Shock and Vibration

Volume 2016 (2016), Article ID 4780181, 17 pages

http://dx.doi.org/10.1155/2016/4780181

## Experimental Study of Active Vibration Control of Planar 3-__R__RR Flexible Parallel Robots Mechanism

^{1}Department of Mechatronics, Foshan University, Foshan, Guangdong 528000, China^{2}Key Laboratory of Precision Equipment and Manufacturing Technology of Guangdong Province, Wushan Road, Tianhe District, Guangzhou 510641, China

Received 21 April 2016; Revised 15 July 2016; Accepted 20 July 2016

Academic Editor: Francesco Ripamonti

Copyright © 2016 Qinghua Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

An active vibration control experiment of planar 3-__R__RR flexible parallel robots is implemented in this paper. Considering the direct and inverse piezoelectric effect of PZT material, a general motion equation is established. A strain rate feedback controller is designed based on the established general motion equation. Four control schemes are designed in this experiment: three passive flexible links are controlled at the same time, only passive flexible link 1 is controlled, only passive flexible link 2 is controlled, and only passive flexible link 3 is controlled. The experimental results show that only one flexible link controlled scheme suppresses elastic vibration and cannot suppress the elastic vibration of the other flexible links, whereas when three passive flexible links are controlled at the same time, they are able to effectively suppress the elastic vibration of all of the flexible links. In general, the experiment verifies that a strain rate feedback controller is able to effectively suppress the elastic vibration of the flexible links of plane 3-__R__RR flexible parallel robots.

#### 1. Introduction

Recently, flexible parallel robots have been intensively studied because of their excellent performance. However, the elastic vibration problem of such robots is serious due to their inertial and driving forces. As a result, suppressing the unwanted elastic vibration is currently a very significant and challenging problem.

There are basically five design philosophies for suppressing unwanted elastic vibration that are classified into passive vibration control and active vibration control [1]. Passive vibration control means that unwanted vibration is suppressed by using advanced composite materials, optimizing the cross-sectional geometry of the elements, or using additional damping materials. Active vibration control means that unwanted vibration is suppressed by introducing a microprocessor-controlled actuator into the original mechanism or embedding some smart structures (actuators and sensors) into flexible elements. The surface-bonded lead zirconate titanate (PZT) sensor/actuator has been widely used for active vibration control of the flexible structure [2, 3]. PZT materials are able to dampen vibration and measure the vibration of distributed parameter systems because of their direct and inverse piezoelectric effect [4]. Photostrictive actuator as a new promising noncontact photoactuation technique was used for suppressing vibration of flexible structures [5].

During the past few decades, the question of active vibration control of flexible robots has been paid considerable attention, as indicated in survey papers [6–9]. Although there are many theoretical and experimental studies of active vibration control in space-based flexible structures and simple flexible beams [10–13], PZT materials are introduced into the vibration control of manipulators or mechanisms that only have a single link with a single actuator and sensor bonded [14, 15]. Meanwhile, compared with numerical simulation, the experimental study of the active vibration control of flexible manipulators is a more challenging work, especially for flexible parallel robots. Zhang et al. [16] have performed an experimental study of active vibration control of 3-PRR flexible parallel robots, for which the elastic vibration of flexible links during motion is suppressed by an SRF controller. The KED assumption is utilized. Zhang et al. [17] address the dynamic modeling and efficient modal control of a planar parallel manipulator (PPM) with three flexible linkages actuated by linear ultrasonic motors (LUSM). We performed studied dynamic modeling and dynamic analysis of planar 3-__R__RR flexible parallel robots [18–21].

The residual vibrations of the flexible links of planar 3-__R__RR flexible parallel robots are studied in this paper. In fact, the amplitudes of the residual vibration are greater than the elastic vibration during the motion and will be verified in the experimental study of this paper, and these residual vibrations influence the repeat positioning accuracy of the system seriously. Thus, it is important to suppress the residual vibration of flexible links to improve the repeat positioning accuracy of the system. Essentially, the dynamic responses measured by PZT sensors include unmodeled or unknown dynamics. Therefore, the vibrations of the flexible elements are very complicated and contain many frequency components that are closely spaced. To measure the modal coordinates or velocities more precisely, it is desirable to use as many PZT sensors as possible. However, the number of PZT sensors is limited by the available space of the flexible links and the real-time computing power of the computer. A trade-off must be made between the real-time computing power of the computer hardware and the number of sensors [22].

To solve the above problems, an experimental study of the active vibration control of planar 3-__R__RR parallel robots with three flexible links, each of which bonds with two pairs of PZT actuators and one PZT sensor film, is implemented in this paper. The moving platform of the planar parallel robots moves along a given trajectory, control experiments are implemented in real-time, and the controller is activated when rigid motion stops. The active control system consists of an industrial computer, a dSPACE controller with many I/O ports, smart flexible links with PZT actuators and PZT sensors, and a PZT driven power and charge amplifier. A strain rate feedback control algorithm is adopted to suppress the elastic vibration of the flexible links of the system [20].

The remainder of the paper is organized as follows. The rigid-elastic coupling dynamic model is established in Section 2. According to the direct and inverse piezoelectric equation of the PZT material, a strain rate feedback control algorithm is given. Section 3 introduces an experimental system setup and describes the working principle of the controller system and each component. In Section 4, the results of an experiment of vibration control are presented. Four experiment schemes are designed. The control performance of the four experiment schemes is analyzed; the experimental results verify the validity of the strain rate feedback controller for suppressing the residual vibration of flexible links. Finally, the conclusions are given in Section 5.

#### 2. Dynamic Model of Planar 3-__R__RR Flexible Parallel Robots

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##### 2.1. Sketch of Planar 3-RR Parallel Robots

A sketch of the planar flexible 3-__R__RR parallel robots is constructed by the regular triangle moving platform , the static platform, and three symmetrical kinematic chains, , , and , as shown in Figure 2. Each kinematic chain has one active revolute (__R__) joint, followed by two consecutive passive revolute (R) joints. The active revolute joints are installed at , . , , and are the regular triangle’s three vertices. and . The vertices and are centers of the regular triangles and , respectively.* O-XY* is the global fixed frame. The parameters and are the angles between the -axis of the fixed frame and linkages and , respectively. is the angle between the -axis of the fixed frame and side of the regular triangle . , , , and are the lengths of the segments , , , and .

##### 2.2. Motion Equations of the Beam Element

The flexible link can be modeled by connecting a series of beam elements. Figure 2 shows a beam element before and after deformation.* O-XY* is the global fixed frame, and the* A-xy* is the local moving frame, with the -axis coincident with the neutral line of the beam element. The original point is located at one node of the beam element before deformation.* B* is another node of the beam element. The - system is an intermediate coordinate frame, the origin of which is rigidly attached to the origin of the* O-XY* and the axes of which are parallel to the axes of the local moving frame* A-xy*. is the angle between the global fixed frame* O-XY* and the intermediate coordinate frame -.

Considering the general point in the element, let point be the corresponding point on the neutral line. Points and are their respective positions after deformation. The elastic deformation of the point in - is given by where is the nodal displacement vector, in which and are the axial displacements of two nodes* A and B*, respectively; and are the lateral displacements; and are the elastic rotational angles; and are the section curvatures; and is the shape function matrix. Assuming that the axial displacement of point is a linear function, the lateral displacement is a fifth-order hermit function, yieldingwhere

The deformation displacements of in* A-xy* can be written as [23]

Let . Then, the displacement of can be expressed in the global fixed frame bywhere is the direction cosine matrix, that is, the - system with respect to the - system, and is given by , where the vector is the location coordinates of the point in the - system and is given by , where the superscript T indicates the matrix transpose and the coupling quantity is the axial displacement caused by the transverse displacement of beam. The parameter can be obtained by taking the first derivative onwhere .

Equation (6) can be expressed compactly aswhere .

###### 2.2.1. Kinetic Energy of the Beam Element

The kinetic energy of the beam element is mainly composed of translational energy and rotational energy. According to (7), the kinetic energy of the beam element is written as where is the material density, is the volume of the beam element, is the absolute rotation angle of small block in the global fixed frame -, is the moment of inertia about the center of mass of the beam element, and are the lumped masses of at the ends of the beam element, and are the lumped moment of inertia at the ends of beam element, and and are the absolute rotation angle of the two ends of the beam element in the global fixed frame -., , , , , , , , , , , , is a skew symmetric matrix defined as is the section moment of inertia of the beam element, and is the two-order unit matrix.

###### 2.2.2. Strain Energy of the Beam Element

Nonlinear terms in the strain-displacement relationship are neglected [24]. Thus, the strain energy of the beam element can be written aswhere , , and is the elastic modulus of materials. and are the cross-sectional moment of inertia and the cross-sectional area of the beam element, respectively.

###### 2.2.3. Motion Equations of the Beam Element

According to Lagrange’s equation, the dynamic equation of the beam element can be derived aswhere and are the generalized external forces and the quadratic velocity vector that contains the gyroscopic and the Coriolis force components, respectively.

The elemental dynamic equation has been established in the* A-xy* system. Before forming the dynamic equation of the system, (13) must be expressed in the* O-XY* system. Defining the coordinate transformation matrix Letwhere is the elemental nodal coordinate vector in the* O-XY* system. By taking the first and the second derivatives of (16) with respect to time,where and . Substituting (16) and (17) into (13) and premultiplying by matrix , the dynamic equation of the beam element can be expressed in the* O-* system bywhere and and are the element generalized mass and the stiffness matrices in the* O-XY* system, respectively. is the quadratic velocity vector in the* O-XY* system. is the generalized external forces vector in the* O-XY* system.

##### 2.3. Constraint Equations

Different from the simple structure, the constraint relationships of the planar 3-__R__RR of flexible parallel robots, which include the rigid-body motion constraints, elastic deformation motion constraints, and dynamic constraints of the moving platform, are very complex and can be used to eliminate the correlation of the generalized coordinates.

###### 2.3.1. Constraint Equations of the Rigid-Body Motion

The generalized coordinates are formed by rigid-body motion coordinates and elastic coordinates. As shown in Figure 1, the rigid-body motion coordinates include drive joint rotation angles , passive joint rotation angles , and translation displacement and rotation angle of the moving platform , and rigid-body motion coordinate vectors , , and are not independent. Because three kinematic chains of 3-__R__RR parallel robots are full symmetrical chains, only one chain is studied. As shown in Figure 3, a closed-loop vector equation can be established:Projecting (19) into and components yieldswhere and are the coordinates of the points and in the* O-XY* system, respectively.