MATHEMATICAL MODEL OF A CLOSED-LOOP QUALITY CONTROL SYSTEM FOR FDM 3D PRINTING

Abstract

Ensuring consistent quality in FDM printing is challenging due to stochastic thermal behavior, non‑Newtonian melt rheology, and the strong sensitivity of bead geometry to feed and motion parameters. This paper presents a closed‑loop mathematical model for quality control that tightly integrates physics‑based submodels with state estimation, defect detection, and optimal control. The process is formulated as a nonlinear discrete‑time system , comprising: (i) a thermal model of heater block and nozzle (energy balance with conduction, convection, and flow‑induced cooling), (ii) rheology/flow (Hagen–Poiseuille with power‑law viscosity), (iii) layer geometry (mass conservation ), and (iv) latent quality indicators (interlayer adhesion , porosity , in‑plane shift ). The sensing layer covers temperature, inertial, and vision channels with explicit noise models. An EKF/UKF observer fuses model and data; consistency is monitored via NIS/NEES. The detection block combines computer‑vision features with residual statistics to produce a defect probability and a fused confidence . Layer quality is quantified by a cost aggregating deviations in geometry, temperature, and latent indicators. Control is performed by MPC: over a receding horizon it minimizes the predicted quality loss subject to plant dynamics and technological constraints, while lightweight adaptive corrections handle fast deviations (flow/speed trims). A safety supervisor implements event‑based logic (soft corrections or pause/stop) if thresholds are exceeded. We provide a structural diagram, numbered equations, a symbol table, time‑scale separation guidelines, and practical tuning advice for weights, thresholds, and noise covariances. The proposed model acts as a deployable template for firmware/external controllers, improves FDM reliability through timely defect detection and compensation, and forms a rigorous basis for digital twins and subsequent process optimization.