Robust controller design for PTP motion of vertical

IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 8, NO. 1, MARCH 2003

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Robust Controller Design for PTP Motion of Vertical Positioning Systems With a Flexible Beam

XY

Bong Keun Kim, Sangdeok Park, Wan Kyun Chung, Member, IEEE, and Youngil Youm

Abstract—This paper presents a point-to-point (PTP) motion control method to get accurate positioning and to suppress vibrapositioning system with a flexible beam. The tion of a vertical proposed method is composed of a feedforward and feedback controller. The input preshaping based on the analytic modeling and the frequency equation of the system is proposed as a feedforward controller to produce desired responses. The feedback controller based on a robust internal-loop compensator is designed to meet specified performance and to stabilize the whole system in the presence of uncertainties and disturbances. By integrating the input preshaping controller and feedback controller, it is shown that the system is controlled to be stable and the vibration of the flexible beam is suppressed. The proposed algorithm is demonstrated expositioning system which consists of a base perimentally on a cart, elastic beam, and moving mass. Index Terms—Flexible positioning system, input preshaping, residual vibration suppression, robust internal-loop compensator (RIC).

I. INTRODUCTION

R

ECENTLY, many kinds of robust motion control schemes for precision positioning systems have been proposed. Disturbance observer [1]–[7], time delay control [8]–[10], adaptive robust control [11], [12], internal model control with enhanced robustness [13], model based disturbance attenuation control [15], [16] are good examples [14], and nonlinear of model based disturbance compensating methods. To obtain their inherent structural equivalence, a generalized disturbance compensating framework was also proposed [17]–[19], and unified analyses were performed [20], [21]. The basic common assumption of these algorithms is that the uncertain disturbance satisfies an important structural property called the matching condition, namely, it enters the state equation exactly at the point where the control variable enters [22]–[27]. However, if there is an output disturbance caused by structural flexibility such as the mechanical vibration of an elastic beam, these methods cannot guarantee the stability and specified performance of the closed-loop system because it does not satisfy Manuscript received August 20 2001; revised august 29, 2002. Recommended by Technical Editor K. Ohnishi. B. K. Kim was with the Department of Mechanical Engineering, Pohang University of Science and Technology (POSTECH), Pohang 790-784, Korea. He is now with the Department of Mechanical Engineering, University of California at Berkeley, Berkeley, CA 94720 USA (e-mail: [email protected]). S. Park is with the Mechanical and Electrical Engineering Research Team, Research Institute of Industrial Science and Technology (RIST), Pohang 790-330, Korea (e-mail: [email protected]). W. K. Chung and Y. Youm are with the Department of Mechanical Engineering, Pohang University of Science and Technology (POSTECH), Pohang 790-784, Korea (e-mail: [email protected];[email protected]). Digital Object Identifier 10.1109/TMECH.2003.809163

the matching condition. In practice, when a high-rise structural system carries heavy loads, vibrational motion due to the flexibility of the beam structure is unavoidable and not easily damped out, even though they have truss-structured beams. Thus, it takes a long time to move heavy loads using these kinds of positioning systems if the vibration is not suppressed. In this paper, we discuss the precision motion control method positioning system, which consists of a base for a flexible cart, elastic beam, and moving mass. Mechanical systems in industrial fields which have structural flexibility such as heavy load handling robots, stacker cranes in the automated warehouse, aerial ladder trucks, and overhead cranes can be exampositioning systems. The problem of the ples of flexible above systems is caused by the fact that the vibration of the flexible beam perturbs the base cart continuously. And also, when the moving mass moves along a high-length flexible beam, it changes the frequency equation of the system and this makes the problem more difficult [28], [29]. In order to solve these problems, an auxiliary control method such as a vibration suppression algorithm should be integrated into the robust motion controller. Therefore, the objective of this paper is to achieve highsystem accuracy positioning performance of the vertical and to suppress the vibration of the flexible beam while the system follows given trajectories. This requires the design of two-loop structures: one is the design of a feedforward controller for vibration suppression of the elastic beam fixed on a moving cart, and the other is the design of a feedback controller for input disturbance compensation and robustness. An input preshaping method, which reduces the residual vibration by altering the shape of command of the actuator without sensory information, is designed as the feedforward controller. This feedforward control input is also used as a reference command of a feedback controller. To design a feedback controller, the generalized disturbance compensating framework, known as a robust internal-loop compensator (RIC), is introduced based on a control input redesign. And then, the feedback controller is designed in the RIC framework. By integrating the input preshaping controller and feedback controller, the system can be controlled to be stable and the vibrations of the flexible beam can be suppressed. In the next section, mathematical modeling of the flexible positioning system is shown. In Section III, a convertical trol method for the horizontal motion of the system is proposed based on the input preshaping and the RIC. In Section IV, a control method for the vertical motion is proposed based on the dual RIC loop. In Section V, the point-to-point (PTP) motion control plane is discussed. The proposed methods method in the

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IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 8, NO. 1, MARCH 2003

where is the th root of the frequency equation changing with the weight ratios of the system as well as the position of the moving mass as follows:

(6)

Fig. 1.

XY positioning system with a flexible beam.

, and

are evaluated through experiments in Section VI, and the conclusion follows.

II. MODELING OF A FLEXIBLE VERTICAL SYSTEM

,

where

POSITIONING

Consider a Positioning system with a Bernoulli-Euler beam fixed on a moving cart and carrying a moving mass as shown in Fig. 1. The equations of motion and the boundary conditions for this system are written in (1-3) at the bottom of the page (see [29]) and as follows: (4) where is the mass of the cart, is the mass per unit length of the unloaded beam, is the mass of the moving mass, is the Dirac delta-function, is the flexural rigidity of the beam, is the length of the beam, is the position of the cart, is the deflection of the beam at . and Natural frequencies of the flexible positioning system in Fig. 1 are represented as

(5)

,

,

,

[28]. III. HORIZONTAL MOTION CONTROL

A. Feedforward Control Various kinds of feedback controllers have been developed to reduce the residual vibration of flexible mechanical systems [30]–[33]. However, feedback control systems require sensory devices such as strain gauges, accelerometers, and CCD cameras to measure vibration, which are difficult to use in many cases of implementation. On the other hand, an input preshaping method, which reduces the residual vibration by altering the shape of command of the actuator without sensory devices is relatively simple to implement [34]–[36]. And also, the shaped input can be used as a reference control input for feedback controllers. However, since this method is based on the accurate model of the system, it is difficult to meet the performance specifications if there are modeling uncertainties, nonlinear friction, and external disturbances. Even though the input preshaping controller is based on the accurate model without any disturbances, the accurate modeling of the system itself is very difficult, especially for moving mass case. We need a frequency equation and equation of motion for the system at each time of control since the natural frequency varies as the mass moves. This is the reason why the input preshaping controllers have been designed only for fixed structure systems. The method by Teo et al. [36] determines the interval and duration time of each pulse with respect to the maximum acceleration, maximum velocity, and target position. On the other

(1) (2) (3)

KIM et al.: ROBUST CONTROLLER DESIGN FOR PTP MOTION OF VERTICAL

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, and the integer of each pulse is given as [36]

to determine the interval

if (9) if and are the maximum acceleration and vewhere in Fig. 2(a) locity of the base cart, respectively. In this case, is given as (10) and the peak velocity of the base cart is represented as (11) If the natural frequency of the system is changed and the period of the first mode of vibration is changed to , and conse, then the duration time of maximum vequently, if locity to reach the target position must be determined as shown in Fig. 2(b). In this case, the magnitude of acceleration to be apis given by plied to reach the target position

(a)

(12) Since the duration time

is given as (13)

is determined as (14) B. Feedback Control

(b) Fig. 2.

v