## 4 Application
### Results of the Tolerance Analysis  
Since all necessary information for solving the vector loop equations is now available, the motion behaviour of the mechanism can be statistically analysed using _sampling techniques_ (see Section 3.3). For the integration of the predictive model, using the SVM algorithms, the objective values, viz. the tolerance value of the linkage deviation and the resulting joint clearance are used for sampling. These values are therefore assumed to be normally distributed (\(\sigma=t/6\)). Furthermore the number of samples \(N\) is set to 10,000 with which valid results could be achieved in prior studies in combination with reasonable computing time [7]. This enables the generation of \(N\) different virtual mechanisms, while the predictive models create a single discrete value [32]. For the \(N\) virtual non-assembly mechanisms, this results in 10,000 different coupling curves which represents the mobility band of the mechanism. Figure 10 illustrates the coupling curve and the mobility band of the kinematic joint \(J_{24}\) as a result of the statistical tolerance analysis using sampling technique. Thereby, the mobility band in comparison to the ideal coupling curve of the 3D-printed mechanism for different process parameter sets (cf. Table 4) can be calculated. In order to evaluate the results in this context, two parameter sets were defined according to a best case scenario (low layer height, optimal build orientation \(Z\) and seam style) in parameter set 1 and a worst case scenario (higher layer height, unfavourable build orientation \(Z\) and seam style) in parameter set 2 for 3D-printing non-assembly mechanisms [8]. In order to evaluate the functionality of the presented mechanism for the different parameter sets, the movement accuracy for the coupling curve of kinematic joint \(J_{24}\) is defined as the FKC as a function of the time \(\tau\) (see Section 3.3):  
\[FKC(\tau) =J_{24}=L_{1}\cdot e^{j\cdot\theta_{1}}+c_{12}\cdot e^{j\cdot \gamma_{12}}+e^{j\cdot\theta_{2}}(L_{2a}+L_{2b}\cdot e^{-i\cdot\theta_{2a2b}}) +c_{23}\cdot e^{j\cdot\gamma_{23}}. \tag{6}\]  
Consequently, the coupling curve of joint \(J_{24}\) and thus the maximum deviation from the nominal movement accuracy of the FKC is calculated using a 95% quantile for the two different sets in order to evaluate their influence on the defined key characteristic. \begin{table}
\begin{tabular}{l c c} \hline \hline  & **Parameter Set 1** & **Parameter Set 2** \\ \hline Layer height & 0.1778 mm & 0.2540 mm \\ Build orientation \(Z\) & 0\({}^{\circ}\) & 90\({}^{\circ}\) \\ Seam Style & random & align \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Chosen process parameters for both parameter sets. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Empirical Predictive Model** & **CoP Value** & **RMSE Value [mm]** \\ \hline SVM Joint clearance & 0.9216 & 0.0356 \\ GP Joint clearance & 0.9183 & 0.0362 \\ SVM Tolerance value & 0.8906 & 0.0666 \\ GP Tolerance value & 0.8926 & 0.0653 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Coefficient of Prognosis (CoP) and root mean squared error (RMSE) values for the different empirical predictive models. Figure 10: (**a**) Coupling curve of kinematic joint \(J_{24}\) as a result of the statistical tolerance analysis (Parameter set 2). (**b**) Section of whole coupling curve for parameter set 1. (**c**) Section of whole coupling curve for parameter set 2. The results of the statistical tolerance analysis comparing both parameter sets are illustrated in Table 5.  
The results in Table 5, in combination with Figure 10, indicate that both comparatively large joint clearances and the influence of the geometrical part deviations significantly affect the coupling curve of the regarded additively manufactured kinematic joint. The comparison of the results for the different parameter sets shows that the choice of process parameters has a greater impact on the motion accuracy of the mechanism as parameter set 1 indicates a smaller deviation and thus a smaller mobility band (see Figure 10).