Fluctuation evaluation and identification model for small-batch multistage machining processes of complex aircraft parts

Abstract

The key to improve the machining quality of workpiece is to decrease the process fluctuation, which requires identifying the fluctuation sources first. For small-batch multistage machining processes of complex aircraft parts, how to identify the fluctuation sources efficiently has become a difficult issue due to the limited shop-floor data and the complicated interactive effects among different stages. Aiming at this issue, a fluctuation evaluation and identification model for small-batch multistage machining process is proposed based on the sensitivity analysis theory. In order to improve the data utilization, an analytical structure of the fluctuation evaluation and identification model for small-batch multistage machining process is presented, which comprises four levels, namely, part level, multistage level, single-stage level and quality feature level. Corresponding to the four levels in the analytical structure, four fluctuation analysis indices are proposed to quantitatively evaluate the fluctuation level of different parts and identify the weak stages and elements that result in the abnormal fluctuation in the process flow. A five-stage deep-hole machining process of aircraft landing gear is used as a case to verify the proposed model.

Introduction

Efficient and effective quality control and monitoring of multistage processes remains a challenge due to the increasing complexity, ¹ especially for small-batch multistage machining processes (SMMPs). SMMP usually refers to the machining process flow of small-batch parts which are of high-value, difficult-to-cut and have strict quality requirement. The aircraft parts are the typical small-batch parts and are usually manufactured in the SMMP pattern. Their blanks need large amount of material removal and are extremely difficult to be processed, resulting in poor machining conditions and low processing efficiency. Thus, SMMP of aircraft parts is difficult to achieve a steady state, and its quality control is more complex.

Actually, in a multistage process, the quality outputs of the downstream stage are comprehensively influenced by the service state of the machining equipment for this stage and the machining quality of the upstream stage.^2–5 The satisfaction for the final quality requirements of a complex high-value part requires a series of stages, each of which needs different machining elements (MEs) to satisfy the demanded quality target. The interactions between multistage processes and multiple MEs will lead to more complex machining error propagation relationships. Therefore, the key to ensure the dynamic stability of machining process is to reduce and control the process fluctuation of SMMP.

Conventionally, statistical process control (SPC) is adopted to monitor the process fluctuation and identify the fluctuation sources in single-stage manufacturing process. However, large amounts of sample data are needed to ensure the credibility of SPC. Obviously, as there are few samples available in small-batch production and inadequate quality information of the parts in an SMMP, traditional SPC methods based on normal distribution cannot accurately identify the decisive error sources, and thus is unable to evaluate the process fluctuation level and identify the fluctuation sources. In other words, the process fluctuation of the small-batch parts is difficult to be quantitatively analyzed through the measuring index based on statistical features.

Considering the quality and machining state information in the machining process, the sensitivity analysis is introduced to quantitatively analyze the process flow fluctuation in SMMP and identify the weak stage and the potential influential factor in the machining system, and correspondingly a fluctuation evaluation and identification model for SMMP (SMMP-FEIM) is proposed. Based on that a sensitivity analysis model is built for multistage processes. Furthermore, due to a wide variety of error sources in SMMP and the cumulative effects of error propagation among the stages, this article proposes a four-level analytical structure, including quality feature (QF) level, single-stage level, multistage level and part level, to measure the fluctuation of multistage processes from multiple perspectives.

The rest of this article is arranged as follows. A brief review about process fluctuation analysis and error source identification is addressed in section “Literature review.” The logic framework and the analytical structure of the SMMP-FEIM are put forward in section “Logic framework of SMMP-FEIM.” In section “Sensitivity analysis mode,” the sensitivity analysis model is established. In section “Process fluctuation evaluation for multiple parts,” the synthesized fluctuation index (SFI) of different parts is calculated based on the result of the sensitivity analysis model. Section “A demonstrative case” provides a case study to demonstrate the availability of the above proposed models. Finally, the discussions and conclusions of this article are presented in sections “Discussions” and “Conclusion.”

Literature review

A large number of researches have been done on the fluctuation analysis and the error source identification for the multistage machining processes (MMPs). The existing methods for identifying error source can be grouped into two kinds: pattern matching methods and statistical estimation-based methods.^1,6

For statistical estimation-based methods, the variances contributed by the process faults were estimated in the physical coupling relationship model, and the coefficient relationship matrix was established by statistical method. ⁷ Apley et al. ⁸ used a least square method to estimate the faults and its variance. Zhou et al. ⁹ used a maximum likelihood estimator and also provided confidence intervals for the estimated variance. Jin and Zhou ¹⁰ presented a variation source identification method based on the analysis of the covariance matrix of process quality measurements. The identification procedure utilized the fact that the eigenspace of the quality measurement covariance matrix can be decomposed into a subspace due to variation sources and a subspace purely due to system noise. Based on the stream of variation (SoV) model, Zhang et al. ¹¹ proposed a sensitivity analysis method to evaluate the sensitivities of dimensional features of a cylinder head to its fixture deviations (the departures of the actual values from the nominal values). Davoodi et al. ¹² modeled multistage Poisson count processes using a first-order integer-valued autoregressive time series (INAR(1)). And the maximum likelihood method was employed to estimate the out-of-control sample along with the out-of-control stage. However, due to lack of sufficient historical data in SMMP, it is infeasible to adopt these statistical methods here to identify error fluctuation sources and the weak stages.

For most pattern matching methods, the abnormal-pattern diagnosis requires a diagnostic model to obtain the characteristics of potential errors. Meanwhile, the characteristics of errors can be extracted from plenty of historical data. And then the present error can be identified by the matching between the patterns of the error symptom and the error signature.^13–15 Ceglarek and Prakash ¹⁶ proposed an enhanced piecewise least squares (EPLS)-based approach for dimensional fault failure diagnosis of ill-conditioned multistation assembly systems. In this approach, an inverse stiffness matrix of assembly structures was used to represent predetermined fault patterns. On the basis of this, the fault patterns obtained from product measurement data are used to detect and isolate dimensional failures caused by fixturing errors. Alaeddini and Dogan ¹⁷ developed a Bayesian network for capturing the cause and effect relationship among control chart patterns, process information and possible root causes/assignable causes. The proposed method provided a real-time identification of single and multiple assignable causes of failures as well as false alarms while improving its performance by learning from mistakes. Du et al. ¹⁸ developed a novel robust root cause identification approach in machining process using hybrid learning algorithm and engineering-driven rules to identify the part variation motion patterns. In a batch manufacturing process, Miao et al. ¹⁹ proposed a multivariate exponentially weighted moving average (MEWMA) chart to identify abnormal error source patterns and detect small deviations. Aimed at the impact of large-scale assembly deformation on dimensional and shape accuracy, Zhang et al. ²⁰ provided an effective pattern mapping method between bulk stress fluctuation and overall component or assembly deformation. In other manufacturing field, Wee et al. ²¹ proposed a root cause analysis method using Bayesian belief network (BBN) and fuzzy cognitive map (FCM). On the basis of BBN-based causal knowledge, the FCM provided a numerical interface to intuitively represent the causal strength in BBN. However, exact modeling for the error propagation relationships is a critical foundation for the description of the interactions between process errors and the final quality of workpiece in MMP. And thus these linear error propagation description models cannot capture all the complex relationships between these dynamic error sources and the quality directly.

To conclude, previous researches have focused mainly on the statistical method on the basis of sufficient sample data or the linear error propagation description models. Besides, there are few contributions on the identification of the process fluctuation and the error sources in SMMP with little quality information. Liu and Jin ²² proposed a new Bayesian networks modeling approach under the condition of small data sets. Zhou and Jiang ²³ proposed a systematic variation source identification method for deep-hole boring process based on multi-source information fusion using Dempster–Shafer (D-S) evidence theory. Loganathan et al. ²⁴ presented a function event digraph model for functional cause analysis of complex manufacturing system by considering its input and output functions and their interrelations. However, it still seems to be a significant challenge to realize the process fluctuation analysis and the identification of fluctuation sources in small-batch production. Therefore, this article establishes a systemic SMMP-FEIM based on the sensitivity analysis method, which can be used to evaluate the process fluctuation level and to identify the weak stages and elements of the process flow in the SMMP for multiple workpiece.

Logic framework of SMMP-FEIM

In order to identify the weak stages and elements in SMMP and further ensure quality consistency, an SMMP-FEIM is established. The SMMP-FEIM focuses on the sensitivity analysis of different elements of a single part, as well as the process fluctuation evaluation for the final quality and its evolution process flow of multiple parts.

Figure 1 shows the logic flow of SMMP-FEIM, including four steps. The first step is to build an error propagation equation linking the quality with error sources on the basis of the mapping relations of machining error propagation network (MEPN). This equation can describe the error propagation relationships in an SMMP based on measurement data. Based on this equation, Step 2 proposes two sub-sensitivity indices to evaluate the effects of the running state of MEs and upstream errors to downstream quality. After that, in Step 3, a sensitivity analysis model with four levels is built based on different partitions for machining stages and elements in manufacturing process. Finally, an SFI is proposed in Step 4 to evaluate the fluctuation level of different parts. These four steps will be investigated in detail in next section.

Figure 1.

The logic flow of SMMP-FEIM.

An analytical structure of the SMMP-FEIM is established to further evaluate the process fluctuation of multiple parts by using the sensitivity indices. According to partition granularity of the MMP, the structure can be divided into four levels: QF level, single-stage level, multistage level and part level, as shown in Figure 2.

1. QF level. QF is the key attribute that can quantitatively evaluate the machining process. By analyzing the effect of the running state of MEs and upstream errors on the quality of downstream QFs, the comprehensive sensitivity of the QFs can be obtained.

2. Single-stage level. In a machining process, machining form feature (MFF) is the key entity to connect the QFs with the machining process. We assume that there is only one MFF that can be machined in a single stage. Thus, the single-stage level describes the sensitivity analysis from the perspective of the MFF. And it is a kind of comprehensive sensitivity analysis, considering multiple QFs attached to the MFF.

3. Multistage level. According to the sensitivity analysis of the single-stage level, the analysis of multistage level can be implemented by comparing the sensitivity changes in different stages that an MFF has experienced.

4. Part level. By comprehensively analyzing the sensitivity fluctuation of the final QFs of all MFFs as well as the QFs’ evolution process flow, this level horizontally compares the sensitivity fluctuations caused by different machining conditions of multiple parts.

Figure 2.

The analytical structure of the SMMP-FEIM.

Sensitivity analysis model

This section proposes an error propagation equation for describing the coupling relationships among nodes in the MEPN. Based on that two sub-sensitivity indices are discussed in detail. Corresponding to the four levels of MMPs in Figure 2, four sensitivity measuring indices are proposed to evaluate the process fluctuation.

MEPN

As the base of following error propagation equation, an MEPN^25–27 is constructed by abstracting MFFs, MEs and their relationships in different stages. The MEPN qualitatively describes the horizontal error propagation among the MFFs and the vertical error propagation from the MEs to the MFFs. In other words, the MFF is influenced by MFFs from upstream procedures as well as MEs in the current stage. As shown in Figure 3, the quality of deep-hole B5 of stage 28 is affected not only by the deep-hole B4 of stage 27 but also the left end face A1 of stage 5 and the excircle C2 of stage 12.

Figure 3.

The MEPN for describing the MMP of deep-hole B5.

In order to make a quantitative description for the error propagation relationship in an SMMP, the running state of ME (namely, state element (SE)) and the QF of MFF are proposed to quantitatively describe the MEs and the MFFs in different stages. A QF is meditated to describe the precision information of dimension, form, position and surface roughness in each procedure. In the extended MEPN, a QF node cannot exist individually and must be attached to one or more MFF nodes with undirected edges. Similarly, an SE is used for storing running state information of MEs for machining MFFs. And the SE node is attached to the ME nodes with undirected edges. Several typical QF items and their demonstrative instances are shown in Table 1.

Table 1.

Typical QF items and instances.

QF item	Demonstrative instances	Formulations
Form dimension
Circular degree
Position dimension
Coaxiality

QF: quality feature.

Error propagation equation

To formulate the error propagation relationships of MEPN, an error propagation equation could be preliminarily simplified as $x_{\mathrm{ij}}=f(x_{i-1},u_i)$ . This equation depicts the horizontal error propagation relationships among the MFFs, as well as the vertical error propagation relationships between the MEs and the MFFs. Table 2 offers an overview of the notations in the error propagation equation.

Table 2.

Notations in the error propagation equation.

Notation	Definition
$x_{\mathrm{ij}}$	The QF value of the jth QF after machining in stage i
$x_{i-1}=\{x_{(i-1)1},\dots ,x_{(i-1)r},\dots ,x_{(i-1)R}\}$	The quality set of QFs affecting xij after machining in stage ( $i-1$ )
$u_i=\{u_{i1},\dots ,u_{\mathrm{is}},\dots ,u_{\mathrm{iS}}\}$	The running state set of MEs in stage i
$\Delta x_{\mathrm{ij}}$	The departure of the actual QF value $x_{\mathrm{ij}}$ from the ideal QF value $x_{\mathrm{ij}}^{*}$ in stage i
$\Delta x_{(i-1)r}$	The departure of the actual QF value $x_{(i-1)s}$ from the ideal QF value $x_{(i-1)s}^{*}$ in stage ( $i-1$ )
$\Delta u_{\mathrm{is}}$	The departure of the actual SE value $u_{\mathrm{ir}}^k$ from the ideal SE value $u_{\mathrm{ir}}^{*}$ in stage i
$x_{\mathrm{ij}}^k$	The QF value of the jth QF in stage i for workpiece k
$x_{(i-1)r}^k$	The QF value of the rth QF in stage ( $i-1$ ) for workpiece k
$u_{\mathrm{is}}^k$	The value of the sth SE in stage i for workpiece k
$x_{\mathrm{ij}}^{\mathrm{tol}}$	The allowed tolerance of the jth QF in stage i
$x_{(i-1)r}^{\mathrm{tol}}$	The allowed tolerance of the rth QF in stage ( $i-1$ )
$u_{\mathrm{is}}^{\mathrm{tol}}$	The allowed tolerance of sth SE in stage i

QF: quality feature; ME: machining element; SE: state element.

Assume that the MEs are under complete control in an ideal point and the QFs’ values in stage ( $i-1$ ) ( $x_{i-1}$ ) are their nominal values. On the basis of this, the QF in stage i is assumed to be in a full control state, and the ideal QF value in stage i ( $x_{\mathrm{ij}}$ ) is its nominal value. Now, let us consider the second-order Taylor expansion of the error propagation equation around the given ideal point $(x_{i-1}^{*},u_i^{*})$

$$\begin{array}{l}{x}_{ij}=f(x_{i-1},u_i)\cong f(x_{i-1}^{*},u_i^{*})+\nabla f(x_{i-1}^{*},u_i^{*})\\ {(x_{i-1}-x_{i-1}^{*},u_i-u_i^{*})}^T+\frac{1}{2}\nabla (x_{i-1}-x_{i-1}^{*},u_i-u_i^{*})\\ H(x_{i-1}^{*},u_i^{*}){(x_{i-1}-x_{i-1}^{*},u_i-u_i^{*})}^T\end{array}$$

(1)

where

$$\begin{array}{l}\nabla f(x_{i-1}^{*},u_i^{*})=\left[\begin{array}{cc}\frac{\partial f(x_{i-1}^{*},u_i^{*})}{\partial x_{i-1}}& \frac{\partial f(x_{i-1}^{*},u_i^{*})}{\partial u_i}\end{array}\right],\\ H(x_{i-1}^{*},u_i^{*})=\left[\begin{array}{cc}\frac{{\partial }^{2}f(x_{i-1}^{*},u_i^{*})}{\partial x_{i-1}^2}& \frac{{\partial }^{2}f(x_{i-1}^{*},u_i^{*})}{\partial x_{i-1}\partial u_i}\\ \frac{{\partial }^{2}f(x_{i-1}^{*},u_i^{*})}{\partial u_i\partial x_{i-1}}& \frac{{\partial }^{2}f(x_{i-1}^{*},u_i^{*})}{\partial u_i^2}\end{array}\right]\end{array}$$

To facilitate the description, we henceforth write as

$$\begin{array}{l}{\alpha }_{i-1}=\frac{\partial f(x_{i-1}^{*},u_i^{*})}{\partial x_{i-1}},{\beta }_i=\frac{\partial f(x_{i-1}^{*},u_i^{*})}{\partial u_i},\\ {{\alpha }^{\prime }}_{i-1}=\frac{{\partial }^{2}f(x_{i-1}^{*},u_i^{*})}{\partial x_{i-1}^2},{{\beta }^{\prime }}_i=\frac{{\partial }^{2}f(x_{i-1}^{*},u_i^{*})}{\partial u_i^2},\\ {\gamma }_i=\frac{{\partial }^{2}f(x_{i-1}^{*},u_i^{*})}{\partial x_{i-1}\partial u_i}\end{array}$$

Note that these parameters, which determine the error propagation relationships, are constant if the ideal work condition keeps unchanged. Thus, we can deduce the matrix representation of error propagation equations of different workpieces as shown in equation (2). And by solving this linear equation, we transfer error propagation relationship from the nonlinear problem to a linear one. With respect to the parameters’ (including ${\alpha }_{i-1}$ , ${\beta }_i$ , $\alpha {\prime }_{i-1}$ , $\beta {\prime }_i$ and ${\gamma }_i$ ) estimation of equation (2), we obtain it from the calculation of the historical quality data and machining condition data based on genetic algorithm ²⁸ and least squared method

$$\begin{array}{l}\left[\begin{array}{ccccc}\Delta x_{i-1}^1& \Delta u_i^1& {(\Delta x_{i-1}^1)}^2/2& {(\Delta u_i^1)}^2/2& \Delta x_{i-1}^1\Delta u_i^1\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ \Delta x_{i-1}^k& \Delta u_i^k& {(\Delta x_{i-1}^k)}^2/2& {(\Delta u_i^k)}^2/2& \Delta x_{i-1}^k\Delta u_i^k\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ \Delta x_{i-1}^n& \Delta u_i^n& {(\Delta x_{i-1}^n)}^2/2& {(\Delta u_i^n)}^2/2& \Delta x_{i-1}^n\Delta u_i^n\end{array}\right]\\ \left[\begin{array}{c}{\alpha }_{i-1}\\ {\beta }_i\\ {{\alpha }^{\prime }}_{i-1}\\ {{\beta }^{\prime }}_i\\ {\gamma }_i\end{array}\right]=\left[\begin{array}{c}\Delta x_{ij}^1\\ ⋮\\ \Delta x_{ij}^k\\ ⋮\\ \Delta x_{ij}^n\end{array}\right]\end{array}$$

(2)

Furthermore, the partial derivative of error propagation equation with respect to error sources can be used to calculate the two sub-sensitivity indices proposed in the next section. In order to descript the sub-sensitivity, a polynomial form of the error propagation equation is also given as equation (3)

$$\begin{array}{l}\Delta x_{ij}={\displaystyle \sum }_r{\alpha }_{(i-1)r}\Delta x_{(i-1)r}+{\displaystyle \sum }_s{\beta }_{is}\Delta u_{is}\\ +\frac{1}{2}\Delta {\displaystyle \sum }_r{\displaystyle \sum }_s{{\alpha }^{\prime }}_{(i-1)rs}\Delta x_{(i-1)r}\Delta x_{(i-1)s}\\ +\frac{1}{2}\Delta {\displaystyle \sum }_r{\displaystyle \sum }_s{{\beta }^{\prime }}_{i,rs}\Delta u_{ir}\Delta u_{is}\\ +{\displaystyle \sum }_r{\displaystyle \sum }_s{\gamma }_{(i-1)r,is}\Delta x_{(i-1)r}\Delta u_{is}\end{array}$$

(3)

where ${\alpha }_{(i-1)r}=(\partial f(x_{i-1}^{*},u_i^{*}))/(\partial x_{(i-1)r})$ , ${\beta }_{\mathrm{is}}=(\partial f(x_{i-1}^{*},u_i^{*}))/(\partial u_{\mathrm{is}})$ , $\alpha {\prime }_{(i-1)\mathrm{rs}}=({\partial }^{2}f(x_{i-1}^{*},u_i^{*}))/(\partial x_{(i-1)r}\partial x_{(i-1)s})$ , $\beta {\prime }_{i,\mathrm{rs}}=({\partial }^{2}f(x_{i-1}^{*},u_i^{*}))/(\partial u_{\mathrm{ir}}\partial u_{\mathrm{is}})$ and ${\gamma }_{(i-1)r,\mathrm{is}}=({\partial }^{2}f(x_{i-1}^{*},u_i^{*}))/(\partial x_{(i-1)r}\partial u_{\mathrm{is}})$ .

The sub-sensitivity measuring index

Based on the error propagation equation, two sub-sensitivity indices, namely, quality feature sensitivity (QFS) and state element sensitivity (SES), are proposed to analyze the vertical and horizontal influencing relationships.

QFS

The achievement of the final quality specification of complex aircraft part needs many machining processes. Take the deep-hole machining of outer cylinder part for example, many different processes need to be completed to achieve its final quality, such as drilling, boring and grinding. Therefore, considering the evolution process of MFF, the QFS is defined to describe the effects of the upstream QF error of an MFF to its downstream QF quality.

Definition 1

QFS is used to evaluate the sensitivity of the QF in the current stage with respect to the upstream QF errors, which is a fundamental sensitivity analysis index. Based on equation (3), the rth QFS of the jth QF in stage i ( $S{q}_{\mathrm{ijr}}$ ) can be described as follows

$$S{q}_{\mathrm{ijr}}=\frac{\partial x_{\mathrm{ij}}/x_{\mathrm{ij}}^{\mathrm{tol}}}{\partial x_{(i-1)r}/x_{(i-1)r}^{\mathrm{tol}}}=\frac{\partial x_{\mathrm{ij}}}{\partial x_{(i-1)r}}·\frac{x_{(i-1)r}^{\mathrm{tol}}}{x_{\mathrm{ij}}^{\mathrm{tol}}}$$

(4)

where $((\partial x_{\mathrm{ij}})/(\partial x_{(i-1)r}))={\alpha }_{(i-1)r}+\sum _s{\alpha \prime }_{(i-1)\mathrm{rs}}\Delta x_{(i-1)s}+\sum _s{\gamma }_{(i-1)r,\mathrm{is}}\Delta u_{\mathrm{is}}$ denotes the partial derivatives of the QF $x_{\mathrm{ij}}$ with respect to upstream QF $x_{(i-1)r}$ .

SES

The MEs are the direct error sources in manufacturing systems. And the service state of MEs directly determines the working accuracy. Therefore, SE is defined as the description of the service state and the manufacturing capability of ME. Aiming at the dynamic and kinematic error sources, this article focuses on the effects of machine tool, cutting tool and fixture on the machining quality. Considering the effect of SE on quality and the measurability of SE, Table 3 describes different SEs of the ME in detail.

Table 3.

Common SEs related to machine tool and cutting tool.

ME	SE	Description
Machine tool	Kinematic error	The homogeneous coordinate transformation principle is used to analyze the transformation of the built-in signal of feed axis in CNC machine tool.
	Thermal deformation error	Consider the impacts of temperature changes in the spindle system on the machining accuracy.
	Vibration of machine tool	When machining, the strong flutter of the machine tool makes its operation condition in an unstable region, leading to the surface-quality deterioration and the geometric errors of the workpiece.
Cutting tool	Cutting tool wear	Motor power of cutting tool is used to indirectly describe the cutting tool wear.
	Thermal distortion	Thermal extension of cutting cool is caused by its thermal deformation.
	Forced deformation	Analyze the deformation of cutting tool caused by the cutting force.
Fixture	Run-out error	For turning and boring of the landing gear part, the radial circular run-out error of fixture is the main source

ME: machining element; SE: state element; CNC: computer numeric control.

Definition 2

SES is used to evaluate the sensitivity of the QF in the current stage with respect to the running state of MEs, which is a fundamental sensitivity analysis index. Based on equation (3), the sth SES of the jth QF in stage i ( $S{u}_{\mathrm{ijs}}$ ) can be described as follows

$$S{u}_{\mathrm{ijs}}=\frac{\partial x_{\mathrm{ij}}/x_{\mathrm{ij}}^{\mathrm{tol}}}{\partial u_{\mathrm{is}}/u_{\mathrm{is}}^{\mathrm{tol}}}=\frac{\partial x_{\mathrm{ij}}}{\partial u_{\mathrm{is}}}·\frac{u_{\mathrm{is}}^{\mathrm{tol}}}{x_{\mathrm{ij}}^{\mathrm{tol}}}$$

(5)

where $((\partial x_{\mathrm{ij}})/(\partial u_{\mathrm{is}}))={\beta }_{\mathrm{is}}+\sum _r{\beta \prime }_{i,\mathrm{rs}}\Delta u_{\mathrm{ir}}+\sum _r{\gamma }_{(i-1)r,\mathrm{is}}\Delta x_{(i-1)r}$ denotes the partial derivatives of the QF $x_{\mathrm{ij}}$ with respect to SE $u_{\mathrm{is}}$ .

Metrics of process fluctuation

Corresponding to the four-level analytical structure shown in Figure 2, four sensitivity indices are extracted to quantitatively evaluate the process fluctuation level in this section. These four indices are obtained from these two aspects: the final QF value and the QF’s evolution. Besides, each aspect of analytical view contains two elements: the single QF and the multiple QFs attached to an MFF. Figure 4 shows the structure of these four sensitivity analysis indices.

Figure 4.

The analytical structure of the sensitivity analysis indices.

In order to quantize the error propagation relationships and quantitatively evaluate the process fluctuation level, the definitions of four sensitivity indices are given as follows.

Sensitivity of single quality feature

The sensitivity of single quality feature (SSQF) is defined to evaluate process fluctuation in the QF level. Actually, in manufacturing process, the final quality of the workpiece always depends on the weakest stages and elements in the machining system. Therefore, the SSQF can be indicated by the maximum QFS and SES in the current stage, as shown in equation (6). It is important to note that the SSQF is just a primary-level quantitative analysis

$$s_{\mathrm{ij}}^k=\max \left(\{{\mathrm{Sq}}_{\mathrm{ijr}}^k(r=1,2,\dots ,R),{\mathrm{Su}}_{\mathrm{ijs}}^k(s=1,2,\dots ,S)\}\right)$$

(6)

where $s_{\mathrm{ij}}^k$ stands for the sensitivity index of the jth QF in stage i for workpiece k, while ${\mathrm{Sq}}_{\mathrm{ijr}}^k$ and ${\mathrm{Su}}_{\mathrm{ijs}}^k$ represent the rth QFS and sth SES of this QF, respectively.

Sensitivity of multiple quality feature

The sensitivity of multiple quality feature (SMQF) is proposed to evaluate the process fluctuation in single-stage level by integrating all QFs attached to the MFF of this stage. And the SMQF can be denoted by the maximum SMQFs attached to the MFF in stage i, as shown in equation (7)

$$s_i^k=\underset{j\in M_i}{max}(s_{\mathrm{ij}}^k)$$

(7)

where $s_i^k$ stands for the SMQF in the critical machining stage i for workpiece k, while $M_i$ denotes the QF set in the stage.

Evolution sensitivity of single quality feature

The evolution sensitivity of single quality feature (ESSQF) is proposed to express the fluctuation of the single QF in constant evolution process, as shown in equation (8). And a fluctuation index H is introduced to quantitatively evaluate the fluctuation of the single QF in different stages

$$s_{i\in N_j}^k=H_{i\in N_j}^k·\underset{i\in N_j}{\mathbf{max}}(s_{\mathrm{ij}}^k)$$

(8)

where $s_{i\in N_j}^k$ stands for the evolution sensitivity index of QF j for workpiece k, while $H_{i\in N_j}^k$ stands for the fluctuation index of these sensitivity indices in the process flow, as shown in equation (9)

$$H_{i\in N_j}^k=-\frac{1}{\ln t_j}\sum _{i=1}^{t_j}{s}_{\mathrm{ij}}^{k\prime }\ln s_{\mathrm{ij}}^{k\prime }$$

(9)

where $s_{\mathrm{ij}}^{k\prime }$ represents the normalized SSQF of the QF j in stage i for workpiece k, $N_j$ denotes the machining procedure set that the jth QF involved and $t_j$ stands for the number of the machining procedures.

Evolution sensitivity of multiple quality feature

The evolution sensitivity of multiple quality feature (ESMQF) is defined to capture the fluctuation of the multiple QFs in constant evolution processes, as shown in equation (10). Similar to the sensitivity index in section “Evolution sensitivity of single quality feature,” a fluctuation index H is introduced to quantitatively evaluate the fluctuation of multiple QFs. It is important to note that the ESMQF is the highest fluctuation evaluation

$$s_{i\in N}^k=H_{i\in N}^k·\underset{i\in N}{max}(s_i^k)$$

(10)

where $s_{i\in N}^k$ stands for the evolution sensitivity index of multiple QFs in the critical machining stage i for workpiece k, while $H_{i\in N}^k$ stands for the fluctuation index of these process sensitivity indices for workpiece k, as shown in equation (11)

$$H_{i\in N}^k=-\frac{1}{\ln t}\sum _{i=1}^t{s}_i^{k\prime }\ln s_i^{k\prime }$$

(11)

where $s_i^{k\prime }$ represents the normalized SMQF in stage i for workpiece k, while N stands for the evolving stage set for achieving the quality of QFs in stage i and t stands for the number of the evolving stage set.

Process fluctuation evaluation for multiple parts

To further evaluate the fluctuation for multiple parts, this section proposes an information aggregation model to horizontally evaluate the process fluctuation of multiple parts. As shown in Figure 5, this model makes horizontal fluctuation analysis on different parts from both the final QF value and the QF’s evolution process flow aspect. And four sensitivity analysis vectors for multiple parts are established based on the four sensitivity indices. Then, through the aggregation analysis on four sensitivity analysis vectors of multiple parts, the SFI of the parts in a certain batch can be obtained so as to identify the weak stages and elements in the process flow and to determine which stage or element should be given the priority of improvement.

Figure 5.

The complexity of process fluctuation for multiple parts.

Sensitivity matrix

As seen in Table 4, these four sensitivity analysis vectors can reflect the effects of the machining condition on the machining quality comprehensively. Based on that a sensitivity matrix S can be obtained through the aggregation of these four sensitivity vectors, as shown in equation (12).

$$S=\{S_{\mathrm{ij}}(j\in J),S_i(i\in I),S_{i\in N_j}(j\in J),S_{i\in N}(i\in I)\}$$

(12)

where $J$ represents the key QF set, while $I$ represents the critical stage set.

Table 4.

The four sensitivity indices in the sensitivity matrix.

Concept	Notation	Definition
Vector of SSQF	$S_{\mathrm{ij}}=\{s_{\mathrm{ij}}^1,s_{\mathrm{ij}}^2,\dots ,s_{\mathrm{ij}}^k,\dots ,s_{\mathrm{ij}}^n\}$	A collection of the SSQFs $s_{\mathrm{ij}}^k$ (k = 1, …, n) of multiple parts. The vector for evaluating the fluctuation of the QF level.
Vector of SMQF	$S_i=\{s_i^1,s_i^2,\dots ,s_i^k,\dots ,s_i^n\}$	A collection of the SMQFs $s_i^k$ (k = 1, …, n) of multiple parts. The vector for evaluating the fluctuation of the single-stage level.
Vector of ESSQF	$S_{i\in N_j}=\{s_{i\in N_j}^1,s_{i\in N_j}^2,\dots ,s_{i\in N_j}^k,\dots ,s_{i\in N_j}^n\}$	A collection of the SMQFs $s_{i\in N_j}^k$ (k = 1, …, n) of multiple parts. The vector for evaluating the fluctuation of the multistage level.
Vector of ESMQF	$S_{i\in N}=\{s_{i\in N}^1,s_{i\in N}^2,\dots ,s_{i\in N}^k,\dots ,s_{i\in N}^n\}$	A collection of the SMQFs $s_{i\in N}^k$ (k = 1, …, n) of multiple parts. The vector for evaluating the fluctuation of the part level.

SSQF: sensitivity of single quality feature; SMQF: sensitivity of multiple quality feature; ESSQF: evolution sensitivity of single quality feature; ESMQF: evolution sensitivity of multiple quality feature.

Information aggregation model of multiple parts

On the basis of the analysis on the sensitivity matrix built in section “Sensitivity matrix,” this section proposes an information aggregation model to realize comprehensive fluctuation evaluation of the highly correlated and interdependent manufacturing process.

Figure 6 shows the operation procedure of the information aggregation model. The first step is to normalize the sensitivity matrix generated by equation (12) as the analysis matrix that will be used to evaluate the fluctuation of different workpieces. The second step is to utilize equations (13)–(15) to calculate the weight vector $\mathit{w}=\{w_1,w_2,\dots ,w_m\}$ ( $m$ represents the number of the sensitivity vectors) for every sensitivity vector

$$E_j=-\frac{1}{\ln n}\sum _{i=1}^n{\stackrel{·}{r}}_{\mathrm{ij}}\ln {\stackrel{·}{r}}_{\mathrm{ij}}$$

(13)

$${\stackrel{·}{r}}_{\mathrm{ij}}=\frac{r_{\mathrm{ij}}}{\sum _{i=1}^n{r}_{\mathrm{ij}}}$$

(14)

$$w_j=\frac{1-E_j}{\sum _{l=1}^m(1-E_l)}$$

(15)

where $r_{\mathrm{ij}}$ represents the sensitivity value of the jth index for ith workpiece, and n stands for the number of the workpieces. Third, two ideal coefficients ²⁹ ( $D_i^{+}$ and $D_i^{-}$ ) built by equations (16) and (17), are used to evaluate the ideal condition and the limiting condition of the process fluctuation of different parts, respectively. Finally, equation (18) is utilized to calculate the SFI for assessing the distance from the actual conditions of different parts to their ideal conditions. Comparing the SFI of different parts, the part with maximum fluctuation can be obtained

$$D_i^{+}={\left(\sum _{j=1}^m{(|{\stackrel{·}{r}}_{\mathrm{ij}}-1|·w_j)}^2\right)}^{1/2}$$

(16)

$$D_i^{-}={\left(\sum _{j=1}^m{({\stackrel{·}{r}}_{\mathrm{ij}}{w}_j)}^2\right)}^{1/2}$$

(17)

$$\mathrm{SF}{I}_i=\frac{D_i^{-}}{D_i^{+}+D_i^{-}}$$

(18)

Figure 6.

The operation procedure of the model.

A demonstrative case

The landing gear of aircraft always stands a transitory impact load when an aircraft is making a landing. Thus, the transitory load capacity of aircraft directly depends on the machining quality and the assembly quality of landing gear. To meet the load capacity requirement of the landing gear, the high-impact, high-hardness, difficult-to-cut material is commonly chosen for the outer cylinder part of the landing gear. In addition, hundreds of machining stages are needed to meet the final quality requirement of the outer cylinder part. Therefore, it is essential to analyze the process fluctuation in the SMMP of the outer cylinder part and identify the fluctuation sources in its manufacturing system. As shown in Figure 7, the MFFs of the outer cylinder part include excircle, deep-hole, end face, top groove and so on. Here, to simplify the analysis problem and highlight the critical quality fluctuation, eight major machining stages in deep-hole machining, including drilling, rough boring, fine boring and grinding, of the outer cylinder part are chosen to verify the proposed method.

Figure 7.

The 2D part drawing and machining form features of the outer cylinder part.

Constructing the extended MEPN

An extended MEPN of the eight-stage deep-hole machining of the outer cylinder part is established to describe the error propagation paths and relationships. And 168 nodes, including 18 MFF nodes, 23 ME nodes, 37 QF nodes and 90 SE nodes, are extracted in the extended MEPN as shown in Appendices 1 and 2. Figure 8 illustrates the extended MEPN of two-stage grinding process in the deep-hole machining.

Figure 8.

The extended MEPN structure graph for two-stage machining.

Solving the error propagation equation

The input data in this article are basically collected from a machining batch of 15 pieces in a landing gear manufacturing enterprises. Detailed as shown in Table 5, the process data, including kinematic error and vibration of machine tool, vibration and temperature of cutting tool and radial run-out error of fixture, were collected by various sensors and devices for evaluating running state of the MEs in the deep-hole machining. To guarantee the validity of the data, this article adopts mature methods of acquisition and processing of working condition data from the literature.^30–32Appendix 3 shows the partial quality data of QFs and machining condition data of MEs.

Table 5.

The sensor configuration solution of SE collection in the deep-hole machining.

MEs	SEs	Sensor	Brand	Type	Pictures
Machine tool	Kinematic error	Dynamical testing and analysis system	Self-developed	MT02-HDT
	Vibration of machine tool	Three-axis accelerometer	Beetech	A301/302
Cutting tool	Vibration of cutting tool	Accelerometer	PCB	ICP352C34
	Cutting tool temperature	Temperature sensor	Beetech	T801/802
Fixture	Run-out error	Eddy current displacement sensor	MICRO-EPSILON	DT3010-M

ME: machining element; SE: state element.

Take the coaxiality (QF120203) of the deep-hole feature (MFF120002) for example, the error propagation equation (it was omitted here for concise reason) can be established. And then the equations of 15 workpieces can be constructed based on equation (2). As shown in Table 6, these parameters in the error propagation equation are solved by using Genetic Algorithm and Direct Search Toolbox (GADS) in MATLAB.

Table 6.

The partial parameters in error propagation equation of QF120203.

	QF010402	QF050502	QF060502	QF120101	QF120102	QF120103	QF120104
$\alpha$	2.378	−1.5653	0.6118	−0.9276	0.7053	−0.2624	−1.8834
$\alpha \prime$	−0.0072	0.8841	−1.176	1.7549	0.357	2.4243	1.8675
	SM053901	SM053902	SC333901	SC333902	SF053901
$\beta$	−0.0245	1.775	−0.7418	−1.969	1.0158
$\beta \prime$	1.5462	0.6222	0.653	1.0804	1.411

Constructing the sensitivity matrix

On this basis, the QFS and SES can be calculated based on equations (4) and (5). According to the analytical structure of the SMMP-FEIM, the machining process fluctuation can be captured by calculating four sensitivity indices from four levels based on equations (6)–(11). In accordance with the quality requirements of deep-hole machining, 10 sensitivity analysis indices, including $s_{12,01}^k$ , $s_{12,02}^k$ , $s_{12,03}^k$ , $s_{12,04}^k$ , $s_{12}^k$ , $s_{12\in N_{01}}^k$ , $s_{12\in N_{02}}^k$ , $s_{12\in N_{03}}^k$ , $s_{12\in N_{04}}^k$ and $s_{12\in N}^k$ , are obtained. And Figure 9 shows the distribution of these 10 analysis indices based on machining data of different parts. Then, a sensitivity matrix is set up. Appendix 4 shows the sensitivity matrix of the final quality of MFF120002 of 15 workpieces.

Figure 9.

The distribution of these 10 analysis indices for 15 parts.

Calculating SFI

The information aggregation model can be built according to the four steps in Figure 6. As mentioned above, the sensitivity matrix involving 15 parts is introduced as the sensitivity matrix for the model. Based on the normalized sensitivity matrix, the index weight vector of each sensitivity vector, as shown in Table 7, can be obtained according to the calculations from equations (13)–(15). The SFI of different parts are calculated based on two ideal coefficients, as shown in Figure 10.

Table 7.

Index weight vector of process fluctuation analysis.

Index	$s_{12,01}^k$	$s_{12,02}^k$	$s_{12,03}^k$	$s_{12,04}^k$	$s_{12}^k$	$s_{12\in N_{01}}^k$	$s_{12\in N_{02}}^k$	$s_{12\in N_{03}}^k$	$s_{12\in N_{04}}^k$	$s_{12\in N}^k$
Weight $w_j$	0.1829	0.1235	0.1252	0.0946	0.1328	0.1044	0.0533	0.0963	0.048	0.039

Figure 10.

The SFI of process fluctuation analysis for different parts.

The comparison of the SFIs shows that the workpiece 07 has larger fluctuation in this batch of 15 workpieces, which need a deeper identification for the potential error sources in its machining process.

Figure 11 shows the process sensitivity distribution of evolving process for seventh workpiece. According to Figure 11, we can conclude that the finishing machining for grinding deep-hole (MFF120002) has bigger sensitivity than the rough machining stages. Moreover, by analyzing the sub-sensitivity of QF120203 as shown in Figure 12, we can see that the roundness error of two excircle feature (MFF050005/MFF060005) and the radial run-out error of fixture (SF053901) have bigger sensitivity than other nodes. The reason is that the feature of excircle MFF050005 and MFF060005 is the key locating datum of finishing grinding machining, and thus the roundness of datum and the radial run-out error of fixture determine the final quality of coaxiality between the deep-hole feature (MFF120002) and excircle feature (MFF050005/MFF060005). The results in Figures 11 and 12 show that the weakest stage (8) and element (QF50502) in the process flow should be given the priority of improvement.

Figure 11.

The process sensitivity distribution of evolving process for seventh workpiece.

Figure 12.

The QFS and SES of QF120203 for seventh workpiece.

Based on the analysis results above, for seventh workpiece, the eighth stage of the deep-hole machining has the maximum sensitivity in stage level, and the maximum sensitivity in QF level comes from the QF120203 (coaxiality). In terms of the QF120203, the sensitivities of the effects of roundness (QF050502/QF060502) error of the excircle feature (MFF050005/MFF060005) and the radial run-out error of fixture (FT05) on the QF are higher than other elements. As shown in Figure 13, the sensitivity trace path of weak stages and elements in SMMP is constructed in part level, stage level, QF level and element level, progressively. This path can clearly identify the potential error sources and the weak stages and elements that have effects on the final quality in the SMMP.

Figure 13.

The trace path chart of the sensitivity.

After analyzing the machining process, it can be found that the deep-hole feature in stage 7 and stage 8 have higher machining accuracy, which determines the performance quality of the landing gear. Through analyzing the grinding process of seventh workpiece, the kinematic error and vibration of machine tool are smaller than other workpieces, and the radial run-out error is larger than other part. In the grinding process of deep-hole, the roundness (QF050502/QF060502) error of excircle datum (MFF050005/MFF060005) and the fixture (FT05) error are the determining factors, which is consistent with the analysis of the case study.

Discussions

To illustrate the effectiveness of the proposed methods, a partial comparison is conducted as shown in Table 8. Some typical methods about error propagation modeling and process fluctuation evaluation including radial basis function artificial neural network (RBF-ANN), state-space modeling, ¹¹ and first-order Taylor expansion are chosen to be applied in the case in section “A demonstrative case.” The result of each method is also shown in Table 8.

Table 8.

Comparison between SMMP-FEIM and other similar methods.

Indices		SMMP-FEIM	First-order Taylor expansion	RBF-ANN	State-space modeling
Relationship		Nonlinear	Linear	Nonlinear	Linear
Parameters in error propagation modeling	$\alpha$ , $\beta$	{−0.0245, 1.775, −0.7418, −1.969, 1.0158, 2.378, −1.5653, 0.6118, 0.7053}	{0.064, 0.8, 0.025, 0.65, 0.31, 0.29, 0.33, 0.2, 0.53}	—	—
	$\alpha \prime$ , $\beta \prime$	{1.5462, 0.6222, 0.653, 1.0804, 1.411, −0.0072, 0.8841, −1.176, 0.357}	—	—	—
	$W^{\mathrm{input}}$	—	—	$W^{13\times 17}$	—
	$W^{\mathrm{output}}$	—	—	$W^{1\times 13}$	—
	$T$	—	—	—	${\Re }^{127\times 127}$
Fluctuation analysis level	QF level	{6.74, 4.3, 8.07, 2.08}	{0.685, 0.82, 0.80, 1.15}	—	—
	Stage level	8.07	1.15	—	{0.06, 0.06, 0.1, 0.06, 0.06, 0.07, 0.28, 0.31}
	Part level	{5.82, 4.2, 7.25, 5.99}	—	—	—
Remarks		Multi-dimensional analysis	QF-level analysisStage-level analysis	Not applicable	Stage-level analysis

SMMP-FEIM: fluctuation evaluation and identification model for small-batch multistage machining process; RBF-ANN: radial basis function artificial neural network; QF: quality feature.

Due to the introducing of the hidden layer, RBF-ANN was usually used in estimating the parameters between the input and output, which is not applicable for direct functional relations. With respect to the state-space modeling, it can be described by a regression matrix T, which represents the transfer function between inputs and outputs. Based on the regression matrix T, the error decomposition can be achieved in the station-level. However, we have not found the application of state-space modeling in upper level multi-workpieces and lower level QFs. The error propagation equation by first-order Taylor expansion can achieve the results only in QF level and stage level.

As for SMMP-FEIM proposed in this article, it systematically reflects the sensitivity of SMMP and its fluctuation level from different levels. The sensitivity trace path can intuitively show the weak stages and elements in SMMP. Actually, there are several advantages using sensitivity analysis to evaluate process fluctuation, and they are listed as follows:

• Through analysis of the influence factors of quality related, the historical data are fully used, thus the effective analysis of error propagation relationship can be captured in the case of small sample size.

• Considering the approximate precision, the error propagation equation derived by second-order Taylor expansion, the nonlinear relation between the error and error sources can be depicted.

• Through multi-dimensional analysis of four layers, the error propagation and process fluctuation rules are analyzed from the perspectives of workpiece, process and QF with limited quality data.

However, its accuracy of the parameter estimation is largely affected by the scale of sample dataset. Owing to the second-order Taylor expansion of error propagation equation, the dimension of the parameters increases exponentially. It directly leads to the complexity of the parameter estimation. In other words, the more parameters there are, the lower accuracy of parameter estimation (with limited measurement data) will be. To solve this problem here, the measurement data of each segment in deep-hole machining are treated as different samples to expand the measurement data. In a word, although the comparison is incomplete, it could imply the advantages of the proposed SMMP-FEIM to some extent.

Conclusion

An SMMP-FEIM has been proposed in this article. The model integrates error propagation MEPN with sensitivity analysis theory so as to evaluate the process fluctuation and identify the weak stages and elements in SMMP. In this model, the nonlinear error propagation relationship between error sources and part quality have been presented based on the error propagation equation derived by second-order Taylor expansion, which shows that the relative quality fluctuation can be estimated by measuring the coupling relationships. Therefore, the impact of each error source on the part quality can be estimated. Furthermore, an analytical structure of the SMMP-FEIM is proposed to multi-dimensionally evaluate the process fluctuation. It involves not only the sensitivity analysis of error sources on part quality in the SMMP of a single part but also the process fluctuation evaluation for the part quality and its evolution process flow of multiple parts. It is a comprehensive reflection of the process capability to control the influential factors of fluctuation and the process quality consistency. Future work on this model will focus on the improvement in the error propagation model, considering more measurable error sources in the nonlinear error propagation model so as to accurately describe the error propagation relationships in the machining process of complex parts.