Chinedum E. Okwudire

Hybrid Modeling of Ball Screw Drives With Coupled Axial, Torsional, and Lateral Dynamics

It has been a common practice to assume that the torsional and axial dynamics are totally decoupled from the lateral dynamics of the screw when modeling ball screw drives. However, experiments show that there is a considerable coupling between them, which could adversely affect the positioning accuracy and fatigue life of the drive. In this paper, the lateral dynamics of the screw is explicitly incorporated into the hybrid finite element model of ball screw drives. The ball screw is modeled by Timoshenko beam elements, and the balls, joints, bearings, and fasteners are modeled as pure springs. Rigid components are modeled as lumped masses. The proposed screw-nut interface model, which includes the effects of lateral vibrations, is shown to predict the coupling between axial, torsional, and lateral dynamics of ball screw drives. The effects of this dynamic coupling on the positioning accuracy of the drive are also presented with experimental proof The proposed model provides a more realistic platform for a designer to optimize the drive parameters for high speed-high acceleration machine tool applications, where the ball screw vibrations limit the fatigue life of the mechanism, bandwidth of the servo systems, and positioning accuracy of the machine.

Minimum Tracking Error Control of Flexible Ball Screw Drives Using a Discrete-Time Sliding Mode Controller

This paper presents modeling, identification, and discrete-time sliding mode control of ball screw drives with structural flexibility. The mechanical system of the drive is modeled by a two degree-of-freedom system dominated by the coupled longitudinal and torsional dynamics of the drive assembly whose parameters are identified. A mode-compensating disturbance adaptive discrete-time sliding mode controller is then designed to actively suppress the vibrations of the drive. However, it is shown theoretically that, without using minimum tracking error filters, the tracking errors of the drive do not go to zero when sliding mode is reached. Therefore, a method for designing stable and robust minimum tracking error filters, irrespective of the identified open-loop behavior of the drive is proposed. The identification and control of flexible ball screw drives are experimentally tested, and the tracking accuracy of the drives is shown to improve considerably as a result of the designed minimum tracking error filters. DOI: 10.1115/1.3155005

Improved Screw–Nut Interface Model for High-Performance Ball Screw Drives

Emerging applications of ball screw drives such as semiconductor inspection, fiber optic alignment, medical equipment, and miniature robotic actuators typically make use of ball screws that are compact, stiff, and precise. Existing models for the screw–nut interface stiffness of ball screw drives are however unable to accurately describe the dynamics of compact and stiff ball screws because they are derived based on the assumption that the portion of the screw within the nut is rigid. This paper proposes a new screw–nut interface stiffness model, which incorporates the elastic deformation of the screw within the nut using Timoshenko beam shape functions. The new model is shown, via simulation and experiments, to provide more accurate predictions of the natural frequencies of compact and stiff ball screw/nut assemblies compared to the existing models. It is therefore more suitable for use in the design simulation/evaluation of high-performance ball screw drives where compactness and rigidity are required.

Energy-Efficient Controller Design for a Redundantly Actuated Hybrid Feed Drive With Application to Machining

This paper presents a method for designing a controller that achieves the best positioning performance while maximizing the energy efficiency of a redundantly actuated hybrid feed drive. A two-degree-of-freedom controller, consisting of a feedforward (FF) controller for tracking and a feedback (FB) controller for regulation, is assumed. It is shown that the ideal FF controller, which achieves perfect tracking with maximum efficiency, is not always stable. Therefore, a method for designing a stable FF controller that achieves perfect tracking with near optimal efficiency is proposed. Furthermore, the optimal relationship between FB control inputs, which guarantees maximum efficiency for any specified regulation performance, is derived. Two approaches for using the derived optimal relationship to synthesize a FB controller that achieves the best positioning performance while maximizing efficiency are proposed. Simulations and machining experiments are conducted to demonstrate the effectiveness of the proposed energy-efficient FF and FB controller design methods. Significant improvements in energy efficiency (without sacrificing positioning performance) are reported.

Tracking control of non-minimum phase systems using filtered basis functions: A nurbs-based approach

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 B-spline (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.Copyright

Linear quadratic design of passive vibration isolators

Design of passive vibration isolators using the popular Linear Quadratic (LQ) control design method is discussed. A single degree of freedom (DOF) example is used to illustrate the main concepts (i.e., skyhook damping, tradeoff between response to base excitation and external disturbances). The LQ design results are also presented for two DOF isolators to investigate two alternative approximate solution strategies, selection of weights, and location of isolators. The LQ approach is shown to be a suitable and convenient approximate design method for locating and tuning passive vibration isolators that approximate skyhook damping.Copyright

Effects of Non-Proportional Damping on the Residual Vibrations of Mode-Coupled Ultra-Precision Manufacturing Machines

Ultra-precision manufacturing (UPM) machines are used to fabricate and measure complex parts having micrometer-level features and nanometer-level tolerances/surface finishes. Therefore, low-frequency residual vibrations that occur during the motion of the machines’ axes must be minimized. Recent work by the authors has revealed that coupling vibration modes of passively-isolated UPM machines can provide conditions for drastic reduction of residual vibrations vis-a-vis the recommended practice of modal decoupling. This paper presents an investigation into the effects of non-proportional (NP) damping on the conclusions reached in the authors’ prior work. With NP damping added, the conditions under which mode coupling is beneficial relative to decoupling are seen to remain largely the same. However, NP damping is shown to significantly influence the conditions under which the system’s response is most sensitive to mode coupling. Design guidelines for maximally exploiting the benefits of mode coupling are presented and demonstrated experimentally on a UPM machine.Copyright

Reduction of Torque-Induced Bending Vibrations in Ball Screw-Driven Machines via Optimal Design of the Nut

As a result of the push for sustainable machine designs, efforts are constantly being made to reduce the mass/inertia of moving machine components so as to minimize material usage and energy consumption. However, the reduction of structural stiffness that often accompanies such efforts gives rise to unwanted vibrations which must be effectively mitigated to ensure satisfactory performance of the designed machine. The ball screw mechanism (BSM) is commonly used in machines for motion and force transmission. Recent research has shown that, due to the coupling introduced by the nut, a torque applied to the shaft of a ball screw mechanism causes undesirable lateral (bending) vibrations of the screw, which adversely affect the fatigue life and positioning accuracy of ball screw-driven machines. In this paper, an analysis of the stiffness matrix connecting the screw to the nut is used to show that the entry/exit angle of the balls and the lead angle of the screw have the greatest influence on the coupling between the torsional and lateral directions. An objective function is proposed to minimize the static coupling between the applied torque and lateral deformations of the screw. The existence of local minima in the objective function is shown to be dependent on the cyclical characteristics of cross-coupling terms in the screw-nut interface stiffness matrix as a function of the entry/exit angle of the balls. Moreover, the sensitivity of the local minima to other nut/screw parameters is shown to highly depend on the lead angle. Simulations conducted on the finite element (FE) model of a single-axis ball screw-driven machine demonstrate that the optimally selected entry/exit angles result in a significant reduction of the low-frequency torque-induced vibrations of the machine compared to the unoptimized case, particularly when the lead angle is small. The proposed method is therefore suitable for reducing the torque-induced lateral vibrations of ball screws without increasing the diameter (i.e., inertia) of the screw, thus leading to more sustainable designs of ball screw-driven machines.

Tracking Control of Linear Time- Invariant Nonminimum Phase Systems Using Filtered Basis Functions

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. [DOI: 10.1115/1.4034367]

A comparison of two methods for energy efficient tracking control using filtered basis functions

This paper presents and compares two methods for trading off energy efficiency and feedforward tracking accuracy using filtered basis functions. In the regular filtered basis function (FBF) approach, 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 system and their coefficients selected such that a performance index which penalizes tracking errors is minimized. It is shown that, in the regular FBF approach, varying the number of basis functions can be used to tradeoff energy efficiency (control energy) and tracking accuracy. Another way of achieving the same goal using FBFs is to minimize a weighted sum of tracking and control energy objectives (i.e., linear quadratic tracking) using a fixed number of basis functions. Simulation-based examples are used to compare both approaches with regard to their degree of tradeoff between tracking accuracy and energy efficiency.