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The structural flexibility of industrial robot arms makes them vibrate when they are commanded to move at fast operation speeds. Among the control strategies, feedforward control stands out as an interesting approach to suppress vibration since it does not create stability issues and works for repeating and non-repeating tasks. Currently, the state-of-the-art feedforward controller dedicated to suppressing residual vibration in robot arms is time-varying input shaping (TVIP). However, TVIP falls short in trajectory tracking tasks since the method adds delays in the commands creating errors in tracking and thereby contouring trajectories. Therefore, this paper proposes the use of an alternate feedforward method, known as the filtered B-splines (FBS) approach, to suppress vibration in six DOF robots while maintaining tracking accuracy. Since time-varying FBS (TVFBS) requires full frequency response functions (FRFs), compared to only natural frequencies and damping ratios for TVIP, we propose a framework for estimating the FRFs of serial kinematic chain 6-degree-of-freedom robots. Residual vibration reduction experiments and trajectory tracking experiments, in which the dynamics of a UR5e collaborative robot change considerably, were carried out to validate the model prediction framework. TVFBS reduced the end-effector vibration by 87% while improving tracking performance in both the y (22%) and z (29%) directions. On the other hand, TVIP worsened the tracking performance (-683.43% for the y and -662.37% for the z direction) despite the excellent vibration reduction (98%). Hence, TVFBS demonstrated significantly better tracking performance than TVIP while retaining comparable vibration reduction. 

Delta 3D printers can significantly increase throughput in additive manufacturing by enabling faster and more precise motion compared to conventional serial-axis 3D printers. Further improvements in motion speed and part quality can be realized through model-based feedforward vibration control, as demonstrated on serial-axis 3D printers. However, delta machines have not benefited from model-based controllers because of the difficulty in accurately modeling their position-dependent, coupled nonlinear dynamics. In this paper, we propose an efficient framework to obtain accurate linear parameter-varying models of delta 3D printers at any position within their workspace from a few frequency response measurements. We decompose the dynamics into two sub-models–(1) an experimentally-identified sub-model containing decoupled vibration dynamics; and (2) an analytically-derived sub-model containing coupled dynamics–which are combined into one using receptance coupling. We generalize the framework by extending the analytical model of (2) to account for differing mass profiles and dynamic models of the printer’s end-effector. Experiments demonstrate reasonably accurate predictions of the position-dependent dynamics of a commercial delta printer, augmented with a direct drive extruder, at various positions in its workspace. Note to Practitioners—This work aims to equip high-speed 3D printers, like delta machines, with model-based controllers to complement their speed with high-accuracy. Due to the coupled kinematic chains of the delta, complex control methodologies, some of which require real-time state measurements, are often used to achieve satisfactory control performance. Our modeling approach provides an efficient methodology for obtaining accurate linear models without the need for real-time measurements, thus enabling practitioners to design linear model-based feedforward controllers to achieve the high throughput and accuracy desired in additive manufacturing (AM). The models we develop in this paper are intended for use with feedforward vibration compensation methods, which can be beneficial for both industrial-scale AM machines that have high-powered servo motors and feedback controllers, as well as consumer-grade AM machines which use stepper motors in feedforward control.