# Process-Oriented Tolerance and Variation Management: Review and Classification
## 3 Classification Results and Main Approaches  
### Analysis of Process Chains  
A total of 72 publications can be assigned to the \"analysis of process chains\" class. The majority of these publications describe the modeling of variation propagation in the  
Figure 3: Classification of PTVM. process chain. This can be achieved in various ways. [PERSON] et al. identify two primary modeling approaches for variation propagation in manufacturing--the previously mentioned SoV and the Model of Manufacture Part. The SoV method is particularly suited for process-oriented tolerancing, among other applications. The Model of Manufacture Part, on the other hand, is preferable for product-oriented activities [31; 32]. [PERSON] et al. compared the two approaches in a publication [32], to which reference is made here for a detailed differentiation. The significance of the SoV approach and the state space model for this class is also demonstrated by the fact that 52 out of 72 publications employ this approach or this form of modeling. The concept of SoV was introduced by [PERSON] et al. [33] about 25 years ago [14]. The original goal of the concept was the diagnosis and prediction of dimensional variations in a multistage automotive body assembly [33]. The most common form of mathematical modeling is the state space model, which was first used in the SoV context by [PERSON] and [PERSON] [34] in 1999. The state space model was adopted from control theory. It describes the mathematical relationship between the sources of variation and the product deviations in multistage processes [12]. The kinematic relationships of the production are represented here by transformations. The matrix entries are constants defined by the product and process design. This modeling allows the simultaneous consideration of multistage production systems and different process routes [35; 36]. The following is the basic mathematical representation of the state space model by [PERSON] [35]:  
\[\mathbf{x}_{k}^{(i)}=\mathbf{A}_{k-1}^{(i)}\mathbf{x}_{k-1}^{(i)}+\mathbf{B}_{k}^{(i)}\mathbf{u}_{ k}^{(i)}+\mathbf{\xi}_{k}^{(i)}\,, \tag{1}\]  
\[\mathbf{y}_{k}^{(i)}=\mathbf{C}_{k}^{(i)}\mathbf{x}_{k}^{(i)}+\mathbf{y}_{k}^{(i)}\,, \tag{2}\]  
\(\{k\}\subset\{1,\ldots,N\}\) and \(\{i\}\subset\{1,\ldots,R\}\) according to [35];  
\(\mathbf{x}_{k}^{(i)}\)--deviation of a part characteristic;  
\(\mathbf{y}_{k}^{(i)}\)--deviation of a quality characteristic of the product;  
\(\mathbf{u}_{k}^{(i)}\)--deviations induced by the process;  
\(\mathbf{A}_{k-1}^{(i)}\)--state transition matrix;  
\(\mathbf{B}_{k}^{(i)}\)--input matrix;  
\(\mathbf{C}_{k}^{(i)}\)--observation matrix;  
\(\mathbf{\xi}_{k}^{(i)}\), \(\mathbf{\eta}_{k}^{(i)}\)--error terms that include, for example, other non-modeled errors or sensor noise;  
\(i\)--process route;  
\(k\)--process stage [35; 36]. The SoV approach, as well as the state space modeling itself, is first detailed and complemented by a series of publications around [PERSON] et al. [2; 37; 38; 39; 40; 41]. Furthermore, the approach is extended to multistage manufacturing systems by [PERSON] et al. [42] and [PERSON] et al. [43]. The approach is improved by [PERSON] et al. [44] in the context of introducing the differential motion vector. This has its origin in robotics and represents the geometric deviations [12]. Extensions by [PERSON] et al. [45], [PERSON] et al. [7; 12; 31; 46], and [PERSON] et al. [47] consider additional fixture layouts and manufacturing errors [19]. A further addition to the approach is made by [PERSON]'s et al. [48] consideration of components with a variable stiffness structure. In a series of publications [4; 17; 18; 19], [PERSON] et al. present an additional approach to modeling the variation propagation in a multistage manufacturing process utilizing dual quaternions. The approach considers deviations in fixtures in various layouts [4; 19] and manufacturing deviations [17; 18]. It is associated with the SoV, among others, and also uses the skin model shapes [19]. Furthermore, [PERSON] et al.