Adaptive nonlinear vibration control of a Cartesian flexible manipulator driven by a ballscrew mechanism

Abstract

A flexible Cartesian manipulator is a coupling system with a moving rigid body and flexible structures. Thus, vibration suppression problem must be solved to guarantee the stability and control accuracy. A characteristic model based nonlinear golden section adaptive control (CMNGSAC) algorithm is implemented to suppress the vibration of a flexible Cartesian smart material manipulator driven by a ballscrew mechanism using an AC servomotor. The system modeling is derived to recognize the dynamical characteristics. The closed loop stability is analyzed based on the model. Also, an experimental setup is constructed to verify the adopted method. Experimental comparison studies are conducted for modal frequencies' identification and active vibration control of the flexible manipulator. The active vibration control experiments include set-point vibration control responses, vibration suppression under resonant excitation and simultaneous translating and vibration suppression using different control methods. The experimental results demonstrate that the controller can suppress both the larger and the lower amplitude vibration near the equilibrium point effectively.

Introduction

Space manipulators should be as light as possible in order to reduce their launching cost [1]. They are usually made of lightweight materials to help launch. Space robots usually have very low damping ratios, high dimensional order and parametric uncertainties in dynamics. Thus, unwanted vibrations of the flexible links will be caused unavoidably [2], [3]. In order to meet the increasing demand for high end-position accuracy robotic systems coupled with needs of space application, the issue on modeling and active vibration control for such a flexible Cartesian robot system is rather difficult and challenging.

The dynamical modeling and active control of a flexible Cartesian robot have attracted attention of many researchers in the last few decades [5], [6], [7], [8], [9], [10], [11]. In most of previous research works, servomotors and piezoelectric patches are used as actuators to achieve desired end-point position while suppressing unwanted vibration. PZT (lead zirconate titanate) as a kind of smart material was effectively implemented to control the mechanical vibration of flexible structures [4]. In the vibration control of flexible robot arms, boundary feedback schemes are employed in order to damp the vibrations and dissipate energy [5]. Hou and Tsui [6] formulated a mathematical model for a flexible robot arm on a moving base with a payload at the tip end using a fourth order partial differential equation with boundary conditions. They show that such a system is both controllable and observable in an infinite dimensional Hilbert space, through the state-space formulation. Luo et al. [7] proposed a shear force feedback control method to suppress vibrations arising from structural flexibility of Cartesian type robots. Dadfarnia et al. [8], [9] presented theoretical and experimental results of an observer-based control strategy and a Lyapunov-based controller for regulating problem of a flexible Cartesian robot. Ge et al. [10], [11] presented a robust distributed controller and an asymptotically stable end-point regulation controller for a single-link Cartesian smart materials robot. Both simulation and experimental results verified the good performance in suppression of residual vibrations under the environment of disturbances.

Although great progress has been achieved in this field, the issue is still far from completely solved. The flexible robot system is of infinite dimensionality. Once using the technique of modal truncation, undesired residual vibrations make flexible robots difficult to control with high precision [10]. For flexible Cartesian robot, the ballscrew is a most popular driving mechanism utilized in high-speed and long stroke positioning stages. However, due to the backlash and nonlinear friction force between the ballscrew and nut, it is difficult to obtain sub-micrometer resolution based on a ballscrew mechanism [12]. The non-linear behavior of a single-link flexible visco-elastic Cartesian manipulator was studied by Pratiher and Dwivedy [13]. As to nonlinear control, Yang et al. [14] proposed a feedback nonlinear control law for the endpoint control of a flexible macro–micro manipulator system. Their dynamic behavior presents a coupling between rigid body displacements and flexible modes of vibration.

These model-based controllers, originally designed for the demands of high performance, may not be easy to implement due to uncertainties in design models, large variations of loads on the robot's end-effector, ignored high frequency dynamics and the high order of the designed controllers [7]. In order to design a satisfactory control system, according to traditional control theory, a mathematical model that describes system dynamics must be determined beforehand. However, it is difficult to establish their mathematical models accurately for some plants, as their characteristics and environment may change unpredictably [15]. Even if a precise mathematical model can be established, the order of the model is extremely high and the structure is very complicated. Unfortunately, the high-order controller is very difficult to realize.

In general, the controller design in previous researches for flexible structures control depends on the reduced order models by modal truncation. Therefore, the problems of “observation spillover” and “control spillover” are unavoidable due to the ignored high-frequency dynamics [7]. To solve the “spillover” problem in traditional modal truncation methods, a characteristic model (CM) based adaptive control method was proposed by Wu et al. [15], [16]. The high order system of flexible structures can be equivalently modeled as a second order time-varying difference equation, named as “characteristic model”. It shows a new way to control large flexible structures with low order controllers and avoids the “spillover” problem at the same time. CM is to implement the controller from the engineering point of view. The CM based nonlinear golden section adaptive control (CMNGSAC) algorithm was introduced in Ref. [16], [17]. Using this control law, the low amplitude residual vibration near the equilibrium point will be suppressed quickly. In recent years, several researchers have employed the CMNGSAC algorithm for vibration control for flexible structures, spacecraft attitude control and tracking control of robotic manipulators [15], [16], [17], [18], [19]. In order to deal with complex nonlinear system, Luo et al. [20] presented a novel neuro-fuzzy dynamic characteristic modeling method by introducing neural network into the fuzzy characteristic modeling control. Among the aforementioned researches, in many cases, only simulation results were obtained. Few experiments are conducted to verify this method.

The major contribution of this paper rests on two aspects. The first is stability analyses of the CMNGSAC algorithm via equivalent gain adaptive regulating control algorithm, using the model of the Cartesian robot. The second is experimental implementation of the CMNGSAC algorithm to suppress the elastic vibration of a flexible smart Cartesian manipulator. In this regard, a kind of flexible Cartesian smart materials robot using a ballscrew mechanism is designed and constructed for experimental studies. Experimental results are provided to demonstrate the satisfactory control performance and robustness of the CMNGSAC method.

The rest of this article is organized as follows. Section 2 introduces the system and the governing equations for a flexible Cartesian smart materials robot. In Section 3, the CMNGSAC algorithm is discussed. The stability of CMNGSAC algorithm for the closed-loop system is analyzed. In Section 4, the experimental setup of the flexible Cartesian smart materials robot is designed and constructed. And experimental comparison studies are conducted using several different control algorithms, including proportional derivative (PD) control, positive position feedback (PPF) control and the CMNGSAC method with respect to the flexible Cartesian robot system. The experimental validation contents include the following cases: set point vibration active control, vibration suppression under resonant excitation and simultaneous translating motion and vibration suppression. The paper ends with conclusions in Section 5.