ, eA, eC) and tool tip deviation (ΔX, ΔY, ΔZ) can be described as Eq. 10.

10 $$\left\{\begin{array}{c}∆X=X_{\text{tool}}\left(X_{\text{axes}}+e_X,Y_{\text{axes}}+e_Y,Z_{\text{axes}}+e_Z,A_{\text{axes}}+e_A,C_{\text{axes}}+e_C\right)-X_{\text{tool}}\left(X_{\text{axes}},Y_{\text{axes}},Z_{\text{axes}},A_{\text{axes}},C_{\text{axes}}\right)\\ ∆Y=Y_{\text{tool}}\left(X_{\text{axes}}+e_X,Y_{\text{axes}}+e_Y,Z_{\text{axes}}+e_Z,A_{\text{axes}}+e_A,C_{\text{axes}}+e_C\right)-Y_{\text{tool}}\left(X_{\text{axes}},Y_{\text{axes}},Z_{\text{axes}},A_{\text{axes}},C_{\text{axes}}\right)\\ ∆Z=Z_{\text{tool}}\left(X_{\text{axes}}+e_X,Y_{\text{axes}}+e_Y,Z_{\text{axes}}+e_Z,A_{\text{axes}}+e_A,C_{\text{axes}}+e_C\right)-Z_{\text{tool}}\left(X_{\text{axes}},Y_{\text{axes}},Z_{\text{axes}},A_{\text{axes}},C_{\text{axes}}\right)\end{array}\right)$$

Graph

The dynamic tracking error of five-axis machine tool can be seen as the coupling result of motion axes' tracking errors. It can be seen that the dynamic tracking error of five-axis machine tool has the following characteristics:

Firstly, the tracking error of servo control process is different from a general sense of "error". Figure 6 is the basic response mode of servo system, and it is very obvious that the tracking error can be treated as a kind of delay rather than the "error" in the conventional sense. The tracking delay Δt can be defined that can make Eqs. 11 and 12 true.

11 $$x_i\left(t-\mathrm{\Delta }t\right)=x_i\left(t\right)+{\xi }_{\mathrm{v}x}\cdot x_i\text{'}\left(t\right)+{\xi }_{\mathrm{a}x}\cdot x_i\text{'}\text{'}\left(t\right)+o\left(t\right)$$

Graph

12 $${\theta }_i\left(t-\mathrm{\Delta }t\right)={\theta }_i\left(t\right)+{\xi }_{\mathrm{vA}}\cdot {\theta }_i\text{'}\left(t\right)+{\xi }_{\mathrm{aA}}\cdot {\theta }_i\text{'}\text{'}\left(t\right)+o\left(t\right)$$

Graph

Graph: Fig. 6 Basic response mode of servo system

Thus, the machine tool's dynamic error can be seen as the result caused by multi-delay of motion axes, and all the research about dynamic tracking error should be conducted in the movement process of machine tools.

Secondly, as shown in Eqs. 9 and 10, the relationship between motion axes' tracking errors and the whole machine tool's dynamic tracking error is non-linear and non-orthogonal, which means focusing on the compensation or testing of single motion axis' tracking error is meaningless. Thus, during dynamic performance test, all the five motion axes should be driven to move together, and as many axes' motion states as possible should be covered.

The DBB is a typical dynamic performance test device, which has a good capability real-time test, which is suitable for dynamic performance test. However, the DBB test in general only includes simple circle or arc test trajectories, which can be conducted with only three axes' movements and meanwhile the other two axes keep still, so it is not enough to reflect the dynamic performance. The cone test is the most popular improved DBB test presently, which draws attention of researchers [[44]–[47]] and is included in the ISO standard [[43]]. During the cone test, the machine tool should be programmed to move along with a cone. However, represented by the ISO-BK3 test, cone, or cone-like tests only include regular trajectory, whose axes' motion and velocity change of this test are relatively smooth [[35]]. Thus, it cannot be sure that cone or cone-like test is the perfect method for the DBB dynamic performance test.

The 8-shape trajectory is a typical trajectory for RTCP test, as shown in Fig. 7, and previous research has suggested that this trajectory is valid and efficient for dynamic performance test [[36]]. Thus, the motion states of 8-shape RTCP test can be considered to cover the most of typical motion modes. The 8-shape trajectory is defined in spherical coordinates as Eq. 13.

13 $$\left\{\begin{array}{l}\alpha =N\cdot sin\left(t\right)\\ \beta =N\cdot sin\left(2t\right)\end{array}\right)$$

Graph

Graph: Fig. 7 8-shape trajectory for RTCP test

Herein, α and β are the tool posture angles in spherical coordinates, as shown in Fig. 8, and N is the core parameter which decide the size of the 8-shape. In general, the set of parameter N usually depended on the limit of the rotary axis position, which is always set between π/3 and π/12. However, it should be noted that, with the increase of N, the 8-shape trajectory may become bigger but smoother, and its velocity coverages may become narrower. Thus, as an example, the size parameter N is set to π/9 in this paper.

Graph: Fig. 8 Definition of the tool posture angles α and β

To improve the dynamic performance test based on DBB, the rotary axes motions in the 8-shape trajectory are applied to rebuild the DBB test process. During the improved DBB test, which is named circle-8 test, while the tool tip is moving in a circle in XY-plane, the two rotary axes move by 8-shape trajectory twice, as Eq. 14 shows.

14 $$\left\{\begin{array}{l}\alpha =\frac{\pi }{9}\cdot sin\left(t\right)\\ \beta =\frac{\pi }{9}\cdot sin\left(2t\right)\\ X_{\text{tool}}=R\cdot sin\left(t/2\right)\\ Y_{\text{tool}}=R\cdot cos\left(t/2\right)\\ Z_{\text{tool}}=0\end{array}\right)$$

Graph

where R is the radius of the tool tip motion trajectory, which can be also seen as the length of DBB. The DBB should keep in XY-plane during the test process. The testing process is shown in Fig. 9.

Graph: Fig. 9 Circle-8 test based on double ballbar (DBB)

In this section, the motion axes' movements of the circle-8 test are compared with the BK3 test, and the influences of different dynamic performance deficiency levels and feedrates are analyzed separately and compared. By these comparisons, the validity of circle-8 test can be verified. It should be noted that, in the ISO standard [[43]], the angle between the base circle of the programmed cone and the table surface, and the apex angle of the programmed cone, shall be respectively either 10° and 30°, or 30° and 90°, and some researchers tried more adjustment about the apex angle [[45]]. In this paper, the BK3 with 10° and 30° angles, which is the most popular, is taken as an example in the comparison and analysis of this section.

In this paper, a B-type five-axis machine tool, as shown in Fig. 4, is considered as an example. Herein, the motions of two rotary axes during the circle-8 test can be can be determined depend on the structure of the machine tool, as shown in Eq. 15

15 $$\left\{\begin{array}{l}A=\left\{\begin{array}{l}arccos\left(cos\alpha \cdot cos\beta \right)\begin{array}{cc}& \end{array}\left(\alpha \ge 0\right)\\ -arccos\left(cos\alpha \cdot cos\beta \right)\begin{array}{cc}& \end{array}\left(\alpha <0\right)\end{array}\right)\\ C=arctan\left(\frac{tan\beta }{sin\alpha }\right)\end{array}\right)$$

Graph

where A and C are the rotary axis positions. Figures 10, 11, 13, and 14 show the comparison of the axes' movements of the circle-8 test and the BK3 test. Herein, for both of the circle-8 test and the BK3 test, the rotary table side ball of DBB is set still at (40, 0, 60), which is fit for the requirement of BK3 test in ISO standard [[43]] (the same in the below of this paper). It is obvious that the movement ranges and velocity ranges of X-, Y-, Z-, and A-axes' movements of the circle-8 test are larger than the ones of the BK3 test, and the speed changes of each axis of the circle-8 test are more drastic.

Graph: Fig. 10 The axes' movements of the circle-8 test

Graph: Fig. 11 The axes' movements of the BK3 test

Graph: Fig. 12 The axes' movements of the 8-shape RTCP test

Graph: Fig. 13 The axes' velocity changes of the circle-8 test

Graph: Fig. 14 The axes' velocity changes of the BK3 test

In addition, Figs. 12 and 15 shows the axes' movements and velocity changes of 8-shape RTCP test. It is obvious that the movement ranges and velocity ranges of all the five axes of the circle-8 test are larger than the ones of the 8-shape RTCP test, and the speed changes of each axis of the circle-8 test are more drastic, which means the circle-8 DBB test can offer stronger testing capability than 8-shape RTCP test.

Graph: Fig. 15 The axes' velocity changes of the 8-shape RTCP test

To exhibit the dynamic performance test sensitivity of circle-8 test, the simulation model of the five-axis machine tool is established, and the simulation test result of the circle-8 test and BK3 test in different dynamic inaccuracy cases are analyzed and compared.

The flowchart of machine tool simulation model is shown in Fig. 16. The ideal tool path and posture changes are converted into NC commands, then the commands of each motion axis are sent into the simulation model of servo control system. The position of each motion axis, which are calculated by control system simulation model, can be used to calculate the real tool position and posture by kinematic transform. By calculation the distance between tool tip and rotary table side ball and subtracting the DBB length, the DBB test result can be simulated.

Graph: Fig. 16 Flowchart of machine tool simulation model

According to Eqs. 2, 4, 5, and 6 in Section 2.1, it can be seen that the tracking error of each axis is effected by many parameters, including position gain Kpp, moment of inertia Jw, friction fw, and so on. However, at least for now, in the simulation and experiment processes, most of these parameters are not allowed to modify because the changes of parameters may cause unpredictable security risk. besides, some kinds of dynamic performance deficiencies, such as deficiency caused by performance degradation of control system or external electromagnetic interference, are difficult to describe or simulate. The position gain Kpp is one of greatest influencing factors of dynamic performance, which is adjustable in the simulation model and experiment. Thus, in the simulations and experiments of this paper, we tried to create the describable simulative dynamic performance deficiency cases according to adjustment of Kpp of each axis.

Firstly, the situations of single axis' dynamic performance deficiency are simulated and analyzed. The position gains of each axis are set as follows:

• X-axis mismatch: KppX:KppY:KppZ:KppA:KppC = 0.9:1:1:1:1;
• Y-axis mismatch: KppX:KppY:KppZ:KppA:KppC = 1:0.9:1:1:1;
• Z-axis mismatch: KppX:KppY:KppZ:KppA:KppC = 1:1:0.9:1:1;
• A-axis mismatch: KppX:KppY:KppZ:KppA:KppC = 1:1:1:0.9:1;
• C-axis mismatch: KppX:KppY:KppZ:KppA:KppC = 1:1:1:1:0.9.

Herein, the feed rate is set to 1500 mm/min (the same in the below of this section). The simulation results, respectively of the circle-8 test and BK3 test, are shown in Fig. 17, and the statistical indexes of the simulation results, including positive and negative maximum, standard deviation, and DBB test value (the difference between the positive maximum and negative maximum), are shown in Table 4. It can be seen that, for the case of X-, Y- A- or C-axis, dynamic performance is lacking, the result of the circle-8 test has more diverse changes, and a larger variation range than BK3 test, and for Z-axis mismatch, the changing characteristics and variation ranges of these two tests are similar.

Graph: Fig. 17 DBB test simulation results of single axis' dynamic performance deficiency situations. aX-axis, bY-axis, cZ-axis, dA-axis, and eC-axis

Statistical indexes of the DBB simulation results in the cases of single axis' dynamic performance deficiency situations

Secondly, the situations of multi-axis' dynamic performance deficiency are analyzed. The position gains of each axis are set as follows:

• X and A mismatch: KppX:KppY:KppZ:KppA:KppC = 0.9:1:1:0.9:1;
• Y and C mismatch: KppX:KppY:KppZ:KppA:KppC = 1:0.9:1:1:0.9

The simulation results are shown in Fig. 18, and the statistical indexes are shown in Table 5. Obviously, when the dynamic performance of rotary axis and linear axis are simultaneously set deficient, the result of the circle-8 test has more diverse changes and a larger variation range than BK3 test.

Graph: Fig. 18 DBB test simulation results of multi-axis' dynamic performance deficiency situations. aX and A and bY and C

Statistical indexes of the DBB simulation results in the cases of multi-axis' dynamic performance deficiency situations

And then the effects of different dynamic performance deficiency levels are analyzed. The position gains of each axis are set as follows:

• X-axis mismatch:
• Level 1: KppX:KppY:KppZ:KppA:KppC = 0.9:1:1:1:1;
• Level 2: KppX:KppY:KppZ:KppA:KppC = 0.8:1:1:1:1;
• Level 3: KppX:KppY:KppZ:KppA:KppC = 0.7:1:1:1:1;
• A-axis mismatch:
• Level 1: KppX:KppY:KppZ:KppA:KppC = 1:1:1: 0.95:1;
• Level 2: KppX:KppY:KppZ:KppA:KppC = 1:1:1: 0.9:1;
• Level 3: KppX:KppY:KppZ:KppA:KppC = 1:1:1: 0.85:1;

The simulation results are shown in Fig. 19, and the statistical indexes are shown in Table 6. According to the summary and analysis simulation results, it can be concluded that:

• For both of the circle-8 test and BK3 test, higher deficiency level will cause a similar shape of test results but larger variation ranges, which is consistent with theoretical analysis;
• In different deficiency levels, the result of the circle-8 test always has more diverse changes and a larger variation range than BK3 test.

Graph: Fig. 19 DBB test simulation results of different dynamic performance deficiency levels. a Circle-8 test (A-axis' dynamic performance is deficient). b BK3 test (A-axis' dynamic performance is deficient). c Circle-8 test (A-axis' dynamic performance is deficient). d BK3 test (A-axis' dynamic performance is deficient)

Statistical indexes of the DBB simulation results in different dynamic performance deficiency levels

To sum up, in different cases of different dynamic performance deficiency situations, the circle-8 test always has more powerful capability of dynamic performance test than BK3 test.

According to Eqs. 2, 4, 5, and 6 in Section 2.1, while the servo gains remain constant, the different motion velocities (feedrates) can cause different tool tip error performances. The position gains of each axis are set as follows:

• K ppX:KppY:KppZ:KppA:KppC = 1:1:1:0.95:1;

The feedrates are separately set to 500 and 1500 mm/min. The simulation results are shown in Fig. 20, and the statistical indexes are shown in Table 7. It can be seen that, with the increase of feedrate, the shape of the test results are in similar basic configurations but stretched radially, which is consistent with theoretical analysis, and in different feedrates, the result of the circle-8 test always has more diverse changes and a larger variation range than BK3 test, which means the circle-8 test always has more powerful capability of dynamic performance test than BK3 test.

Graph: Fig. 20 DBB test simulation results of different feedrates. a Circle-8 test and b BK3 test

Statistical indexes of the DBB simulation results in different feedrates

To verify the efficiency of the circle-8 test and correctness of the above simulations, the circle-8 test and BK3 test were conducted by DBB in a five-axis machine tool with a tilting rotary table, as Fig. 21 shows. The experimental setups are list as follows:

• With the same situation and setting, such as feedrate or room temperature, the X-axis and A-axis were set to mismatch separately, and the X-axis and A-axis were set to multi-mismatch in order to create describable dynamic performance deficiency cases;
• With the same situation and setting, the A-axis was set to mismatch and double mismatch separately in order to simulate different dynamic performance deficiency levels;
• The A-axis was set to mismatch, and the feedrate was set to 500 and 1000 mm/min separately.

Graph: Fig. 21 Experiment of circle-8 and BK3 test based on double ballbar (DBB)

Herein, because the parameters in CNC system are processed by normalization, the position gain ratio is not meaningful and replaced by "deficiency" or "double deficiency."

The experiment results are shown in Figs. 22, [23], 24, and the statistical indexes are shown in Table 8. According to the experiment results, the summary and analysis can be listed:

• Although the shapes and order of magnitudes of experiment results are not completely same as simulations in Section 3.2, the change tendencies and topological structures are very similar. As to the difference between simulation and experimental results, the simulation model cannot be built totally the same as the experimental machine tool, especially some parameters of NC system and servo motors have been uniformized or normalized so the simulation and experimental results cannot perfectly fit.
• For different dynamic performance deficiency cases, the circle-8 is always more sensitive than BK3, which is fit for the analysis in Section 3.1 and simulations in Section 3.2.2;
• According to the figures of experiment results in different deficiency levels, it can be seen that, for both of circle-8 test and BK3 test, higher deficiency level will cause a similar but radial-stretched test result shape. However, the statistical indexes of BK3 cannot show the tendency clearly, which means the circle-8 test has more powerful testing capability than BK3 in the situations of different deficiency levels;
• According to the figures of experiment results in different feedrates, it can be seen that, for both of circle-8 test and BK3 test, higher feedrate will cause a similar but radial-stretched test result shape. The statistical indexes of circle-8 test can make the tendency more clearly than BK3, which means the circle-8 test has more powerful testing capability than BK3 in the situations of different feedrates.

Graph: Fig. 22 Experiment results of different dynamic performance deficiency cases. aX-axis, bA-axis, and cX and A

Graph: Fig. 23 Experiment results of different deficiency degrees. a Circle-8 test and b BK3 test

Graph: Fig. 24 Experiment results of different federates. a Circle-8 test and b BK3 test

Statistical indexes of the experiment results

To sum up, the circle-8 test always has more powerful capability of dynamic performance test than BK3 test.

In this paper, the mechanism and characteristics of dynamic tracking error of five-axis machine tools are summarized, which shows that the essence of this error type is similar to delay or mismatch rather than an error. Aiming at the above characteristics, a novel dynamic performance test based on DBB, which is named circle-8 test, is proposed. In this new test, the five motion axes are driven together to finish a complicated motion process, and the dynamic tracking error is acquired during the movement, which is more suitable for dynamic performance test than exist methods. The above-improved dynamic performance test can be applied to provide effective data for the research of modeling and error reduction of five-axis machine tools.

To exhibit the improvement of the circle-8 test, this test and the BK3 test of ISO standard, which is considered as a comparison, are conducted in a five-axis machine tool with a tilting rotary table. In different dynamic inaccuracy situations, the test result of the circle-8 test and BK3 test from ISO standard are compared by simulation and experiment separately. According to the simulation and experiment results, the conclusion can be drawn that, for different kinds of dynamic inaccuracy situations, the circle-8 test results always have more distinctive shape, larger variation ranges, and easier-to-observe change tendencies, which means the circle-8 test has more powerful capability of dynamic performance test than BK3 test.

In the future research, with the perfection of the simulation model and the improvement of experimental conditions, analysis of the effects of more specific error sources on circle-8 test results is a meaningful topic, which can offer a validity method for machine tool error diagnosis or tracing.

This work was supported by the (04) National Science and Technology Major Projects of China (Grant No. 2017ZX04002001-002).

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