Active Vibration Control Using Adaptive Positive Position Feedback

Abstract

An active vibration control strategy is developed for a flexible manipulator with a collocated piezoelectric sensor/actuator pair. Dynamic modeling of the flexible manipulator is first shown and then a control law is developed. The positive position feedback (PPF) control law is augmented with an adaptive parameter estimator based on the recursive least squares method which updates the natural frequencies of the structure online. Accurate targeting of the vibrational modes is critical for damping of the vibrations using PPF and simulations are conducted to demonstrate its eectiveness. I. Introduction Active vibration control of a single-link flexible manipulator is a relatively complex problem and has had numerous papers published on it. From a robotics standpoint, a flexible manipulator is desirable due to the fact that existing manipulators are large and heavy due to the need for the manipulator to be sti in order for the precise location of the tip to be known. The flexible manipulator on the other hand, would have the advantages of lighter weight, lower power consumption, and better maneuverability due to smaller size. The drawback to the flexible manipulator is that it faces large vibrations, leading to positional inaccuracy of the tip during and after movement. The same problems occur in large flexible structures for use in space, such as solar panel arrays. Since the size and mass of an object that can be put into space is severely limited, most structures are flexible due to their smaller mass. This can lead to a variety of problems in areas such as high precision attitude control and stability of motions. A significant number of papers have been written on PPF 1‐3 demonstrating its eectiveness in active vibration control. However, the eectiveness of PPF deteriorates when the natural frequencies of the structure are poorly known or changing due to, for example, the presence of a tip mass. There has been significantly less work done on adaptive PPF than there has been on just PPF. Kwak et al. 4 present a method for adaptively tuning the controller frequency of the PPF controller for a single mode of a cantilever beam, and also illustrate their implementation of a real-time controller. Rew et al. 5 propose an adaptive PPF controller for a plate in a fixed-free configuration in which estimated natural frequencies are adjusted at every time step. Baz and Hong 6 present an adaptive modal PPF controller where the AMPPF controller parameters are adjusted in an adaptive manner in order to follow the performance of an optimal reference model (for a cantilever beam). In the first section, development of a finite element model for an Euler-Bernoulli beam to take into account the eect of the sensor and actuator dynamics is presented. This model is created for use in the simulations and is based upon the work of Bandyopadhyay et al. 7 with some modifications. In the second section, a positive position feedback controller is designed for damping of the beam’s vibrations, and then an adaptive online parameter estimator based upon the recursive least squares method with forgetting factor is designed for the first two modes of vibration. Next, the results of using the adaptive estimator to update the estimates of the beam’s first two natural frequencies are presented and their eectiveness demonstrated. Finally, conclusions and possible further areas of investigation are discussed.