Quality characteristic association analysis of computer numerical control machine tool based on meta-action assembly unit

Abstract

As everyone knows, assembly quality plays a very important role in final product quality. Since computer numerical control machine tool is a large system with complicated structure and function, and there are complex association relationships among quality characteristics in assembly process, then it is difficult and inaccurate to analyze the whole computer numerical control machine tool quality characteristic association at one time. In this article, meta-action assembly unit is proposed as the basic analysis unit, of which quality characteristic association is studied to guarantee the whole computer numerical control machine tool assembly quality. First, based on “Function-Motion-Action” decomposition structure, the definitions of meta-action and meta-action assembly unit are introduced. Second, manufacturing process association and meta-action assembly unit quality characteristic association are discussed. Third, after understanding the definitions of information entropy and relative entropy, the concrete meta-action assembly unit quality characteristic association analysis steps based on relative entropy are described in detail. And finally, the lifting piston translation assembly unit of automatic pallet changer is taken as an example, the association degree between internal leakage and the influence factors of part quality characteristics and mate-relationships among them are calculated to figure out the most influential factors, showing the correctness and feasibility of this method.

Introduction

It is known that computer numerical control (CNC) machine tool is the most basic equipment of manufacturing industry, especially the key important equipment of national defense industry, and its quality and reliability have already become the important symbols to measure the modernization level of a nation’s industry. Since assembly process is a significant part in manufacturing process, and assembly operation is a major factor in determining the whole machine’s quality, then assembly process quality analysis is very necessary in CNC machine tool quality control. Beiter et al. ¹ and Li et al. ² used assembly quality methodology (AQM) to analyze the product potential assembly way under multinational resource production mode in design stage and evaluate the product assembly quality with design for assembly (DFA) in order to improve the assembly quality in design stage. Khodaygan and Movahhedy ³ presented a novel method based on fuzzy concepts for process capability analysis of assembly dimensions in mechanical assemblies. Andolfatto et al. ⁴ proposed a method to select an assembly technique for each joint of a product and to allocate geometrical tolerances accordingly.

In manufacturing process, product quality characteristics are not independent; they affect each other directly or indirectly. Recently, a lot of scholars have been working hard on quality characteristic association from different angles. Guo et al. ⁵ and Zhang et al. ⁶ studied on decoupling technology of product quality characteristics. Yang and Duan ⁷ put forward the quality characteristic association network with the quality characteristic association unit as the basic unit to realize the quantitative modeling and analysis of the correlative relationships among quality characteristics. Ouyang et al. ⁸ combined the quality characteristics analysis chart (QCAC), entropy method, and technique for order of preference by similarity to ideal solution (TOPSIS) to measure quality characteristics and rank improvements for all substandard quality characteristics of a product in light of resource requirements and potential for performance improvements.

However, all of the researches above analyzed quality characteristics based on the whole CNC machine tool from a macroscopic angle, which is very difficult and inaccurate for assembly process quality control. ⁹ The recent popular method is to cut through complexity: first, decompose the whole system into relatively simple sub-systems, components, and even basic parts by the method of “Top-Down,” and study quality on the “part layer”; then analyze the whole machine’s comprehensive performance by the way of “Bottom-Up,” integrating each component property layer by layer from bottom to top. And the three commonly used decomposition methods are “Subassembly-Component-Part (SCP),”“Function-Behavior-Structure (FBS),” and “Component-Suite-Part (CSP),” all of which focus only on the static physical structure, not fitting for quality and reliability analysis.

In this article, a new decomposition structure of “Function-Motion-Action (FMA)” and the definition of meta-action are introduced, and then meta-action assembly unit (MAU) is proposed as the basic analysis unit, of which quality characteristic association is studied to guarantee the whole CNC machine tool assembly quality, and thus improving the final quality of CNC machine tool. Based on the large systems cybernetics, structural decomposition theory, and system dynamics theory, analysis on quality characteristic association of MAU from a dynamic fine-grained angle is simpler and more effective, which can not only help assembly process quality control more targeted, but also play an important role in assembly quality tracing and prediction. ¹⁰

Basic theories

MAU

Meta-action

System functional decomposition in reliability driven assembly technology (RADT) is to break one complex system down into many basic actions according to the “FMA” structure, ¹¹ while this structurally independent basic action unit can realize a certain action or achieve a certain purpose, and also can be controlled and analyzed without once-more breakdown need, which is called meta-action. The main difference between structural decomposition and functional decomposition is that the former one’s research object is hardware composition such as parts based on precision, and the latter one’s is functional motion such as meta-actions, considering not only precision, but also quality and reliability, as shown in Figure 1. From system dynamics theory, it can be seen that CNC machine tool working process is accomplished by the relative motions between tools and workpieces. CNC machine tool has various motion forms, while the accuracy and stability of different motions determine the machining quality of CNC machine tool.

Figure 1.

System decomposing process.

A large complex system being decomposed into several small simpler systems is a common scientific research method in large systems cybernetics. ¹² The whole structure and function of CNC machine tool are too complicated to be analyzed at one time, so it is necessary to decompose CNC machine tool working process down into meta-action, which is the basic unit of action, according to “FMA” decomposition structure. Since meta-action’s quality and reliability is the basis of the whole CNC machine tool’s, the system quality control technology should be studied first on the “meta-action layer.” Only the combined effect of multiple meta-actions can achieve different motions, and only the combined effect of one or multiple motions can realize the functions, which means that only quality improvement of each meta-action can guarantee the motions’ quality and finally the whole CNC machine tool’s quality. Therefore, it is very urgent and necessary to analyze meta-action quality before implementing the whole CNC machine tool quality analysis.

Although CNC machine tool working process is a complex synthesize motion, the composed meta-actions are very simple. Usually, the meta-motion of general part is only simple rotation or translation in the CNC machine tool design process in accordance with the no-coupling design principle, while rotation is the most common meta-action in working process, such as motor rotation, screw rotation, gear rotation, and shaft rotation, and translation almost plays a role of feed motion auxiliary, such as oil cylinder piston translation, screw-nut translation. The automatic pallet changer (APC) of THM6380 is taken as an example, as shown in Figure 2. The APC functional component has four motions and eight meta-actions to help accomplish its function.

Figure 2.

“FMA” decomposition of APC functional component.

MAU

From dynamics angle, CNC machine tool is a system of transforming the input power of motor, pump, and so on into working motion through some ways. As the most basic element of the system, the basic function of meta-action is to transfer motion and force. Typical meta-action structural unit includes supporting part, power source, executive components, transmission parts, and fasteners, as shown in Figure 3. Meta-action is the basic unit composing CNC machine tool working process, and it has quality characteristics closely related to system performance, which are guaranteed by its structure being correctly assembled; therefore, MAU needs to be analyzed.

Figure 3.

Meta-action structural unit.

Since MAU is defined based on the definition and structural unit of meta-action, then it should be decomposed exactly according to the mate-relationships in assembly process on the basis of correct meta-action decomposition. ¹³ MAU is an independent component assembled by a certain group of parts based on one reference part of meta-action, which can meet the technical requirements of the meta-action, and can realize the required function of the meta-action in assembly process.

MAU is an independent assembly component, which can exist independently from other parts or components, and all experiments, tests, and analyses can be done on MAU. ¹⁴ In this article, the quality characteristic association of MAU is analyzed to guarantee the whole CNC machine tool assembly process quality and reliability. The MAU decomposition should follow the following principles:

• MAU can be assembled independently and meet the technical requirements of meta-action function. Usually, meta-action (rotation or translation) is required to be stable, reliable, and flexible.

• MAU should include all the parts which can correctly realize the motion function of meta-action except for the power source (external part driving the basic parts of MAU).

• Independent experiments, tests, and other analyses can be done on MAU after the power source being added.

• One MAU should be relatively independent to the others, and there should be as few relations as possible among them. Different MAUs can share one or several parts. Based on the difference of meta-actions, MAU may be one part of a functional component or may include the whole functional component or the connection parts of several functional components.

MAU quality characteristic association

Manufacturing process association

CNC machine tool manufacturing process is a complex dynamic system composed of multiple factors (5M1E): men, machine, material, method, measurement, environment, and so on. ¹⁵ In the system, the multiple factors in each process are interactional, and so are the processes. Because of the interactions among factors and different processes, each process cannot be independent, which is called manufacturing process association. The manufacturing process association embodies in the influence of multiple factors on quality characteristics and the influence of the quality characteristics generated in the former process on the quality characteristics generated in the latter one, which means that one quality characteristic may be influenced simultaneously by multiple factors and other quality characteristics in former processes, as shown in Figure 4. If there is fluctuation in one quality characteristic, it can be blamed on both the fluctuations of the multiple influential factors in this process and the fluctuations of quality characteristics in closely related former processes. It can be seen that the association relationships among quality characteristics and between quality characteristics and the multiple influential factors make the quality characteristic fluctuation reasons more complex. In this article, only the association relationships among quality characteristics are discussed.

Figure 4.

Manufacturing process association.

MAU quality characteristic association

In the assembly process, all parts are assembled together to form units even products by many concrete processes according to different principles. To simplify the product quality characteristic association analysis, the assembly unit composed of parts should be analyzed first. It can be seen that any process forming the parts needed in this process is the associated former process, meaning that each process has many associated former processes simultaneously. The quality association between the former process and the latter one embodies not only in the influence of the part quality characteristic fluctuations generated in the former process on the quality characteristics generated in the latter one, but also in the influence of the mate-relationships among parts generated in several associated former processes on the quality characteristics generated in the latter one.

Compared with any other assembly unit, MAU is a basic action unit in CNC machine tool assembly process according to “FMA” decomposition structure, which is more suitable for quality and reliability analysis. Each MAU has a specific function and can independently complete the designed action. Its quality characteristics are shown not only as MAU precision, but also as MAU performance, MAU reliability, and so on. Because MAU is assembled by many parts accordingly to some certain mate-relationship principals, the quality characteristics of MAU are influenced not only by the part quality characteristics themselves, but also by the mate-relationships among them. The association relationships among quality characteristics of MAU in assembly process are shown in Figure 5.

Figure 5.

MAU quality characteristic association.

Main methods

Relative entropy

Information entropy

Information entropy was proposed by C. E. Shannon in “A Mathematical Theory of Communication” in 1948, which is the measurement of information established on probability statistics model. It also expresses the measurement of uncertain.

If there are many discrete events $S=\{E_1,E_2,\dots ,E_n\}$ existed in the system $\mathrm{S}$ , of which each event’s probability distribution is $P=\{p_1,p_2,\dots ,p_n\}$ , while $p=(S=E_i\cap S=E_j)=0,i\ne j,\sum _{i=1}^n{p}_i=1$ . Then each event’s amount of information is $\mathrm{le}=-\mathrm{lo}{g}_K{p}_i$ . The whole system’s average amount of information is

$$H\left(S\right)=H(p_1,p_2,\dots ,p_n)=-K\sum _{i=1}^n{p}_i\log p_i$$

(1)

Here, $H(S)$ means information entropy, which describes the uncertainty of information sources. The constant K is decided by the chosen unit, and usually $K=1$ . Different base of the logarithmic is selected, different unit the entropy will have. When 2 is selected as the base of the logarithmic, the unit is bit; and when e is selected as the base of the logarithmic, the unit is nat. Usually e is selected.

Relative entropy

Assumed $x_i\ge 0,y_i\ge 0(i=1,2,\dots ,n)$ and $\sum y_i\le \sum x_i\le 1$ , then

$$H(X,Y)=\sum x_i\log \frac{x_i}{y_i}$$

(2)

while $H(X,Y)$ is the relative entropy of X to Y.

The main characteristics of relative entropy are shown as follows:

• $\sum x_i\log \frac{x_i}{y_i}\ge 0$ ;

• The necessary and sufficient condition of $\sum x_i\log (x_i/y_i)=0$ is that to any i, $x_i=y_i$ .

It can be seen from the above that when the discrete probability distributions X and Y are equal, the relative entropy can get its minimum value. Therefore, the relative entropy can be used to measure the coincidence degree of the two.

MAU quality characteristic association analysis based on relative entropy

As a basic action unit of CNC machine tool, MAU should be analyzed as a whole dynamic sub-system with more consideration on its quality and reliability from the kinematic point of view. MAU quality characteristic association analysis is to analyze the association relationships between the quality characteristics of MAU and the influence factors of its composed part quality characteristics and mate-relationships among them, by measuring the coincidence degree of the MAU quality characteristics’ probability distributions and the influence factors’ to find out the influence factors which have the closest association with the quality characteristics of MAU.

The concrete steps of CNC machine tool MAU quality characteristic association analysis based on relative entropy are described as follows:

• 1. Determine the quality characteristic set of the MAU being analyzed $Q_n=(q_{n1},q_{n2},\dots ,q_{\mathrm{nm}})$ ;

• 2. Determine the influence factor set composed of all part quality characteristics and mate-relationships among them $Q_{n-1}=(q_{(n-1)1},q_{(n-1)2},\dots ,q_{(n-1)l})$ , which affects each quality characteristic of MAU $q_{\mathrm{ni}}(i=1,2,\dots ,m)$ . Since every assembly product part has many quality characteristics and many mate-relationships among them, for the convenience and correctness of research, those part quality characteristics and mate-relationships having no influence on $q_{\mathrm{ni}}$ will not be listed in $Q_{n-1}$ , according to the professions’ suggestions.

• 3. Collect as much sample data of each quality characteristic and influence factor as possible. Assumed the number of samples is t, then

  $$\begin{array}{l}{q}_{\left(n-1\right)j}=\left\{q_{\left(n-1\right)j}\left(1\right),q_{\left(n-1\right)j}\left(2\right),\mathrm{\dots },q_{\left(n-1\right)j}\left(k\right),\mathrm{\dots },q_{\left(n-1\right)j}\left(t\right)\right\}\\ \left(j=1,2,\mathrm{\dots },l\right)\end{array}$$
  $$\begin{array}{c}{q}_{\mathrm{ni}}=\{q_{\mathrm{ni}}\left(1\right),q_{\mathrm{ni}}\left(2\right),\dots ,q_{\mathrm{ni}}\left(k\right),\dots ,q_{\mathrm{ni}}\left(t\right)\}\hfill \\ (i=1,2,\dots ,m)\hfill \end{array}$$

• 4. Process the data columns $q_{(n-1)j}$ and $q_{\mathrm{ni}}$ in order to meet the calculation condition of relative entropy.

Let each data in columns $q_{(n-1)j}$ and $q_{\mathrm{ni}}$ minus the minimum value in each column, and divided by the sum of all obtained data in each column, then new data columns ${q\prime }_{(n-1)j}$ and ${q\prime }_{\mathrm{ni}}$ can be obtained. ${q\prime }_{(n-1)j}(k)\ge 0$ , ${q\prime }_{\mathrm{ni}}\left(k\right)\ge 0$ and $\sum {q\prime }_{\mathrm{ni}}\le \sum {q\prime }_{(n-1)j}\le 1$ .

• 5. Calculate the relative entropy of the new data columns ${q\prime }_{(n-1)j}$ and ${q\prime }_{\mathrm{ni}}$ referring to equation (2)

$$H({q\prime }_{(n-1)j},{q\prime }_{\mathrm{ni}})=\sum {q\prime }_{(n-1)j}\left(k\right)\log \frac{{q\prime }_{(n-1)j}\left(k\right)}{{q\prime }_{\mathrm{ni}}\left(k\right)}$$

(3)

Rank according to the values of relative entropy. Smaller $H({q\prime }_{(n-1)j},{q\prime }_{\mathrm{ni}})$ is, more coincident ${q\prime }_{(n-1)j}$ and ${q\prime }_{\mathrm{ni}}$ are, more affective ${q\prime }_{(n-1)j}$ is to ${q\prime }_{\mathrm{ni}}$ , and closer their association relationship is.

• 6. Assumed the matrix

  $$\mathrm{SH}=\left[\begin{array}{c}\begin{array}{ccc}\begin{array}{cc}{h}_{11}\hfill & h_{12}\hfill \end{array}\hfill & \cdots \hfill & h_{1l}\hfill \\ \multicolumn{1}{c}{\begin{array}{cc}{h}_{21}\hfill & h_{22}\hfill \end{array}}& \cdots \hfill & h_{2l}\hfill \\ \multicolumn{1}{c}{\begin{array}{cc}⋮\hfill & ⋮\hfill \end{array}}& \ddots \hfill & ⋮\hfill \end{array}\hfill \\ \multicolumn{1}{c}{\begin{array}{ccc}\begin{array}{cc}{h}_{m1}\hfill & h_{m2}\hfill \end{array}\hfill & \cdots \hfill & h_{\mathrm{ml}}\hfill \end{array}}\end{array}\right]$$

  while $h_{\mathrm{ij}}=H({q\prime }_{(n-1)j},{q\prime }_{\mathrm{ni}})$ . Matrix SH is the association matrix between the quality characteristics of MAU and the influence factors, according to which the quality characteristic tracing and prediction can be realized.

Case study

Background

APC is one of the CNC machine tool’s most important functional components, and its quality characteristics are still very hard to be analyzed clearly at one time. From Figure 2, it can be seen that lifting piston translation is one decomposed meta-action of APC functional component. It is an important action in CNC machine tool working process to realize the lifting motion of APC carrier, and it is difficult to control the assembly quality of its parts, so the MAU based on it is taken as an example to show the correctness and feasibility of the method proposed in this article. The structure diagram of APC lifting part is shown in Figure 6.

Figure 6.

Structure diagram of APC lifting part.

The lifting piston translation assembly unit of APC includes mandrel unit, large piston unit, piston sealed cap unit, oil guiding bar unit, and oil import and export unit. And its main parts are mandrel, large piston, sealed cap, oil guiding bar, attachment, and other auxiliary parts, as shown in Figure 7. According to the concrete steps of CNC machine tool MAU quality characteristic association analysis based on relative entropy, the quality characteristic association analysis of APC lifting piston translation assembly unit and its composed parts is as follows.

Figure 7.

Lifting piston translation assembly unit of APC.

Calculating process

• 1. Considering the quality and reliability requirements of APC lifting piston translation assembly unit comprehensively, based on the discussion of the professions (including three enterprise leaders, five enterprise technicians, and four experts), the key working performance indexes of lifting piston translation assembly unit are chosen as internal leakage, minimum starting pressure, and minimum steady speed.

Internal leakage: lifting piston translation assembly unit internal leakage will reduce the volumetric efficiency, aggravate the oil temperature rise, and affect the positional accuracy to hinder the piston from stopping accurately and stably in the cylinder; ¹⁶ Minimum starting pressure: the minimum working pressure of lifting piston translation assembly unit under no-load condition, reflecting the parts manufacturing and assembly precision, and also the seal friction size of piston translation assembly unit; Minimum steady speed: the minimum speed of lifting piston translation assembly unit under full-load condition without crawl phenomenon.

Then, the quality characteristic set of piston translation assembly unit is determined as

$$Q_n=\left(q_{21},q_{22},q_{23}\right)=(\mathrm{Internal\; leakage},\mathrm{Minimum\; starting\; pressure},\mathrm{Minimum\; steady\; speed})$$

• 2. The quality characteristic of internal leakage is taken as an example, it can be seen that each part’s influential quality characteristics and mate-relationship indexes among them are decided as the influence factor set of internal leakage. According to the comprehensive dynamic analysis on the assembly process quality and reliability of APC lifting piston translation assembly unit, the part quality characteristics influencing internal leakage of lifting piston translation assembly unit include the concentricity of large piston and mandrel, the surface roughness of mandrel inner wall, the radial run-out of mandrel, the straightness of mandrel inner wall, the surface straightness of large piston, and the concentricity of large piston and oil guiding bar. The mate-relationships among parts influencing internal leakage of lifting piston translation assembly unit include the clearance of large piston and mandrel inner wall, the clearance of large piston and sleeve, and the clearance of cap and oil guiding bar.

Therefore, the influence factor set of $q_{21}$ (Internal leakage) is determined as

$$\begin{array}{c}\multicolumn{1}{c}{Q_{n-1}=(q_{11},q_{12},q_{13},q_{14},q_{15},q_{16},q_{17},q_{18},q_{19})}\\ \multicolumn{1}{c}{=(\mathrm{the\; concentricity\; of\; large\; piston\; and\; mandrel},}\\ \mathrm{the\; surface\; roughness\; of\; mandrel}\hfill \\ \mathrm{inner\; wall},\mathrm{the\; radial\; run}-\mathrm{out\; of\; mandrel},\hfill \\ \mathrm{the\; straightness\; of\; mandrel\; inner\; wall},\hfill \\ \mathrm{the\; surface\; straightness\; of\; large\; piston},\hfill \\ \mathrm{the\; concentricity\; of\; large\; piston\; and\; oil}\hfill \\ \mathrm{guiding\; bar},\mathrm{the\; clearance\; of\; large\; piston\; and}\hfill \\ \mathrm{mandrel\; inner\; wall},\mathrm{the\; clearance\; of}\hfill \\ \mathrm{large\; piston\; and\; sleeve},\mathrm{the\; clearance\; of\; cap}\hfill \\ \mathrm{and\; oil\; guiding\; bar})\hfill \end{array}$$

• 3. For internal leakage and each influence factor, nine groups of sample data have been collected by experiments, as shown in Table 1.

• 4. After being processed by the method described in step 4 of CNC machine tool MAU quality characteristic association analysis above, new data can be obtained, as shown in Table 2.

• 5. The relative entropy of each influence factor and internal leakage are calculated according to equation (3) where natural logarithm is chosen. Time of 0 probability does not affect the information entropy, meaning $0\log 0=0(\mathrm{when}x\to 0,x\log x\to 0)$ . Because ${q\prime }_{\mathrm{ni}}(k)=0$ after the fourth group of sample data being processed, the fourth group of sample data is determined to be invalid and removed. Calculating process and results are shown in Table 3.

Table 1.

Collected nine groups of sample data.

No.	$q_{21}$ (mL/min)	$q_{11}$ (Φ mm)	$q_{12}$ (um)	$q_{13}$ (mm)	$q_{14}$ (mm)	$q_{15}$ (mm)	$q_{16}$ ( $\Phi$ mm)	$q_{17}$ (mm)	$q_{18}$ (mm)	$q_{19}$ (mm)
1	0.186	0.0139	0.372	0.009	0.009	0.005	0.0021	0.401	0.095	0.112
2	0.171	0.0123	0.286	0.012	0.007	0.007	0.0015	0.355	0.087	0.124
3	0.214	0.0157	0.366	0.003	0.003	0.004	0.0022	0.422	0.144	0.128
4	0.155	0.0122	0.254	0.010	0.006	0.006	0.0019	0.358	0.092	0.108
5	0.201	0.0151	0.233	0.006	0.004	0.006	0.0031	0.412	0.098	0.156
6	0.175	0.0124	0.260	0.005	0.005	0.008	0.0016	0.397	0.112	0.119
7	0.223	0.0181	0.387	0.012	0.004	0.003	0.0039	0.427	0.097	0.148
8	0.190	0.0145	0.258	0.015	0.006	0.009	0.0030	0.409	0.131	0.125
9	0.209	0.0160	0.391	0.008	0.009	0.007	0.0027	0.417	0.100	0.140

Table 2.

Processed sample data.

No.	${q\prime }_{21}$	${q\prime }_{11}$	${q\prime }_{12}$	${q\prime }_{13}$	${q\prime }_{14}$	${q\prime }_{15}$	${q\prime }_{16}$	${q\prime }_{17}$	${q\prime }_{18}$	${q\prime }_{19}$
1	0.0942	0.0833	0.1958	0.1132	0.2308	0.0714	0.0706	0.1141	0.0462	0.0213
2	0.0486	0.0049	0.0746	0.1698	0.1538	0.1429	0.0000	0.0000	0.0000	0.0851
3	0.1793	0.1716	0.1873	0.0000	0.0000	0.0357	0.0824	0.1663	0.3295	0.1064
4	0.0000	0.0000	0.0296	0.1321	0.1154	0.1071	0.0471	0.0074	0.0289	0.0000
5	0.1398	0.1422	0.0000	0.0566	0.0385	0.1071	0.1882	0.1414	0.0636	0.2553
6	0.0608	0.0098	0.0380	0.0377	0.0769	0.1786	0.0118	0.1042	0.1445	0.0585
7	0.2067	0.2892	0.2169	0.1698	0.0385	0.0000	0.2824	0.1787	0.0578	0.2128
8	0.1064	0.1127	0.0352	0.2264	0.1154	0.2143	0.1765	0.1340	0.2543	0.0904
9	0.1641	0.1863	0.2225	0.0943	0.2308	0.1429	0.1412	0.1538	0.0751	0.1702

Table 3.

Calculating process and results.

No.	${q\prime }_{11}\left(k\right)\times \ln \frac{{q\prime }_{11}\left(k\right)}{{q\prime }_{21}\left(k\right)}q\prime$	${q\prime }_{12}\left(k\right)\times \ln \frac{{q\prime }_{12}\left(k\right)}{{q\prime }_{21}\left(k\right)}$	${q\prime }_{13}\left(k\right)\times \ln \frac{{q\prime }_{13}\left(k\right)}{{q\prime }_{21}\left(k\right)}$	${q\prime }_{14}\left(k\right)\times \ln \frac{{q\prime }_{14}\left(k\right)}{{q\prime }_{21}\left(k\right)}$	${q\prime }_{15}\left(k\right)\times \ln \frac{{q\prime }_{15}\left(k\right)}{{q\prime }_{21}\left(k\right)}$	${q\prime }_{16}\left(k\right)\times \ln \frac{{q\prime }_{16}\left(k\right)}{{q\prime }_{21}\left(k\right)}$	${q\prime }_{17}\left(k\right)\times \ln \frac{{q\prime }_{17}\left(k\right)}{{q\prime }_{21}\left(k\right)}$	${q\prime }_{18}\left(k\right)\times \ln \frac{{q\prime }_{18}\left(k\right)}{{q\prime }_{21}\left(k\right)}$	${q\prime }_{19}\left(k\right)\times \ln \frac{{q\prime }_{19}\left(k\right)}{{q\prime }_{21}\left(k\right)}$
1	−0.0102	0.1432	0.0208	0.2068	−0.0198	−0.0204	0.0219	−0.0329	−0.0317
2	−0.0112	0.0320	0.2124	0.1773	0.1541	0.0000	0.0000	0.0000	0.0477
3	−0.0076	0.0082	0.0000	0.0000	−0.0576	−0.0641	−0.0126	0.2005	−0.0555
5	0.0024	0.0000	−0.0512	−0.0496	−0.0285	0.0560	0.0016	−0.0501	0.1537
6	−0.0179	−0.0178	−0.0180	0.0181	0.1925	−0.0193	0.0562	0.1251	−0.0023
7	0.0971	0.0104	−0.0334	−0.0647	0.0000	0.0881	−0.0260	−0.0737	0.0062
8	0.0065	−0.0389	0.1710	0.0094	0.1500	0.0893	0.0309	0.2216	−0.0147
9	0.0236	0.0678	−0.0522	0.0787	−0.0198	−0.0212	−0.0099	−0.0587	0.0062
H	0.0827	0.2049	0.2494	0.3760	0.3709	0.1084	0.0621	0.3318	0.1096

And then, the relative entropy H can be ranked from small value to large value

$$H_4>H_5>H_8>H_3>H_2>H_9>H_6>H_1>H_7$$

Results

From the calculating results above, it can be seen that $H_7,H_1$ , and $H_6$ have smaller value, meaning $q_{17}$ (the clearance of large piston and mandrel inner wall), $q_{11}$ (the concentricity of large piston and mandrel), and $q_{16}$ (the concentricity of large piston and oil guiding bar) have closer relationships with internal leakage, which are the main influence factors to internal leakage. And strict measures should be taken to control the clearance of large piston and mandrel inner wall, the concentricity of large piston and mandrel, and the concentricity of large piston and oil guiding bar in design stage and assembly process to guarantee the quality and reliability of APC or even CNC machine tool.

Also, the association matrix of APC lifting piston translation assembly unit internal leakage and its parts’ quality characteristics can be obtained finally

$$SH={\left(h_{ij}\right)}_{1\times 8}=(0.0827,0.2049,0.2494,0.3760,0.3709,0.1084,0.0621,0.3318,0.1096)$$

SH shows the association degree of internal leakage and each part quality characteristic and the mate-relationship indexes among them, according to which the quality characteristic tracing and prediction of internal leakage can be realized. The association degree of any other quality characteristic such as minimum starting pressure, minimum steady speed, and each part quality characteristic and the mate-relationship indexes among them can also be calculated using the same method.

Compared with any other assembly unit, MAU quality characteristic association analysis considers more on quality and reliability, not only precision, and more on the mate-relationships among parts, not only the part quality characteristics themselves, which is more suitable for the dynamic CNC machine tool system. Using the results above, the manufacturing enterprises find it more simple and accurate to quality defect or fault tracing and quality prediction, and more effective to improve the quality and reliability of CNC machine tool, which verifies the correctness of this method.

Conclusion

As a complicated integrated mechatronic product, CNC machine tool is hard to be analyzed from a macroscopic angle. It is difficult to figure out all the association relationships among the quality characteristics of the whole CNC machine tool at one time. Therefore, based on the large systems cybernetics, structural decomposition theory, and system dynamics theory, a new thought of CNC machine tool working process being broke down into many meta-actions according to the “FMA” decomposition structure to make analysis easier was proposed. Decomposed from the kinematic point of view, MAU focuses more on quality and reliability requirements than any other assembly unit, concentrating both on part quality characteristics themselves and mate-relationships among them in MAU quality characteristic association analysis. Actually, analyzing on the quality characteristics of each MAU separately is proved to be simple, scientific, and targeted for the whole CNC machine tool assembly process quality control.

In this article, a new method of quality characteristic association analysis based on MAU was proposed to provide theoretical and technical support for CNC machine tool assembly quality control. First, the definitions of meta-action and MAU were introduced, and the “FMA” decomposition structure was mentioned. Second, manufacturing process association and MAU quality characteristic association were discussed. Third, the definitions of information entropy and relative entropy were introduced, and then the concrete MAU quality characteristic association analysis steps based on relative entropy were described in detail. Finally, the lifting piston translation assembly unit of APC was taken as an example, and the association degree between internal leakage and the influence factors of part quality characteristics and mate-relationships among them were calculated to figure out the most influential factors, by which this method was proved to be correct and scientific.