Software Compensation of Undesirable Racking Motion of H-frame 3D Printers using Filtered B-Splines

The H-frame (also known as H-Bot) architecture is a simple and elegant two-axis parallel positioning system used to construct the XY stage of 3D printers. It holds potential for high speed and excellent dynamic performance due to the use of frame-mounted motors that reduce the moving mass of the printer while allowing for the use of (heavy) higher torque motors. However, the H-frame's dynamic accuracy is limited during high-acceleration and high-speed motion due to racking -- i.e., parasitic torsional motions of the printer's gantry due to a force couple. Mechanical solutions to the racking problem are either costly or detract from the simplicity of the H-frame. In this paper, we introduce a feedforward software compensation algorithm, based on the filtered B-splines (FBS) method, that rectifies errors due to racking. The FBS approach expresses the motion command to the machine as a linear combination of B-splines. The B-splines are filtered through an identified model of the machine dynamics and the control points of the B-spline based motion command are optimized such that the tracking error is minimized. To compensate racking using the FBS algorithm, an accurate frequency response function of the racking motion is obtained and coupled to the H-frame's x- and y-axis dynamics with a kinematic model. The result is a coupled linear parameter varying model of the H-frame that is utilized in the FBS framework to compensate racking. An approximation of the proposed racking compensation algorithm, that decouples the x- and y-axis compensation, is developed to significantly improve its computational efficiency with almost no loss of compensation accuracy. Experiments on an H-frame 3D printer demonstrate a 43 percent improvement in the shape accuracy of a printed part using the proposed algorithm compared to the standard FBS approach without racking compensation.