2. State space model of assembly error propagation

The hierarchy diagram of assembly errors propagation is established by decomposing the error propagation process hierarchically based on the error propagation carriers that function assembly units, motion assembly units and MUs, is depicted in **Figure 16**.

The small displacement torsor is introduced on the basis of a hierarchy diagram, while the errors between actual geometric characteristics and ideal geometric characteristics are represented by the error vector *R* ¼ ½ � *a; b;c; α; β; γ <sup>T</sup>*, where *a, b, c* and *α, β, γ* mean the translation errors and rotation errors along the three axes, respectively. The relative poses among assembly units are determined by their position and pose parameters, and the feature matrix is established according to the pose parameters among sub-coordinate systems, shown in Eq. (3).

$$A\_k = \begin{bmatrix} \mathbf{1} & -\Delta \mathbf{y} & \Delta \beta & \Delta a + \mathbf{x} \\ \Delta \mathbf{y} & \mathbf{1} & -\Delta a & \Delta b + \mathbf{y} \\ -\Delta \beta & \Delta a & \mathbf{1} & \Delta c + \mathbf{z} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{1} \end{bmatrix} \tag{3}$$

#### **Figure 16.**

*Hierarchy diagram of assembly errors propagation.*

*Reliability Technology Based on Meta-Action for CNC Machine Tool DOI: http://dx.doi.org/10.5772/intechopen.85163*

The link matrix is constructed according to the two-level composite matrix architecture. In this table, the row elements represent the links, the elements in the first level column stand for the assembly parts or components, the second level column elements signify the parts contained in the components, and the center cells are identified by the error source type *k*ð Þ 0≤*k*≤ 5 . However, if there is no error propagation or if there are no effects in assembly quality and accuracy during the

The hierarchy diagram of assembly errors propagation is established by decomposing the error propagation process hierarchically based on the error propagation carriers that function assembly units, motion assembly units and MUs, is

The small displacement torsor is introduced on the basis of a hierarchy diagram, while the errors between actual geometric characteristics and ideal geometric char-

> 1 �Δ*γ* Δ*β* Δ*a* þ *x* Δ*γ* 1 �Δ*α* Δ*b* þ *y* �Δ*β* Δ*α* 1 Δ*c* þ *z* 000 1

*α, β, γ* mean the translation errors and rotation errors along the three axes, respectively. The relative poses among assembly units are determined by their position and pose parameters, and the feature matrix is established according to the pose

*<sup>T</sup>*, where *a, b, c* and

(3)

error propagation, the cells should be empty.

*Reliability and Maintenance - An Overview of Cases*

depicted in **Figure 16**.

**Figure 16.**

**62**

*Hierarchy diagram of assembly errors propagation.*

2. State space model of assembly error propagation

acteristics are represented by the error vector *R* ¼ ½ � *a; b;c; α; β; γ*

parameters among sub-coordinate systems, shown in Eq. (3).

*Ak* ¼

Suppose the geometric characteristics of motion assembly units are affected by single factor of the MUs, thus, by sorting the MUs that affect the *h*th geometric error of the *g*th motion unit according to the assembly sequence number, shown in **Figure 17**.

According to the assembly process, after finishing the assembly of *k*th MU, the assembly error outputs are represented by the small displacement screw *Xgh*ð Þ*k* as below:

$$X\_{gh}(k) = \begin{bmatrix} d\_k \\ \delta\_k \end{bmatrix} = \begin{bmatrix} a\_k & b\_k & c\_k & a\_k & \beta\_k & \gamma\_k \end{bmatrix}^T$$

where, *k* ¼ 1*,* 2*,* …*, i*,*i* is the total number of the MUs that affect the *h*th geometric error of the *g*th motion unit, *dk* is the translation component of geometric error, and *δ<sup>k</sup>* is the rotation component of geometric error.

The errors introduced by the dynamic uncertain factors of assembly force and measurement, etc., are considered in the actual assembly process and are shown in Eq. (4).

$$\begin{cases} X\_{gh}(k) = A\_{gh}(k)X\_{gh}(k-1) + B\_{gh}(k)\mu\_{gh}(k) + \upsilon\_{gh}(k) \\ \qquad T\_{gh}(k) = C\_{gh}(k)X\_{gh}(k) + \xi\_{gh}(k) \end{cases} \tag{4}$$

where, *Agh*ð Þ*k* is the transformation matrix of the geometric error vector among characteristic co-ordinate systems, *Bgh*ð Þ*k* is the error input matrix that reflects the affection of new input geometric characteristic error on assembly units, and *μgh*ð Þ*k* is the geometric error vector introduced by the assembly of the *k*th MU.

The error vector consists of the errors generated by the assembly, grinding and repairing of the MUs; and *vgh*ð Þ*k* is the assembly error introduced by the assembly force, *ξgh*ð Þ*k* is the measurement noise obeying the normal distribution with a mean value of 0. However, it is worth noting that there is no error input if this station

the meta-action fault tree after function decomposition as separate independent

based on the modularization fault model, as *F* ¼ ð Þ *S; P; T; Q* , where:

• *P* stands for the set of assembly units' performance attribute,

• *S* symbolizes the set of assembly units' performance,

*Modularization fault tree model based on the function decomposition.*

*Reliability Technology Based on Meta-Action for CNC Machine Tool*

*DOI: http://dx.doi.org/10.5772/intechopen.85163*

The function implementation is the key performance of the assembly quality; the performances of the assembly units are characterized by using the quadruples

• *Pm* means the performance evaluation index of the assembly units, and the indices constitute the set of the assembly units' performance attribute,

• *T* characterizes the set of all action obtained by the function decomposition,

• *Tij* denotes the cell of *T*, and *Ti j*ð Þ <sup>þ</sup><sup>1</sup> is used to represent the subordinate functional action of *Tij* because of the inclusion relationship among the

• *Q* signifies the mapping function from the functional action to assembly

modularization fault tree is simplified by sorting basic events. Then, the fault tree is transformed into a binary decision diagram (BDD) by using ITE structural analysis methodology. Finally, transforming the meta-action sub-fault tree into a BDD, researching the assembly reliability of meta-action assembly units by combining the BDD with a mapping function, and obtaining the mapping function *Qa* of meta-

where *M* means the mapping matrix of different performance attributes' weight for meta-actions and *F* indicates performance index evaluation results of the MUs'

In addition to the application of reliability modeling, reliability design, and assembly reliability analysis for CNC machine tool based on MU, FMA methodology

*Qa* : *M* � *F* (7)

On the basis of the modularization fault tree, the assembly reliability

modules (**Figure 19**).

**Figure 19.**

functional actions, and

performance, *Qa* : *T* ! *P*.

reliability.

**65**

action assembly reliability are shown in Eq. (7).

*3.3.4 Other reliability application based on meta-action*

#### **Figure 18.**

*The state space model of assembly error propagation.*

does not need to be measured. *Cgh*ð Þ*k* is the output matrix and *Tgh*ð Þ*k* is the geometric error obtained by measuring.

The state space model of assembly error propagation is shown in **Figure 18**.

The definition of the motion assembly units' final output error is the geometric error *Tgh*ð Þ*i* measured after finishing the assembly of the final assembly unit *i*, i.e., *Xh Yg* � � <sup>¼</sup> *Tgh*ð Þ*<sup>i</sup>* . Therefore, the state space models of assembly error propagating from motion assembly units to function assembly units, from function assembly units to the whole machine assembly are deduced for the same reason, shown in Eqs. (5) and (6).

$$\begin{cases} X\_{\varepsilon f}(k) = A\_{\varepsilon f}(k)X\_{\varepsilon f}(k-1) + B\_{\varepsilon f}(k)\mu\_{\varepsilon f}(k) + \nu\_{\varepsilon f}(k) \\ \qquad T\_{\varepsilon f}(k) = C\_{\varepsilon f}(k)X\_{\varepsilon f}(k) + \xi\_{\varepsilon f}(k) \end{cases} \tag{5}$$

$$\begin{cases} X\_{\varepsilon}(k) = A\_{\varepsilon}(k)X\_{\varepsilon}(k-1) + B\_{\varepsilon}(k)\mu\_{\varepsilon}(k) + \nu\_{\varepsilon}(k) \\ \qquad T\_{\varepsilon}(k) = C\_{\varepsilon}(k)X\_{\varepsilon}(k) + \xi\_{\varepsilon}(k) \end{cases} \tag{6}$$

The geometric error *X G*ð Þ¼ *<sup>e</sup> X*1ð Þ *Ge ;X*2ð Þ *Ge ;* …*;Xf*ð Þ *Ge* � � of the function assembly unit *e*, referred as synthesis error of function assembly unit *e*, is obtained by introducing the error of assembly units into the state space model layer-by-layer is shown below:

$$\begin{aligned} E(G\_\epsilon) &= F(X\_1(G\_\epsilon), X\_2(G\_\epsilon), \dots, X\_f(G\_\epsilon)) = \\ F(X\_1(Y\_1), X\_2(Y\_1)\dots X\_1(Y\_2), X\_2(Y\_2)\dots X\_1(Y\_n), X\_2(Y\_n)\dots) &= \\ F(X\_1(D\_1), X\_2(D\_1)\dots X\_1(D\_2), X\_2(D\_2)\dots X\_1(D\_n), X\_2(D\_n)\dots) \end{aligned}$$

#### *3.3.3.2 Assembly reliability modeling based on the MUs*

A large number of attempts had been made in the assembly reliability modeling of MUs, and their respective modeling methodology by the modular fault tree proposed by Li et al. [27].

The FTA is accomplished on the target product first, and decomposes the fault tree into the layer of MUs, then performs the analysis and calculation by regarding *Reliability Technology Based on Meta-Action for CNC Machine Tool DOI: http://dx.doi.org/10.5772/intechopen.85163*

**Figure 19.**

does not need to be measured. *Cgh*ð Þ*k* is the output matrix and *Tgh*ð Þ*k* is the geomet-

The state space model of assembly error propagation is shown in **Figure 18**. The definition of the motion assembly units' final output error is the geometric error *Tgh*ð Þ*i* measured after finishing the assembly of the final assembly unit *i*, i.e.,

� � <sup>¼</sup> *Tgh*ð Þ*<sup>i</sup>* . Therefore, the state space models of assembly error propagating from motion assembly units to function assembly units, from function assembly units to the whole machine assembly are deduced for the same reason, shown in

> *Xef*ð Þ¼ *k Aef*ð Þ*k Xef*ð Þþ *k* � 1 *Bef*ð Þ*k μef*ð Þþ *k vef*ð Þ*k Tef*ð Þ¼ *k Cef*ð Þ*k Xef*ð Þþ *k ξef*ð Þ*k*

*Xz*ð Þ¼ *k Az*ð Þ*k Xz*ð Þþ *k* � 1 *Bz*ð Þ*k μz*ð Þþ *k vz*ð Þ*k Tz*ð Þ¼ *k Cz*ð Þ*k Xz*ð Þþ *k ξz*ð Þ*k*

assembly unit *e*, referred as synthesis error of function assembly unit *e*, is obtained by introducing the error of assembly units into the state space model layer-by-layer

A large number of attempts had been made in the assembly reliability modeling

The FTA is accomplished on the target product first, and decomposes the fault tree into the layer of MUs, then performs the analysis and calculation by regarding

� � <sup>¼</sup> *F X*ð <sup>1</sup>ð Þ *Y*<sup>1</sup> *;X*2ð Þ *Y*<sup>1</sup> …*X*1ð Þ *Y*<sup>2</sup> *;X*2ð Þ *Y*<sup>2</sup> …*X*1ð Þ *Yn ;X*2ð Þ *Yn* …Þ ¼ *F X*ð Þ <sup>1</sup>ð Þ *D*<sup>1</sup> *;X*2ð Þ *D*<sup>1</sup> …*X*1ð Þ *D*<sup>2</sup> *;X*2ð Þ *D*<sup>2</sup> …*X*1ð Þ *Dn ;X*2ð Þ *Dn* …

of MUs, and their respective modeling methodology by the modular fault tree

� � of the function

The geometric error *X G*ð Þ¼ *<sup>e</sup> X*1ð Þ *Ge ;X*2ð Þ *Ge ;* …*;Xf*ð Þ *Ge*

*E G*ð Þ¼ *<sup>e</sup> F X*1ð Þ *Ge ;X*2ð Þ *Ge ;* …*;Xf*ð Þ *Ge*

*3.3.3.2 Assembly reliability modeling based on the MUs*

(5)

(6)

ric error obtained by measuring.

*The state space model of assembly error propagation.*

*Reliability and Maintenance - An Overview of Cases*

(

�

*Xh Yg*

**Figure 18.**

Eqs. (5) and (6).

is shown below:

proposed by Li et al. [27].

**64**

*Modularization fault tree model based on the function decomposition.*

the meta-action fault tree after function decomposition as separate independent modules (**Figure 19**).

The function implementation is the key performance of the assembly quality; the performances of the assembly units are characterized by using the quadruples based on the modularization fault model, as *F* ¼ ð Þ *S; P; T; Q* , where:


On the basis of the modularization fault tree, the assembly reliability modularization fault tree is simplified by sorting basic events. Then, the fault tree is transformed into a binary decision diagram (BDD) by using ITE structural analysis methodology. Finally, transforming the meta-action sub-fault tree into a BDD, researching the assembly reliability of meta-action assembly units by combining the BDD with a mapping function, and obtaining the mapping function *Qa* of metaaction assembly reliability are shown in Eq. (7).

$$Q\_a: \mathcal{M} \times \mathcal{F} \tag{7}$$

where *M* means the mapping matrix of different performance attributes' weight for meta-actions and *F* indicates performance index evaluation results of the MUs' reliability.

#### *3.3.4 Other reliability application based on meta-action*

In addition to the application of reliability modeling, reliability design, and assembly reliability analysis for CNC machine tool based on MU, FMA methodology has also been used in failure classification [24], system motion reliability analysis [28], maintenance decision [29], to name but a few.

reliability based on meta-action method can be built, which would promote the reliability level of CNC machine tool holistically. This basically includes the follow-

*Reliability Technology Based on Meta-Action for CNC Machine Tool*

*DOI: http://dx.doi.org/10.5772/intechopen.85163*

• reliability design technology from bottom to top by regarding the meta-actions as the smallest units, since the meta-actions are decomposed from functions;

• fault mode classification by meta-action, because the fault modes of meta-

• fault mechanism study by meta-action, as the FMA has the function of

action units are relatively fixed and have certain regularity; and

simplifying the CNC machine tools.

ing three aspects:

**Author details**

**67**

Chongqing University, China

Yan Ran\*, Wei Zhang, Zongyi Mu and Genbao Zhang

\*Address all correspondence to: ranyan@cqu.edu.cn

provided the original work is properly cited.

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,
