An Improved Equivalent Fixture Error Model for Machining Process

Although the equivalent fixture error approach used in variation propagation can directly model the process physics regarding how datum-induced, fixture-induced and machine tool-induced errors generate the same error pattern on part features, the developed equivalent fixture error compensation technique did not consider machining-induced variations. Machining-induced variations are often caused by geometric-thermal effects, cutting force-induced variations, and/or cutting-tool wear, etc. Such machining-induced variations are an important factor that influences the part quality. Without considering machining-induced variations, the application of EFE model for error compensation will be limited. In order to overcome this limitation, this paper extends current equivalent fixture error to include machining-induced variations. This paper shows the benefits of the extended model through a case study.


Introduction
For a multistage machining process (MMPs), the part variation at certain operation is often due to two components: 1) the variation brought by current machine stage, such as fixture error and machine tool path error; and 2) the variation brought by datum feature error generated and propagated from previous stages [1]. The above errors account for the majority of machining errors in the machining process. Due to the existence of these main error sources, the features of the workpiece may deviate from their designed values to their actual values.
SPC is a very useful tool for monitoring process variation and reducing product variation [2][3][4]. Tsung et al. [5] gave a thorough review about the application of SPC techniques in MMPs, including MMPs and multistage service operations. However, SPC could not be directly used to model the relationship between the error sources and feature variations. It also cannot handle variation propagation problem in MMPs.
SoV is another useful technology for reducing product variation and can overcome the limitation of SPC. This technology can directly model how the variation in MMPs caused by datum, fixture, and machine path tool propagate and accumulate. The SoV methodology for MMPs can be described by employing the state space concept which has been extensively used in system or automatic control theories. The introduction of the state space modeling structure can successfully describe the relationship between the error sources and feature variations. A huge of literature can be found on SoV modeling. Jin et al. [6] introduced state space model to describe the dimensional variation accumulation and propagation for multistage body assembly processes. To obtain the SoV models for MMPs, lots of efforts have been done [1,[7][8][9][10][11][12]. Huang et al. [7] proposed an implicit non-linear SoV model to predict the variation accumulation and its propagation for MMPs. Djurdjanovic et al. [8] used Taylor series expansion to linearize Huang et al.'s non-linear SoV model. But the linearized model is yet implicit. Using differential motion vectors (DMVs), Zhou et al. [1] developed an explicit linear SoV model. Abellan-Nebot et al. [9] demonstrated that the absence of machining-induced variations in state space modeling of variation propagation for MMPs could be an important factor that influences the accurate in variation prediction. Using DMVs, Abellan-Nebot et al. [10] introduced such machining-induced variations including geometricthermal effects, cutting force-induced variations, and/or cutting-tool wear, etc into the SoV model. Several review papers have introduced SoV modeling methods briefly [11][12][13][14].
To reduce this variation, Wang et al. [15,16] proposed equivalent fixture error (EFE) concept. With this concept, datum error and machine tool error can be transformed to equivalent fixture locator errors at each operation and be compensated. However, only the datum errors and geometric errors of machine tool have been explicitly transformed to equivalent fixture locator variations at each operation in the reported works [15,16], leaving the machining-induced variations unaddressed. Therefore, without considering machininginduced variations, the application of current EFE model for error compensation will be limited. In order to overcome this limitation of current EFE compensation, this paper extends current EFE model to include machining-induced variations. By this extended EFE model, the machining-induced variations can be cancelled out by their corresponding EFE. Then, the machining quality can correspondingly improve. The limitation of current EFE model and the benefits of extending the EFE model through a case study will also be discussed in this paper. 1 The EFE model for datum error, fixture error and machine tool path error can be described as follows. ( 1) are the primary datum error, secondary datum error and tertiary datum error, respectively. The detailed expression of matrix K , Ψ , H and G can be found in [15].

Compensation strategy
To compensate the error sources, corresponding d , ( ) k f and m should be calculated by EFE model.
Denote the total EFE as ( ) [ ( ) In order to cancel out ( ) k t , the amount of adjustment should be applied to the corresponding fixture locators. The detailed compensation process by EFE model is shown in Fig. 3. For example, to compensate the datum error shown in in Fig. 1(b), d should be calculated. When the amount of adjustment d shown in Fig. 4 is applied to the corresponding fixture locators, d can be cancelled out. Then, the datum error can be compensated by this strategy. The same procedure can be performed to compensate the machine tool path error m u .
Ascertain input vectors , and Adjust the length of the fixture locators and start to machine

Improved Equivalent Fixture Error Model
In order to overcome the limitation of current EFE model, this paper extends current EFE model to include machining-induced variations. Abellan-Nebot et al. [10] presented a generic framework for machining-induced variation representation based on DMVs. The final machine tool path error caused by machining-induced variations is mathematically modeled as T are the matrices to describe the transition of differential motion vectors can be found in [10].
It can be seen from Eq. 4 that when C  x 0, the machining tool path error due to cutting-tool wear can be written as Similarly, the machine tool path error due to cutting force-induced variation, spindle-thermal variation and geometric-thermal variation of machine-tool axes can be obtained as Eq. 6, Eq. 7 and Eq. 8, respectively.
The extended EFE model introduces the machininginduced variations into machine tool path error. Therefore, it is feasible to calculate the EFE for machining-induced variations. Then, the limitation of current EFE model can be overcome. The benefits of extending the EFE model and the limitation of current EFE model through a case study will also be discussed in the following section.

Case Study
A workpiece from [15] will be employed to compare current EFE model and improved EFE model. As show in Fig. 5, the origin of the PCS is O p . In this case, FCS is taken the same as PCS for sake of simplicity. Only top surface EFGH is milled in this case. The workpiece is located by using the primary datum ABCD, secondary datum CDHG and tertiary datum BCGF to mill feature EFGH.
The nominal locations of the six locators expressed in 0 FCS are shown in Table. 1. The nominal location and orientation of the features in PCS are listed in Table. 2. In the machining, the input errors are intentionally added to the process. Using the above parameters listed in Table. 1 and Table. 2, corresponding EFE models can be established.    under their corresponding compensation values for the current and improved models, the improved EFE model can cancel out more error sources than current EFE model. Therefore, after compensating errors by the improved model, machining quality can be improved obviously.

Conclusions
Current EFE compensation technique did not consider the machining-induced variations. Without compensating these machining-induced variations, the EFE based error compensation methodology will be limited. In order to overcome this limitation, this paper extends current EFE model that includes the machining-induced variations. A case study shows the benefits of the extended EFE model. The improved EFE model can compensate more errors than current EFE model. Compensating more errors by the improved model means that machining quality can be improved obviously.
Its open structure provides a quantitative framework for further expansion. In the future work, the EFE procedure based on DMVs can be applied to investigate error compensation methodology for a general 3-2-1 fixture setup rather than orthogonal 3-2-1 fixture layout. In addition, the EFE model can also expand to deformed workpiece. All of these potential works using EFE model based on DMVs will be pursued and reported in the future.