# An Outline of Fused Deposition Modeling: System Models and Control Strategies
## 4 Conclusions  
Fused Deposition Modeling (FDM) is an appealing process for numerous applications thanks to its versatility and promptness, which makes it possible to fabricate arbitrary geometries in one single operation. However, extrusion dynamics is a complex physical phenomenon that is not trivial to model. In this review, (i) some formal models of the process and some qualitative considerations have been presented, and (ii) the most widespread strategies adopted in the industry have been reviewed. The goal is to ease the study of FDM from the control viewpoint, which is usually less prioritized with respect to material and hardware.  
Some interesting open challenges have been highlighted, which involve the dynamic modeling of the extrusion and deposition processes. Analytical models exist, but they are focused on static relationships. Differential equations are used solely in Computational Fluid Dynamics (CFD) software, which iteratively solves momentum and energy balances to propagate the solution among each node according to Discrete Element Methods. Despite the usefulness of such tools, their computational requirements make them inapplicable for online prediction.  
The most viable solutions revolve around black or gray box identification from experimental data. In this case, the extruder is usually parametrized as a first-order transfer function with delay, whose poles, gain and delay are selected as the ones that best fit the data. Nevertheless, the accuracy of this type of approximation is acceptable only locally and cannot be generalized to different printing conditions. A different and effective identification approach involves Deep Learning (DL) methods. For instance, in different applications, Recurrent Neural Networks (RNNs) or Long-Short Term Memory (LSTM) networks are able to well approximate the system dynamics, enabling the use of model-based control strategies, such as Model Predictive Control [63] or Internal Model Control [64]. However, DL is extremely data-hungry and thus properly automated acquisition systems are necessary for this approach. Nowadays, inline monitoring has been dealt with in several ways, from the installing of a rheometer on the printing head to geometry estimation via Computer Vision.  
The lack of formal knowledge about the dynamics of the extrudate severely affects the choice of control strategy. The simplest one consists of keeping the extrusion rate constant throughout the print and adapting the path plan to it. Some of the contributions examined have designed synchronization between motion and extrusion along corners by decreasing the extrusion rate proportionally to the linear acceleration of the extruder. Alternatively, an open loop compensation can be tuned to fasten the extrusion response to accelerations of the printing head. A promising approach involves the use of fuzzy controllers, which are designed to translate high-level linguistic rules into suitable control laws.  
Conceptualization, M.M.; methodology, M.M.; formal analysis, M.M.; investigation, M.M.; resources, F.C.; data curation, M.M.; writing--original draft preparation, M.M.; writing--review and editing, M.D., G.M. and H.A.; visualization, M.M.; supervision, M.D., G.M. and H.A.; project administration, M.S. and F.C.; funding acquisition, F.C. All authors have read and agreed to the published version of the manuscript.  
This work is co-funded by POR FESR 2014-2020--Call HUB Research and Innovation of Regione Lombardia's project IRCRAM 4.0 (International Research Center for Robot and Additive Manufacturing 4.0--Project ID 1175193).  
Special thanks to the Camozzi Group for their help and support.  
The authors declare no conflict of interest.