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A hybrid (i.e., physics-guided data-driven) feedforward tracking controller is proposed for systems with unmodeled linear or nonlinear dynamics. The proposed controller is based on the filtered basis functions (FBF) approach, and hence called a hybrid FBF controller. It formulates the feedforward control input to a system as a linear combination of a set of basis functions whose coefficients are selected to minimize tracking errors. To predict the system response and thereby the tracking errors, the basis functions are filtered using a combination of two linear models. The first model is physics-based and remains unaltered during the execution of the controller, while the second is data-driven and is continuously updated during the execution of the controller. To ensure its practicality and safe learning, the proposed hybrid FBF controller is equipped with the abilities to handle delays in data acquisition and to detect impending instability due to its inherent data-driven feedback loop. The effectiveness of the hybrid FBF controller is demonstrated via application to vibration compensation of a 3D printer with unmodeled linear and nonlinear dynamics. Thanks to the proposed hybrid FBF controller, the tracking accuracy of the 3D printer and the print quality are both significantly improved in experiments involving high-speed printing, compared to standard FBF controller that does not incorporate a data-driven model. Furthermore, the ability of the hybrid FBF controller to detect, and hence to potentially avoid, impending instability is demonstrated offline using data collected online from experiments.

Analytical and low-order numerical models are very useful for studying friction behavior of rolling element machine components like ball bearings and ball screws. This is because they provide generalizable insights into friction behavior at much lower computational costs compared with high-order numerical models like finite element analysis (FEA). While analytical and low-order numerical models in the literature are mainly focused on ball-to-groove contact friction, experimental studies have shown that ball-to-ball contact friction is also very important. This is especially true for linear ball bearings/guideways and ball screws which, unlike rotary ball bearings, do not typically make use of caged balls to prevent ball-to-ball contact. Therefore, in this paper, low-order numerical models for ball-to-ball contact friction in linear ball bearings and ball screws are developed. Furthermore, an analytical model for ball-to-ball contact friction in four-point contact linear ball bearing is derived by making simplifications to its low-order numerical model. Compared with ball-to-ball friction predictions from FEA models developed in ansys, the proposed numerical models are shown in case studies to be accurate within 7%, while computing at least three orders of magnitude faster. Moreover, case studies are used to demonstrate how the developed models can be used in practice, e.g., for the mitigation of ball-to-ball contact friction in linear ball bearings and the prediction of friction variation during the operation of a ball screw.

Accurate tracking of nonminimum phase (NMP) systems is well known to require large amounts of control effort. It is, therefore, of practical value to minimize the effort needed to achieve a desired level of tracking accuracy for NMP systems. There is growing interest in the use of the filtered basis functions (FBF) approach for tracking the control of linear NMP systems because of distinct performance advantages it has over other methods. The FBF approach expresses the control input as a linear combination of user-defined basis functions. The basis functions are forward filtered through the dynamics of the plant, and the coefficients are selected such that the tracking error is minimized. There is a wide variety of basis functions that can be used with the FBF approach, but there has been no work to date on how to select the best set of basis functions. Toward selecting the best basis functions, the Frobenius norm of the lifted system representation (LSR) of dynamics is proposed as an excellent metric for evaluating the performance of linear time varying (LTV) discrete-time tracking controllers, like FBF, independent of the desired trajectory to be tracked. Using the metric, an optimal set of basis functions that minimize the control effort without sacrificing tracking accuracy is proposed. The optimal set of basis functions is shown in simulations and experiments to significantly reduce control effort while maintaining or improving tracking accuracy compared to popular basis functions, like B-splines.

The friction behavior of rolling ball machine components like linear ball bearings is very important to their functionality. For instance, differences in linear velocity of balls induces ball-to-ball contact in certain circumstances, resulting in significant increases and variations in friction. In this paper, an improved analytical formula for determining the linear velocity of balls in four-point-contact linear ball bearings is derived as a function of contact angle deviations and contact forces. The analytical formula is validated against a comprehensive friction model in the literature and shown to be in good agreement, while an oversimplified analytical model proposed by the authors in prior work is shown to be inaccurate. A case study is presented where insights gained from the derived analytical formula are used to mitigate velocity difference of balls in a linear ball bearing which otherwise would experience ball-to-ball contact.

The Frobenius norm of the lifted system representation of tracking error dynamics is proposed as a metric for evaluating the tracking performance of discrete-time linear time invariant (LTI) and linear time varying (LTV) controllers. The proposed metric is introduced here in the context of feedforward tracking control of LTI single-input single-output (SISO) nonminimum phase (NMP) systems, though it is more broadly applicable. It is shown that the filtered basis functions (FBF) approach, an LTV tracking control technique studied by the authors in prior work, is the optimal solution to the rank constrained minimization of the proposed metric. Moreover, for the FBF controller, the metric is independent of plant dynamics, which is not the case for most other tracking controllers; it is also independent of the type of basis functions employed in the FBF controller. The effectiveness of the proposed metric as a tracking performance evaluation tool for both LTI and LTV tracking controllers is demonstrated analytically and numerically on LTI plants with different zero locations.

An approach for minimizing tracking errors in linear time-invariant (LTI) single-input single-output (SISO) discrete-time systems with nonminimum phase (NMP) zeros using filtered basis functions (FBF) is studied. In the FBF method, the control input to the system is expressed as a linear combination of basis functions. The basis functions are forward filtered using the dynamics of the NMP system, and their coefficients are selected to minimize the error in tracking a given desired trajectory. Unlike comparable methods in the literature, the FBF method is shown to be effective in tracking any desired trajectory, irrespective of the location of NMP zeros in the z-plane. The stability of the method and boundedness of the control input and system output are discussed. The control designer is free to choose any suitable set of basis functions that satisfy the criteria discussed in this paper. However, two rudimentary basis functions, one in time domain and the other in frequency domain, are specifically highlighted. The effectiveness of the FBF method is illustrated and analyzed in comparison with the truncated series (TS) approximation method.

Four-point contact between ball and raceways is common in machine elements like ball bearings and ball screws. The ideal four-point-contact machine element is designed with pure rolling (i.e., no sliding at contact points) to minimize friction. However, this ideal may not always be achieved, leading to sliding and higher frictional forces. In this paper, a simplified analytical model for rolling/sliding behavior and friction in four-point contact is developed, based on Coulomb friction model and rigid body assumption. It is found that pure rolling is only possible when the contact-point geometry satisfies a certain relationship. When pure rolling condition fails to hold, the sliding contact point(s) can be determined analytically as a function of contact forces and contact angles. Case studies are presented to demonstrate how the proposed model could elucidate the roles of misalignments, manufacturing errors and loading conditions on rolling/sliding behavior and friction.

This paper studies slip in friction drives in the context of a precision servo positioner equipped with a twist-roller friction drive. The servo positioner is modeled using a two-mass model that includes the experimentally-identified nonlinear slip dynamics of the friction drive. The model is then used to design an observer-based compensation scheme aimed at reducing the slip-induced performance degradation of the servo positioner's linear time invariant controller. The compensation scheme is validated in simulations and experiments using state feedback controllers. Tracking performance with the proposed slip compensation scheme, compared with the original controller and an intuitive high pass filter approach, is shown to improve considerably.

Commercial desktop 3D printers are designed with light, but flexible, structures and driven with stepper motors to reduce their cost, size and weight. As a result, parts manufactured on such 3D printers suffer from surface waviness and registration errors (caused by stepper motors skipping counts) due to excessive vibration triggered by the motion of the print head or build platform. Such vibration-induced errors not only mar the aesthetics of 3D printed parts, but often lead to highly distorted and hence scrapped parts. The most common solution to this problem, not involving hardware modifications, is to reduce motion speed at the expense of productivity. This paper demonstrates the use of a software-based method called filtered Bsplines (FBS) for mitigating vibration-induced errors of 3D printers without sacrificing productivity. In the FBS method, the motion command sent to the machine is parameterized using B-splines, whose basis functions are filtered using a model of the machine's dynamics. The control points associated with the filtered basis functions are then selected to minimize tracking errors. The effectiveness of the FBS method in mitigating surface waviness and registration errors is demonstrated on parts printed on a commercial desktop 3D printer.

Linear motor drives (LMDs) are well known to provide significant advantages in terms of positioning speed and precision over traditional screw drives (SDs), making them better suited for high-speed, high-precision machine tools. However, their use in such machine tools is limited by their tendency to consume a lot of electrical energy and cause thermal issues that help drive up costs. A hybrid feed drive (HFD) has been proposed as a possible solution to this dilemma. The HFD combines LMD and SD actuation to achieve speeds and accuracies similar to LMDs while consuming much less energy. This paper explores control strategies to further improve the performance of the HFD without unduly sacrificing its efficiency. First, it highlights two performance limitations of the controller proposed for the HFD in prior work, namely, imperfect tracking and suboptimal feedback gains. Then it compares two approaches for achieving perfect tracking with regard to performance and energy efficiency. Finally, it presents an approach for optimizing the feedback gains of the HFD to achieve the best positioning performance. Simulations and experiments are used to demonstrate significant gains in precise positioning using the methods proposed in this paper, while maintaining superb energy efficiency relative to an equivalent LMD.

This paper proposes an approach for minimizing tracking errors in systems with non-minimum phase (NMP) zeros by using filtered basis functions. The output of the tracking controller is represented as a linear combination of basis functions having unknown coefficients. The basis functions are forward filtered using the dynamics of the NMP system and their coefficients selected to minimize the errors in tracking a given trajectory. The control designer is free to choose any suitable set of basis functions but, in this paper, a set of basis functions derived from the widely-used non uniform rational Bspline (NURBS) curve is employed. Analyses and illustrative examples are presented to demonstrate the effectiveness of the proposed approach in comparison to popular approximate model inversion methods like zero phase error tracking control.

When traversing sharp corners, manufacturing machines are forced to tradeoff speed and accuracy. The most common way of reducing this tradeoff is to smooth the sharp corner using a pre-specified curve (e.g., a circular arc or spline). However, pre-specified curves cannot guarantee optimal performance. This paper presents a preliminary investigation into the potential of using methods from optimal control to minimize this tradeoff. First, a useful simplification is made to the exact cornering problem to make it tractable. Dynamic programming is then used to determine the best free-form curve that minimizes corner traversal time while adhering to path tolerance and machine kinematic constraints. Significant improvements in cornering time are demonstrated compared to two methods that use pre-specified curves. However, dynamic programming is found to be too computationally costly, thus impractical. Less computationally intensive techniques in optimal control are considered for future work.