Mathematical Modelling and Motion Analysis of Cartesian and Polar 3D Printers Driven by Stepper Motors - Marathe, Mayur; Deshmukh, Venkatesh S. (Journal of Applied Nonlinear Dynamics, 2026)
Mathematical models of 3D printers in Cartesian and Polar configurations driven by stepper motors, referred to henceforth as Unified Models, are formulated and derived as combinations of the basic building blocks. The building blocks are mathematical models of stepper motor, single-stage reduction gearbox, timing belt assembly, power screw assembly, turntable assembly and rack & pinion assembly. The Unified Models formulated have DC voltage pulse applied to the two phases of the Hybrid Bipolar stepper motor as the input and the displacement of the printing head or the printing table of the 3D printer as the output. The sin-cos micro-stepping voltage pulse profiles applied to the two phases of the motor are also derived to demonstrate the printer operation in micro-stepping mode. The operation uses an open loop paradigm in the absence of the encoder and phase current measurement feedback. The stepper motor velocity profile is controlled by changing the duration of the pulse or the micro-pulse. An exponential motion profile in a Speed-up Factor, which is the inverse of the pulse time duration, is also developed to facilitate the motion feasibility, as the traditional S-Curve Velocity profile used in closed loop paradigm does not work with the open loop control paradigm in stepper motors. The unified Motor-Gearbox-Mechanism models in Cartesian and Polar configurations, the Unified Models, are simulated with the designed micro-stepping voltage pulses to demonstrate various printing head and printing table motions encountered during 3D printing process. The Unified Models of Cartesian and Polar Configurations are also simulated in the closed loop paradigm to ascertain the best available controlled response. Lastly, the importance of the open and closed loop motion analysis and establishing the motion parameters for the Unified Models in the theoretical development of the non-traditional closed loop controller algorithm is discussed.