## 2 Methodologies and Techniques of the Research Work
### Error Propagation Modelling through SoV Model  
The SoV model is able to estimate the error propagation throughout a MMS by modelling a series of kinematic relationships between fixtures, datum surfaces used for locating parts within the fixtures, and the deviation of the manufacturing action itself, such as welding, machining, etc. The mathematical formulation is based on the well-known state space model representation commonly used in the field of control systems [17]. For a MAS, the SoV model can be mathematically expressed as [18]:  
\[\mathbf{X}_{k}=\mathbf{A}_{k}\cdot\mathbf{X}_{k-1}+\mathbf{B}_{k}\cdot\mathbf{U}_{k}+\mathbf{W}_{k}, \tag{1}\]  
where \(\mathbf{X}_{k}\) is the vector that represents the geometric quality deviations of each part of the assembly after the manufacturing stage, \(k\); \(\mathbf{X}_{k-1}\) represents the same deviations but after stage k\(-\)1, which becomes the input of stage k; \(\mathbf{U}_{k}\) represents the deviations of the fixture locators that hold the parts during the manufacturing operation; matrices \(\mathbf{A}_{k}\) and \(\mathbf{B}_{k}\) are obtained through kinematic analyses and represent the reorientation effect of the assembled parts after stage \(k\)\(-\)1 when placed on stage \(k\) and the effects of the deviations of the locators over the resulting quality deviations at stage \(k\), respectively; \(\mathbf{W}_{k}\) represents the modelling errors since matrices \(\mathbf{A}_{k}\) and \(\mathbf{B}_{k}\) are obtained through kinematic analyses subjected to a certain linearization degree. The geometric quality deviations of each part represented in \(\mathbf{X}_{k}\) is defined by the deviation of the local coordinate system (LCS) of the part, since the parts are assumed to be rigid. Thus, the deviation of the LCS is modelled by a differential motion vector (DMV) in the form \([\mathbf{d},\theta]^{\mathrm{T}}\), where \(\mathbf{d}\) and \(\theta\) are the position and orientation deviation of the LCS, respectively.  
In this multistage process, an inspection stage may be placed along the process for quality control. Assuming an inspection stage after the assembly stage k, the inspection results are described as:  
\[\mathbf{Y}_{k}=\mathbf{C}_{k}\cdot\mathbf{X}_{k}+\mathbf{V}_{k}, \tag{2}\]  
where \(\mathbf{Y}_{k}\) is the vector that represents the deviation of the features inspected after stage \(k\) and, therefore, it is a function of the current deviation of the resulting assembly (\(\mathbf{X}_{k}\)) through the matrix \(\mathbf{C}_{k}\) plus the measurement error represented by the vector \(\mathbf{V}_{k}\). Both Equations (1) and (2) define the error propagation in MAS of N stages, where \(k\) = 1,..., N. An example of MAS and the effect of error propagation is shown in Figure 1, where fixturing is based on 4-way and 2-way locators. As it can be seen, a locator error at stage \(k\) produces an error in the resulting subassembly after stage \(k\). Thus, this subassembly enters stage \(k\)\(+\)1 and, due to relocation error, and even if no fixture or welding error is added, the resulting subassembly after stage \(k\)\(+\)1 presents a geometric error. Later, at the inspection stage, the assembly is inspected and a quality error is identified. Readers interested in a detailed description of SoV model derivation, please refer to references [3; 4; 5; 18].