An observer-based piezoelectric control of flexible Cartesian robot arms: theory and experiment

https://doi.org/10.1016/j.conengprac.2003.09.003Get rights and content

Abstract

An observer-based control strategy is proposed for regulating problem of a Cartesian robot arm, which is modeled as a flexible cantilever beam with a translational base support. The base motion is controlled utilizing an electrodynamic shaker, while a piezoelectric (PZT) patch actuator is bonded on the surface of the flexible beam for suppressing residual arm vibrations. Taking into account both electrical and mechanical fields for the PZT patch actuator, a full-order model is developed using assumed mode model expansion and the Lagrangian approach. Utilizing three measurable quantities (i.e., the base displacement, arm tip deflection and the strain at the root end of the arm), a reduced-order observer is designed to estimate the velocity related variables, which are not directly measurable for the truncated model. A simple PD controller is selected for the moving base regulation, while a Lyapunov-based controller is utilized for the PZT input voltage to make the closed-loop system energy dissipative and hence stable. The PZT input voltage control uses the velocity related signals estimated by the reduced-order observer, which simplifies the control implementation in practice. The feasibility of the controller is validated through both numerical simulations and experimental testing. Significant matching between experimental results and numerical simulation is observed.

Introduction

The demands for high-speed performance and low energy consumption are the main motivations for the use of lightweight robot manipulators in industrial applications. The lightweight and highly flexible nature of these robots, however, leads to a challenging problem in end-point trajectory control. Early studies on control problem of flexible manipulators concentrated on model-based controller design schemes (Cannon & Schmitz, 1984; Magee & Book, 1993; Yuh, 1987; Ge, Lee, & Zhu, 1997). However, these controllers offer limited applicability in practical applications due to uncertainties in design models, unmodeled dynamics and ignored high frequency dynamics (de Querioz, Dawson, Agrawal, & Zhang, 1999; Jalili, Elmali, Moura, & Olgac, 1997; Liu, Jalili, Dadfarnia, & Dawson, 2002). Hence, it is highly desirable to seek simple and robust controllers for the control of flexible robot arms (De Luca & Sicilicano, 1993; De Luca & Di Giovanni, 2001; Book, 1990; Yuan, Book, & Huggins, 1993; Dadfarnia, Jalili, Xian, & Dawson, 2003).

Owing to the distributed flexibility along the robot arm, an improved control technique is required to suppress the residual vibrations in the arm while tracking the desired trajectory at the arm base. Consequently, this control problem has attracted significant attention in the literature which resulted in the development of several control methods (de Querioz, Dawson, Nagarkatti, & Zhang, 2000; Jalili, 2001; Jalili & Olgac, 1998). Many industrial robots, especially those widely used in automatic manufacturing assembly line, are Cartesian types (Ge, Lee, & Zhu, 1998b). A flexible Cartesian Robot can be modeled as a flexible cantilever beam with a translational base support (see Fig. 1). Traditionally, a PD control strategy is used to regulate the movement of the robot arm. In lightweight robots, the base movement will cause undesirable vibrations at the arm tip because of the flexibility distributed along the arm. In order to eliminate such vibrations, the PD controller needs to be upgraded with additional compensating terms. Luo et al. introduced a shear force feedback (Luo, Kitamura, & Guo, 1995) and strain feedback control (Luo, 1993) for flexible robots.

It is shown that a simple strain feedback combined with a PD controller for the force exerted on the arm base can suppress the residual vibration in rotating flexible arms (Ge et al., 1998b; Luo, 1993). Compared with the traditional PD controller, Ge's controller (Ge, Lee, & Gong 1998a, Ge, Lee, & Zhu, 1998b) demonstrated better vibration suppression performance since the link flexibility was taken into the consideration. However, many industrial robots are Cartesian type with translational joints, and it can be demonstrated that a strain feedback control action does not introduce a damping mechanism for such robots as it does for robots with rotational joints (Luo et al., 1995; Luo, 1993). In order to further improve the vibration suppression performance, which is a requirement in the high-precision manufacturing market, a second controller such as a piezoelectric (PZT) patch actuator attached on the surface of the arm can be utilized (Andre’, 1997; Oueini, Nayfeh, & Pratt, 1998; Ge, Lee, & Gong, 1999).

To predict the behavior of the flexible beams incorporating PZT actuators and sensors, many exact (analytical) and approximate (numerical) models have been developed (Chandrashekhara & Varadarajan, 1997; Crawley & Anderson, 1990; Wang & Rogers, 1991; Wang & Quek, 2000; Ge, Lee, & Gong, 1998a). These mathematical models consider the piezoelectric patches in the form of either bonded on the transverse surface or embedded in the beam, and assume linear strain in the actuator and a pure shear in the bond film (Crawley & Anderson, 1990). Wang and Rogers considered the effective moment induced by the PZT actuator and presented the classical theory to predict the deformation of the laminated plates with embedded strain actuator patches (Wang & Rogers, 1991). Taking into account the dynamic reaction and based on the Euler-Bernoulli beam theory, Wang and Quek presented the governing equations for a long and thin beam coupled with embedded piezoelectric actuators (Wang & Quek, 2000). Ge et al. presented a finite element method (FEM) for a robot with a rotating smart material arm and developed the coupled equations for the beam and PZT actuator using Hamilton's principle (Ge et al., 1998a).

The purpose of this paper is to present an observer-based control strategy for regulating a Cartesian robot arm, which is modeled as a flexible cantilever beam with a translational base support. Our model is rather complete and general in the sense that it utilizes both mechanical and electrical properties of the PZT materials, instead of the PZT strain–stress relationship only (Crawley & Anderson, 1990). The control objective here is to regulate the arm base movement, while simultaneously suppressing the vibration transients in the arm. To achieve this, a simple PD control strategy is selected for the regulation of the movement of the base and a Lyapunov-based controller for the PZT voltage signal. The selection of the proposed energy-based Lyapunov function naturally results in velocity related signals which are not physically measurable. To remedy this, a reduced-order observer is designed to estimate the velocity related signals. For this, the control structure is designed based on the truncated two-mode beam model.

The rest of the paper is organized as follows. The system description and modeling efforts are presented in Section 2. The derivation of the equations of motion is given in Section 3. Section 4 presents the design of the proposed Lyapunov-based controller. The control implementation and the design of the reduced-order observer are explained in Section 5. Numerical simulations and experimental results are presented in 6 Numerical simulations, 7 Control experiments, respectively followed by the concluding remarks in Section 8.

Section snippets

Mathematical modeling

For the purpose of model development, a uniform flexible cantilever beam is considered with PZT actuator bonded on its top surface. As shown in Fig. 1, one end of the beam is clamped into a moving base with the mass of mb, and a tip mass, mt, is attached to the free end of the beam. The beam has total thickness tb, and length L, while the piezoelectric film possesses thickness and length tb and (l2l1), respectively. It is assumed that the PZT and the beam have the same width, b. The PZT

Derivation of the equations of motion

An energy method is used to derive the equations of motion. Neglecting the electrical kinetic energy, the total kinetic energy of the system can be expressed asEk=12mbṡ(t)2+120Lρ(x)(ṡ(t)+ẇ(x,t))2dx+12mt(ṡ(t)+ẇ(L,t))2,whereρ(x)=[ρbtb+G(x)ρptp]b,G(x)=H(x−l1)−H(x−l2)and H(x) is the Heaviside function, ρb and ρp are the respective beam and PZT volumetric densities. Neglecting the effect of gravity due to planar motion and the higher order terms of quadratic in w′ (Esmailzadeh & Jalili, 1998),

Derivation of the controller

Utilizing , , the truncated two-mode representation of the beam with PZT model reduces tomb+mt+0Lρ(x)dxs̈(t)+m1q̈1(t)+m2q̈2(t)=f(t),m1s̈(t)+md1q̈1(t)+ω12md1q1(t)−hl21′(l2)−φ1′(l1))βl(l2−l1){(φ1′(l2)−φ1′(l1))q1(t)+(φ2′(l2)−φ2′(l1))q2(t)}=−hlb(φ1′(l2)−φ1′(l1))βlv(t),m2s̈(t)+md2q̈2(t)+ω22md2q2(t)−hl22′(l2)−φ2′(l1))βl(l2−l1){(φ1′(l2)−φ1′(l1))q1(t)+(φ2′(l2)−φ2′(l1))q2(t)}=−hlb(φ2′(l2)−φ2′(l1))βlv(t).The equations of motion given in , , can be written in the following more compact formMΔ̈+KΔ=Fe,

Controller implementation and observer design

The control input v(t) requires the information from the velocity related signals q̇1(t) and q̇2(t), which are usually not measurable. Sun and Mills solved the problem by integrating the acceleration signals measured by the accelerometers (Sun & Mills, 1999). However, such controller structure may result in unstable closed-loop system in some cases. In this paper, a reduced-order observer is designed to estimate the velocity signals q̇1 and q̇2. For this, the following three available signals

Numerical simulations

In order to show the effectiveness of the controller, the flexible beam structure in Fig. 1 is considered with the PZT actuator attached on beam surface. The system parameters are taken as listed in Table 1.

First, the beam is considered without PZT controller. For this, the PD control gains are taken as kp=120 and kd=20. Fig. 3 shows the results for the beam without PZT control (i.e., only with PD force control applied on the base). Next, the PZT controller given by (20) along with the input

Control experiments

In order to better demonstrate the effectiveness of the controller, an experimental setup is constructed and used to verify the numerical results. The experimental apparatus consists of a flexible beam with PZT actuator and strain sensor attachments, data acquisition, amplifier, signal conditioner and the control software. As shown in Fig. 6, the plant consists of a flexible Aluminum beam with a strain sensor and a PZT patch actuator bonded on each side of the beam surface. One end of the beam

Conclusions

A reduced-order observer based controller has been presented for a flexible cantilever beam with PZT patch actuator attachment and subjected to a moving base. Taking into account both electrical and mechanical fields for the PZT patch actuator, a full-order model was developed using assumed mode model expansion and the Lagrangian approach. The control structure was designed based on the truncated two-mode beam model. A simple PD controller was selected for the moving base input force, and the

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