Abstract

The piezoelectric actuators are used to investigate the active vibration control of flexible manipulators in this paper. Based on the assumed mode method, piezoelectric coupling model, and Hamilton’s principle, the dynamic equation of the single flexible manipulator (SFM) with surface bonded actuators is established. Then, a singular perturbation model consisted of a slow subsystem and a fast subsystem is formulated and used for designing the composite controller. The slow subsystem controller is designed by fuzzy sliding mode control method, and the linear quadratic regulator (LQR) optimal control method is used to design fast subsystem controller. Furthermore, the changing trends of natural frequencies along with the changes in the position of piezoelectric actuators are obtained through the ANSYS Workbench software, by which the optimal placement of actuators is determined. Finally, numerical simulations and experiments are presented. The results demonstrate that the method of optimal placement is feasible based on the maximal natural frequency, and the composite controller presented in this paper can not only realize the trajectory tracking of the SFM and has a good result on the vibration suppression.

1. Introduction

In recent years, the robotic technology has been widely used in many areas, such as aerospace, industry, and medical treatment. Due to the characteristics of high speed, high precision, and high loading weight ratio, flexible structures have received great attention in the robot areas [1, 2]. The elastic deformations of flexible manipulators couple with the rigid motions of rigid bodies in the light-duty and high-speed flexible manipulators system [1], and they have obvious influence on the positioning precision of the equipment. Therefore, lots of scholars have done tons of researches on the vibration suppression of flexible manipulators. Smart piezoelectric material, especially the piezoelectric ceramic, has a superior characteristic of dual functions of actuating and sensing with excellent mechanical-electrical coupling characteristics which are widely used in the study of active vibration control of flexible manipulators [3].

The dynamical model of flexible manipulators has the features of being time-varying, strong coupling, and highly nonlinear which brings insuperable difficulties of the single controller design and it is hard to get ideal control effect [4]. Besides, the trajectory tracking and vibration suppression of flexible manipulators need to be controlled at the same time during movement. Therefore, to realize the trajectory tracking and vibration suppression while simplifying the design of controller and improving the robustness properties, the singular perturbation theory has been shown to be a convenient strategy for “reduced-order modelling” in recent years [5]. Based on the singular perturbation theory, the highly nonlinear and strongly coupled dynamic equation of flexible manipulators is decomposed into a slow subsystem corresponding to the rigid body and a fast subsystem describing the flexible motion. Then according to the characteristics of the two subsystems, the controllers are designed independently. For example, Siciliano and Book [4] proposed a state feedback controller based on the singular perturbation model of single flexible manipulator (SFM), and the trajectory tracking and vibration suppression was realized through the way of controlling the motor torque. Analogously, singular perturbation adaptive controller of multiple flexible links manipulators was designed by Yu and Chen [6]. Lin and Lewis [5] carried on the control simulations of flexible link robot arms and the control torque was obtained through two-time scale fuzzy logic controller. Computed torque control and a composite control were utilized to reduce mechanical vibrations of SFM by Jia et al. [7]. In the composite control system, a dynamic sliding mode controller was designed for slow subsystem, and optimal controller was designed for fast subsystem. From the above it can be seen that all the active vibration controls of flexible manipulators were realized by the way of controlling the motor torque, and most scholars less focused on the composite control of SFM with piezoelectric actuators based on the singular perturbation theory.

Moreover, experiments and simulations results have proved that the positions of the actuators have important influence on vibration suppression effect [8], and the research hotspots focus on the optimal criterion and solution algorithm. Qiu et al. [9] and Leleu et al. [10] have conducted research on the control indexes of open-loop control system based on controllability and observability. The performance indexes of the closed-loop control system were studied from the angle of the control energy and vibration energy [11–13]. After establishing optimal criterion, many intelligent search algorithms such as genetic algorithm, particle swarm optimization algorithm, and ant colony algorithm can be applied to solve the optimal problem of actuators or sensors. When the size or number of piezoelectric actuators and sensors reaches a certain extent, the natural frequencies of flexible manipulators will be changed and the system stability is reduced. Zhang et al. [14] analyzed the natural frequencies of a piezoelectric beam by the general finite element software ANSYS, the modal frequencies of the system are changed at a certain extent, and the change of modal frequencies explains that the piezoelectric actuators have some influence on the natural frequencies of the host structure. Therefore, it is necessary to analyze the effect of the piezoelectric actuators or sensors which are bonded on flexible manipulators. Abreu et al. [15] researched the influence of the position of piezoelectric actuators on the natural frequencies by experiment method. Waisman and Abramovich [16] investigated the influence of the actuators by means of the pin-force model, and numerical experiments demonstrate that the natural frequencies and vibration modes of the structure can be actively altered using the piezoelectric actuators. Spier et al. [17] analyzed the locations of actuators and sensors with the fundamental frequency and the frequency gaps between the higher order frequencies. A review of the literature reveals that the actuators or sensors can change the structural characteristics of flexible manipulators. Therefore, the location of optimization of piezoelectric actuator based on the index of natural frequency is useful to avoid resonance when the excitation frequency is around the natural frequency. However, many scholars tend to neglect this in the study of optimal placement of actuators or sensors.

This paper aims to investigate the modelling and composite control of the SFM with piezoelectric actuators and realizes the trajectory tracking and vibration suppression of flexible manipulators based on the composite controller which is designed by singular perturbation theory. Furthermore, the optimal placement of piezoelectric actuators is done through the index of the maximal natural frequency. The paper is organized as follows: “Dynamic Model of the SFM” is illuminated in the following section. “Singular Perturbation Decomposition of Dynamic Equation” presents the reduction of dynamic equation. Subsequently, a compound controller is designed for trajectory tracking and vibration suppression of the SFM in “Design of the Composite Controller” of this paper. Then, the location optimization of piezoelectric actuators, simulation, and experiments results of the composite control method are given in “Optimal Placement and Composite Control of the SFM.” The paper is concluded with a brief summary in last section.

2. Dynamic Model of the SFM

The structural diagram of the SFM in the horizontal plane studied in this paper is shown in Figure 1 and two actuators which are dual surface bonded on the SFM are denoted as a group. The specifications of the actuators are chosen to be the same and they are made of piezoelectric ceramic. In the figure, the coordinates and indicate the inertial frame and the reference frame, respectively. And is the output torque of the speed reducer which is driven by a motor, is the rotational inertia of the hub (motor shaft and fixture), is the radius of the hub, is the angular displacement of the hub, and is the tip mass of the SFM. Besides, , , , , and denote the elastic modulus, density, length, width of cross section, and height of cross section of the SFM, respectively. , , , , and denote the elastic modulus, density, length, width of cross section, and height of cross section of the actuators, respectively.

Figure 1 

Structural diagram of the SFM.

As shown in Figure 1, the position vector of point with respect to the reference frame can be represented as [18]where is the rotational transformation matrix and represents the transversal deformation of the SFM.

The velocity vectors of point can be obtained by (1) as follows:

The kinetic energy of the SFM is written aswhere is the coordinate of the position vector from to the tip mass, is the starting position of the th actuator, and is the ending position of the th actuator.

Based on Euler–Bernoulli theory, the potential energy is given by

As shown in Figure 2, for the model of the SFM with surface bonded actuators, cross section strain distribution along the thickness is linear and using Hooke’s law to obtain an expression for stress in terms of strain:where is the curvature radius of the neutral layer and is the distance from the neutral plane.

Figure 2 

Model of the SFM with surface bonded actuators.

Because the model is symmetrical, taking the upper part of the SFM with surface bonded actuators as an example to empirical analysis and the moment equilibrium equation about the neutral plane is given as [19]

The bending moment of the th actuator can be determined using (12):where

The virtual work carried out by the torque and actuators acting on the SFM can be represented by [20]where is a bending moment generated by the th actuator. is a positive constant representing the bending moment per volt. is the control voltage applied to the th actuator. The term denotes the slope difference between two ends of the th actuator. For simplicity, we define .

In accordance with the assumed mode method, is written aswhere is the th mode shapes and is the th mode generalized coordinates.

The governing equation of motion can be obtained through the application of the Hamilton’s principleBy substituting (4), (5), and (10) into (12), the dynamics equation of the SFM at the element level in compact form can be written aswhere

3. Singular Perturbation Decomposition of Dynamic Equation

Since the inertia matrix M in (13) is positively definite, it can be inverted and denoted by N, which can be partitioned [7].Hence, (13) can be decomposed as follows:

Assuming that is the smallest element of the stiffness matrix , then define new variable . When is big enough, . Then the new variables are defined as [7]Equation (17) is substituted into (16), the dynamics equation of the SFM can be represented in singular perturbation form as

3.1. Slow Subsystem

To derive the boundary layer correction, is set to zero and the model of the rigid manipulator is expressed as

Combining (15), (20), and (21), the slow subsystem is given bywhere is the control torque of slow time scale subsystem and is the control voltage of slow time scale subsystem.

3.2. Fast Subsystem

To derive the fast subsystem, introduce a fast time scale and define boundary layer variables as [21]Then (19) is written as

Because the slow time scale and fast time scale are independent of each other, near boundary layer region and the slow varying component can be regarded as constant (). Assuming that and (21) are substituted into (24a) and (24b), the fast subsystem is expressed aswhere is the control torque of fast time scale subsystem and is the control voltage of fast time scale subsystem.

4. Design of the Composite Controller

The control objective is to track the desired trajectory and suppress the vibration of the SFM. As shown in Figure 3, the control torque and control voltage for trajectory tracking are got by slow subsystem controller and the control torque and control voltage for vibration suppression are got by fast subsystem controller. The total inputs of controller are gained as and . In this section, the slow subsystem controller is designed by fuzzy sliding mode control method, and the linear quadratic regulator (LQR) optimal control method is used to design fast subsystem controller.

Figure 3 

Structure diagram of the composite controller.

4.1. Slow Subsystem Controller Design

Assuming the desired trajectory as , the trajectory tracking error and speed tracking error can be written as [22]Define a sliding surface aswhere is positive proportionality coefficient.

Obviously, if the slow subsystem states stay on the sliding mode surface, it ensures that and and can realize the trajectory tracking control. The exponential reaching law function is shown asTaking the derivative of the sliding surface and (22) which are substituted into it, then (29) is obtained asEquation (28) is substituted into (29); the control law is expressed as

A sliding mode controller has two phases; the first is called the reaching phase. During this phase the controller is guiding the system state onto the sliding surface. The second phase is called the sliding phase, during which the system state “slides” along the sliding surface [23]. Figure 4 shows the generic sliding surface. Therefore, the chattering with high frequency in sliding mode control systems is a barrier for the application to the practice engineering problems. It is known that, based on the fuzzy sets, fuzzy language variables, and fuzzy logic inference, as a popular nonlinear control theory, fuzzy control has been widely used in industrial control systems. Using fuzzy control method, the switching variable of sliding mode control can be adjusted adaptively according to expertise, and it can effectively weaken the chattering phenomenon and strengthen the robustness of system.

Figure 4 

Generic sliding surface.

The adaptive fuzzy sliding mode controller is illustrated in Figure 5. In the sliding mode controller, the switching variable is selected as a time-varying value which is denoted as and realized by fuzzy controller. Based on the two-dimensional fuzzy controller, the sliding surface and its derivative are selected as the inputs of fuzzy controller, and the switching variable is selected as the output of fuzzy controller. Linguistic variables correspond to seven different linguistic terms , abbreviation of “Negative Big,” “Negative Middle,” “Negative Small,” “Zero,” “Positive Small,” “Positive Middle,” “Positive Big,” respectively. In addition, one should define the relevant proportional factors from the experimental test data to linguistic levels. The linguistic variables , , and represent sliding surface , derivative of the sliding surface , and switching variable in the fuzzy set [24]. The universes of discourses of , , and are from −1 to 1, −1 to 1, and 0 to 1, respectively. Then [24]where and is the maximum value of the sliding surface .where and is the maximum value of derivative of the sliding surface .where and is the maximum value allowed for switching variable .

Figure 5 

Adaptive fuzzy sliding mode controller.

, , and are chosen from numerical simulations.

Seven linguistic levels (NB, NM, NS, ZO, PS, PM, and PB) are used to represent the input domain and output domain with their membership values lying between 0 and 1 [24]. The basic triangular and Z forms are chosen for the input and output membership functions [25, 26]. The membership functions are employed to convert these input and output variables into linguistic control variables (NB, NM, NS, ZO, PS, PM, and PB) and the membership functions are shown in Figure 6.

(a)

(b)

(c)

(d)

(a)
(b)
(c)
(d)

Figure 6 

Membership functions of fuzzy controller ((a) linguistic variable ; (b) linguistic variable ; (c) linguistic variable ; (d) schematic of fuzzy inference).

Table 1 shows the generated fuzzy inference rules of fuzzy controller [25], and the surface of the fuzzy rules is shown in Figure 7. The fuzzy inference rules are designed based on the changes of the sliding surface and its derivative which are obtained by simulations.

Figure 7 

Surface of the fuzzy rules.

4.2. Fast Subsystem Controller Design

Define boundary layer variables as ; (25a) and (25b) are rewritten aswhereIt can be seen that the fast subsystem, as described by (34), is in a linear feedback form, for which the fast subsystem controller is designed using LQR method. The optimal control block diagram of the fast subsystem is shown in Figure 8.

Figure 8 

Optimal control block diagram of the fast subsystem.

To damp out the flexible vibration and decrease the control effort, the quadratic objective function is chosen aswhere and are weighting matrix for state variables and control effort, respectively, and represents the input or control variable of the fast subsystem system. Then, the optimum control input for fast subsystem is determined as [21]where is the feedback gain matrix and is the solutions of the Riccati equation as follows: