## 3 Statistical Tolerance Analysis of 3D Printed Non-Assembly Mechanisms in Motion
### Determination of Geometric Part Deviations Using Empirical Predictive Models  
A key aspect for tolerancing in AM is the determination of geometric part deviations resulting from the process. As there is currently no FDM-specific guideline for tolerancing as part deviations are strongly machine-specific, tolerance values have to be determined through a trial-and-error process, which is time- and cost-intensive [31]. In order to avoid this iterative process during the design stage for every new nominal design and set of process parameters, empirically built-up predictive models for considering the influence of process parameters and design features on geometric part deviations and thus determining resulting tolerances are purposeful. Predictive models or meta-models are therefore useful methods to evaluate an approximation of high-fidelity models (e.g., the FDM process). Physics-based equations to determine the influence of process parameters on part deviations can therefore be substituted [26]. These models automatically learn the relationship between input features and output objectives based on a previously gathered training data set [36]. Despite these advantages, there are some challenges that have to be faced when building up a predictive model. Gathering training data is a tedious task and so the training data sets are usually small, whereby the quality of the model can thus be impaired. Another cause of poor performance of the model is the overfitting and underfitting issue [23]. Overfitting means that the algorithm tries to fit every data point in the training set, whereby the model is highly sensitive to noises or outliners. A multitude of input parameters can cause the model to overfit and impair the quality of the model [23]. Consequently, the input parameters have to be reduced to the most significant ones. Preliminary experiments are thus purposeful to determine these parameters to ensure a sufficient model productivity [26]. To identify these most significant parameters in this contribution, preliminary studies were conducted, whereby test joint specimens printed via FDM are optically measured for examining geometric deviations. Subsequently, the results are evaluated in the context of a literature review concerning their influence on the geometric accuracy of FDM printed parts. In further experiments, the process parameters were varied for gathering the training data for the predictive models. The procedure for setting up empirical predictive models is schematically shown in Figure 4.  
The experimentally gathered data set is utilised for training the predictive models. ML therefore offers various algorithms, whose suitability depends upon the amount of available data and their quality. Regression models therefore help to describe basic relationships, whereas neural networks are capable of recognizing complex relationships within the data sets [23; 26]. A requirement for good prediction quality, however, is a suitable evaluation of the model. For the evaluation and thus the selection of the most suitable model, the Root  
Figure 4: Procedure for gathering a suitable training data set to predict geometric deviations of 3D-printed parts. Mean Square Error (RMSE) in combination with the Coefficient of Prognosis (CoP) can be applied. The value for the CoP is calculated as follows [37]:  
\[CoP=1-\frac{SS_{E}^{Prediction}}{SS_{T}}. \tag{3}\]  
\(SS_{E}^{Prediction}\) is thereby defined as the sum of squared prediction errors, which are estimated based on cross validation, whereas \(SS_{T}\) is defined as equivalent to the total variation [37]. In this paper, empirical predictive models are used for an early determination of resulting joint clearances and tolerances of non-assembly mechanisms manufactured via FDM based on chosen process parameters and predefined nominal part characteristics.