This paper studies the combined effect of sampling and dry friction on the dynamics of mechanical systems subjected to position control using discretetime state feedback. This paper aims to highlight that even in case of a single degree-of-freedom mechanical subsystem, the controlled system cannot be characterized by a model with a single dominant frequency and some of the upper harmonics become relevant resulting multi-frequency vibrations. The presented stability analysis of the frictionless system gives insight into
IntroductionPositioning is a basic task in robotics, where the applied controller aims to drive the robot into a desired position. Industrial robotic applications often demand for high accuracy, strict repeatability and at the same time, fast operation. Modern robots are equipped with digital motion controllers. The performance of these with respect to the above requirements are limited by the sampled data nature 1 of the applied digital control. For example, the positioning accuracy of a PD controller can be improved by the proportional control gain [1], but at the same time the system becomes less robust to parameter variations and might get unstable for large gain values [2]. The effect of sampling have been investigated in the literature in detail [2-6].Reference [2] introduces tools and design methods for digitally controlled robots, [3][4][5] focus on the analytical investigation of the dynamics of digital force control and [6] presents also experiments that show how sampling and quantization can cause performance limitations. In these studies the effect of Coulomb friction is considered as a stabilizing effect and it is neglected. This provides sufficffient conservativism in many cases in the control design, but cannot be applied when dry friction is the dominant or the only source of physical dissipation.It is possible to derive an equivalent viscous damping to represent Coulomb friction by using the method of describing function analysis [7,8]. To simplify the analysis and reduce computational costs, it is usual to model the dissipative forces due to the actuators, the motion transmission elements and bearings as viscous damping [9]. Structural damping is often considered in the form of modal damping. This type of viscous damping often captures well the combined effect of different dissipation mechanisms, and as it will be shown later in the paper, it can also be successfully applied to model friction effects in digitally controlled systems. However, in certain parameter ranges viscous damping alone cannot explain the experimentally observed decaying of vibrations.
Dissipation mechanisms and dissipative forces play a pivotal role in the operations and performance of human-machine interfaces and particularly in haptic systems. Dissipation is a very difficult phenomenon to model. Coulomb friction in general can be the most influential element in systems involving multiple direct contact connections such as joints with transmissions or mechanically guided components. Coulomb friction includes nonsmooth discontinuity and can induce complex dynamic behaviors. The effect of Coulomb friction is often neglected in haptics. The part of the literature which deals with friction mainly focuses on friction compensation and/or simulation of friction for haptic rendering. In this paper, the nature of the dynamic behavior caused by Coulomb friction in haptic sampled-data systems is illustrated by experiment, analysis, and simulation. It is also demonstrated that a simple model can represent this behavior and show the effects of the haptic system parameters on this dynamics.
This paper aims to present that the effect of sampling can result in multi-frequency vibration even in the case of a single-degree-of-freedom linear mechanical model. Even though the sampled-data systems have an infinite number of characteristic exponents due to sampling, the vibrations of these systems can still be characterized by an effective system model with a single dominant frequency. However, as this paper shows, additional harmonics become relevant, resulting in multi-frequency vibrations depending on the magnitude of applied control parameters. The vibrations presented by the time histories of vibration and their spectra resulted in numerical simulation of the sampled-data system.
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