# Modelling and Composite Control of Single Flexible Manipulators with Piezoelectric Actuators

Authors: En Lu; Wei Li; Xuefeng Yang; Mengbao Fan; Yufei Liu
Journal: Shock and Vibration
 (2016)

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## 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.


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## Body

## 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 Figure1 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 XY and xy indicate the inertial frame and the reference frame, respectively. And τ(t) is the output torque of the speed reducer which is driven by a motor, Jh is the rotational inertia of the hub (motor shaft and fixture), r is the radius of the hub, θ(t) is the angular displacement of the hub, and mL is the tip mass of the SFM. Besides, Eb, ρb, Lb, bb, and hb denote the elastic modulus, density, length, width of cross section, and height of cross section of the SFM, respectively. Ep, ρp, Lp, bp, and hp 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 Figure1, the position vector of point P with respect to the reference frame xy can be represented as [18](1)r=Θr0+xwx,t,(2)Θ=cos⁡θ-sin⁡θsin⁡θcos⁡θ,where Θ is the rotational transformation matrix and w(x,t) represents the transversal deformation of the SFM.The velocity vectors of pointP can be obtained by (1) as follows:(3)r˙=Θ˙r0+xwx,t+Θ0w˙x,t.The kinetic energy of the SFM is written as(4)T=12Jhθ˙2+12∫0LρbAbr˙Tr˙dx+12∑i=1n∫ai1ai2ρpApr˙Tr˙dx+12mLr˙LTr˙L,where rL is the coordinate of the position vector from O to the tip mass, ai1 is the starting position of the ith actuator, and ai2 is the ending position of the ith actuator.Based on Euler–Bernoulli theory, the potential energy is given by(5)V=12∫0LEbIb∂2wx,t∂x22dx+12∑i=1n∫ai1ai2EpIp∂2wx,t∂x22dx.As shown in Figure2, 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:(6)σb=Ebεb=Eb1ρy,σP=EPεb-εp=EP1ρy-d31hpUi,where ρ is the curvature radius of the neutral layer and y 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](7)∫0hb/2yσbdy+∫hb/2hb/2+hpyσpdy=0,1ρ=Ep12hbhp+12hp2d31/hpUiEbhb3+Ep6hb2hp+12hbhp2+8hp3.The bending moment of theith actuator can be determined using (12):(8)Mi=EbIb+2EpIp1ρ=Ebhb3bb+2Ephp3bpEphbhp+hp2d31/hpEbhb3+Ep6hb2hp+12hbhp2+8hp3Uit=κUit,where(9)κ=Ebhb3bb+2Ephp3bpEphbhp+hp2d31/hpEbhb3+Ep6hb2hp+12hbhp2+8hp3.The virtual work carried out by the torqueτ(t) and actuators acting on the SFM can be represented by [20](10)δW=τtδθ+∑i=1nκUitδθ+∑i=1nκUitΦ˙ai2-Φ˙ai1δq˙,where κUi(t) is a bending moment generated by the ith actuator. κ is a positive constant representing the bending moment per volt. Ui(t) is the control voltage applied to the ith actuator. The term (Φ˙(ai2)-Φ˙(ai1))δq˙ denotes the slope difference between two ends of the ith actuator. For simplicity, we define Φ˙(ai2)-Φ˙(ai1)=Φ˙~(i).In accordance with the assumed mode method,w(x,t) is written as(11)wx,t=∑i=1∞ϕjxqjt,where ϕj(x) is the jth mode shapes and qj(t) is the jth mode generalized coordinates.The governing equation of motion can be obtained through the application of the Hamilton’s principle(12)∫t0tfδT-δV+δWdt=0.By substituting (4), (5), and (10) into (12), the dynamics equation of the SFM at the element level in compact form can be written as(13)Mθθθ,qMθqθ,qMqθθ,qMqqθ,qθ¨q¨+hθθθ,q,θ˙,q˙hθqθ,q,θ˙,q˙hqθθ,q,θ˙,q˙hqqθ,q,θ˙,q˙θ˙q˙+000Kqqθq=τt+gθUgqU,where(14)Mθθ=Jh+13ρbAbr+L3-r3+13∑i=1nρpAPr+ai23-r+ai13+qTMffq+mLr+L2,Mθq=∫0LρbAbr0+xΦxdx+∑i=1n∫ai1ai2ρpAPr0+xΦxdx+mLr0+LΦL,Mqθ=Mθq-1,Mqq=∫0LρbAbΦxTΦxdx+∑i=1n∫ai1ai2ρpAPΦxTΦxdx+mLΦTLΦL,hθθ=2q˙TMqqq,hθq=θ˙qTMqq,hqθ=-Mqqqθ˙,hqq=0,Kqq=∫0LEbIbΦ¨TxΦ¨xdx+∑i=1n∫ai1ai2EpIPΦ¨TxΦ¨xdx,gθ=κ1⋯1,gq=κΦ˙~1⋯Φ˙~n.


## 3. Singular Perturbation Decomposition of Dynamic Equation

Since the inertia matrixM in (13) is positively definite, it can be inverted and denoted byN, which can be partitioned [7].(15)Nθ,q=Nθθθ,qNθqθ,qNqθθ,qNqqθ,q=M-1=MθθMθqMqθMqq-1.Hence, (13) can be decomposed as follows:(16)θ¨=-Nθθθ,qhθθθ,q,θ˙,q˙θ˙-Nθθθ,qhθqθ,q,θ˙,q˙q˙-Nθqθ,qhqθθ,q,θ˙,q˙θ˙-Nθqθ,qhqqθ,q,θ˙,q˙q˙-Nθqθ,qKqqq+Nθθθ,qτ+Nθθθ,qgθ+Nθqθ,qgqU,q¨=-Nqθθ,qhθθθ,q,θ˙,q˙θ˙-Nqθθ,qhθqθ,q,θ˙,q˙q˙-Nqqθ,qhqθθ,q,θ˙,q˙θ˙-Nqqθ,qhqqθ,q,θ˙,q˙q˙-Nqqθ,qKqqq+Nqθθ,qτ+Nqθθ,qgθ+Nqqθ,qgqU.Assuming thatkc is the smallest element of the stiffness matrix K, then define new variable ε=1/kc. When kc is big enough, ε→0. Then the new variables are defined as [7](17)ξ=qε2,K~qq=ε2Kqq.Equation (17) is substituted into (16), the dynamics equation of the SFM can be represented in singular perturbation form as(18)θ¨=-Nθθθ,ε2ξhθθθ,ε2ξ,θ˙,ε2ξ˙θ˙-ε2Nθθθ,ε2ξhθqθ,ε2ξ,θ˙,ε2ξ˙ξ˙-Nθqθ,ε2ξhqθθ,ε2ξ,θ˙,ε2ξ˙θ˙-ε2Nθqθ,ε2ξhqqθ,ε2ξ,θ˙,ε2ξ˙ξ˙-Nθqθ,ε2ξK~qqξ+Nθθθ,ε2ξτ+Nθθθ,ε2ξgθ+Nθqθ,ε2ξgqU,(19)ε2ξ¨=-Nqθθ,ε2ξhθθθ,ε2ξ,θ˙,ε2ξ˙θ˙-ε2Nqθθ,ε2ξhθqθ,ε2ξ,θ˙,ε2ξ˙ξ˙-Nqqθ,ε2ξhqθθ,ε2ξ,θ˙,ε2ξ˙θ˙-ε2Nqqθ,ε2ξhqqθ,ε2ξ,θ˙,ε2ξ˙ξ˙-Nqqθ,ε2ξK~qqξ+Nqθθ,ε2ξτ+Nqθθ,ε2ξgθ+Nqqθ,ε2ξgqU.


### 3.1. Slow Subsystem

To derive the boundary layer correction,ε is set to zero and the model of the rigid manipulator is expressed as(20)θ¨=-Nθθθ,0hθθθ,0,θ˙,0θ˙-Nθqθ,0hqθθ,0,θ˙,0θ˙-Nθqθ,0K~qqξ+Nθθθ,0τs+Nθθθ,0gθ+Nθqθ,0gqUs,(21)0=-Nqθθ,0hθθθ,0,θ˙,0θ˙-Nqqθ,0hqθθ,0,θ˙,0θ˙-Nqqθ,0K~qqξ+Nqθθ,0τs+Nqθθ,0gθ+Nqqθ,0gqUs.Combining (15), (20), and (21), the slow subsystem is given by(22)θ¨=M-θθ-1-h-θθθ˙+τs+gθUs,where τs is the control torque of slow time scale subsystem and Us is the control voltage of slow time scale subsystem.


### 3.2. Fast Subsystem

To derive the fast subsystem, introduce a fast time scaleγ=t/ε and define boundary layer variables as [21](23)η1=ξ-ξ-,η2=εξ˙.Then (19) is written as(24a)dη1dγ=η2,(24b)dη2dγ=-Nqθθ,ε2η1+ξ-hθθθ,ε2η1+ξ-,θ˙,εη2θ˙-εNqθθ,ε2η1+ξ-hθqθ,ε2η1+ξ-,θ˙,εη2η2-Nqqθ,ε2η1+ξ-hqθθ,ε2η1+ξ-,θ˙,εη2θ˙-εNqqθ,ε2η1+ξ-hqqθ,ε2η1+ξ-,θ˙,εη2η2-Nqqθ,ε2η1+ξ-K~qqη1+ξ-+Nqθθ,ε2η1+ξ-τ+Nqθθ,ε2η1+ξ-gθ+Nqqθ,ε2η1+ξ-gqU.Because the slow time scale and fast time scale are independent of each other, near boundary layer regionε→0 and the slow varying component can be regarded as constant (dξ/dγ=εξ=0). Assuming that ε=0 and (21) are substituted into (24a) and (24b), the fast subsystem is expressed as(25a)dη1dγ=η2,(25b)dη2dγ=-N-qqK~qqη1+N-qθτf+gθUf+N-qqgqUf,where τf is the control torque of fast time scale subsystem and Uf is the control voltage of fast time scale subsystem.


## 3.1. Slow Subsystem

To derive the boundary layer correction,ε is set to zero and the model of the rigid manipulator is expressed as(20)θ¨=-Nθθθ,0hθθθ,0,θ˙,0θ˙-Nθqθ,0hqθθ,0,θ˙,0θ˙-Nθqθ,0K~qqξ+Nθθθ,0τs+Nθθθ,0gθ+Nθqθ,0gqUs,(21)0=-Nqθθ,0hθθθ,0,θ˙,0θ˙-Nqqθ,0hqθθ,0,θ˙,0θ˙-Nqqθ,0K~qqξ+Nqθθ,0τs+Nqθθ,0gθ+Nqqθ,0gqUs.Combining (15), (20), and (21), the slow subsystem is given by(22)θ¨=M-θθ-1-h-θθθ˙+τs+gθUs,where τs is the control torque of slow time scale subsystem and Us is the control voltage of slow time scale subsystem.


## 3.2. Fast Subsystem

To derive the fast subsystem, introduce a fast time scaleγ=t/ε and define boundary layer variables as [21](23)η1=ξ-ξ-,η2=εξ˙.Then (19) is written as(24a)dη1dγ=η2,(24b)dη2dγ=-Nqθθ,ε2η1+ξ-hθθθ,ε2η1+ξ-,θ˙,εη2θ˙-εNqθθ,ε2η1+ξ-hθqθ,ε2η1+ξ-,θ˙,εη2η2-Nqqθ,ε2η1+ξ-hqθθ,ε2η1+ξ-,θ˙,εη2θ˙-εNqqθ,ε2η1+ξ-hqqθ,ε2η1+ξ-,θ˙,εη2η2-Nqqθ,ε2η1+ξ-K~qqη1+ξ-+Nqθθ,ε2η1+ξ-τ+Nqθθ,ε2η1+ξ-gθ+Nqqθ,ε2η1+ξ-gqU.Because the slow time scale and fast time scale are independent of each other, near boundary layer regionε→0 and the slow varying component can be regarded as constant (dξ/dγ=εξ=0). Assuming that ε=0 and (21) are substituted into (24a) and (24b), the fast subsystem is expressed as(25a)dη1dγ=η2,(25b)dη2dγ=-N-qqK~qqη1+N-qθτf+gθUf+N-qqgqUf,where τf is the control torque of fast time scale subsystem and Uf 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 Figure3, the control torque τs and control voltage Us for trajectory tracking are got by slow subsystem controller and the control torque τf and control voltage Uf for vibration suppression are got by fast subsystem controller. The total inputs of controller are gained as τ=τs+τf and U=Us+Uf. 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 asθd(t), the trajectory tracking error e(t) and speed tracking error e˙t can be written as [22](26)et=θdt-θt,e˙t=θ˙dt-θ˙t.Define a sliding surface s as(27)s=e˙t+λet,where λ is positive proportionality coefficient.Obviously, if the slow subsystem states stay on the sliding mode surface, it ensures thatθ(t)→θd(t) and θ˙(t)→θ˙d(t) and can realize the trajectory tracking control. The exponential reaching law function is shown as(28)s˙=-ψsgn⁡s-δs,ψ>0,δ>0.Taking the derivative of the sliding surface s˙ and (22) which are substituted into it, then (29) is obtained as(29)s˙=e¨t+λe˙t=θ¨dt-θ¨t+λe˙t=θ¨dt-M-θθ-1τs+gθUs-h-θθθ˙+λe˙t.Equation (28) is substituted into (29); the control law is expressed as(30)τs=M-θθθ¨dt+λθ˙dt-θ˙t+ψsgn⁡s+δs-gθUs+h-θθθ˙.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 Figure5. In the sliding mode controller, the switching variable ψ is selected as a time-varying value which is denoted as Ψ and realized by fuzzy controller. Based on the two-dimensional fuzzy controller, the sliding surface s and its derivative s˙ are selected as the inputs of fuzzy controller, and the switching variable ψ is selected as the output of fuzzy controller. Linguistic variables correspond to seven different linguistic terms {NB,NM,NS,ZO,PS,PM,PB}, 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 S, DS, and Ψ represent sliding surface s, derivative of the sliding surface s˙, and switching variable ψ in the fuzzy set [24]. The universes of discourses of S, DS, and Ψ are from −1 to 1, −1 to 1, and 0 to 1, respectively. Then [24](31)S=k1s,where k1=1/smax and smax is the maximum value of the sliding surface s.(32)DS=k2s˙,where k2=1/s˙max and s˙max is the maximum value of derivative of the sliding surface s˙.(33)ψ=k3Ψ,where k3=ψmax/1 and ψmax is the maximum value allowed for switching variable ψ.Figure 5
Adaptive fuzzy sliding mode controller.s m a x, s˙max, and ψmax 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.Figure 6
Membership functions of fuzzy controller ((a) linguistic variableS; (b) linguistic variable DS; (c) linguistic variable Ψ; (d) schematic of fuzzy inference).
(a) (b) (c) (d)Table1 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 s and its derivative which are obtained by simulations.Table 1
Fuzzy inference rules of fuzzy controller.
Ψ DS S NB NM NS ZO PS PM PB NB PB PM PS ZO PS PM PB NM PM PS ZO NS ZO PS PM NS PS ZO NS NM NS ZO PS ZO ZO NS NM NB NM NS ZO PS PS ZO NS NM NS ZO PS PM PM PS ZO NS ZO PS PM PM PB PM PS ZO PS PM PBFigure 7
Surface of the fuzzy rules.


### 4.2. Fast Subsystem Controller Design

Define boundary layer variables asη=η1,η2T; (25a) and (25b) are rewritten as(34)η˙=Afη+Bfuf,where(35)Af=0I-N-qqK~qq0,Bf=00N-qθN-qθgθ+N-qqgq,uf=τf,UfT.It 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 as(36)J=12∫0∞ηTQη+ufTRufdt,where Q and R are weighting matrix for state variables and control effort, respectively, and uf represents the input or control variable of the fast subsystem system. Then, the optimum control input for fast subsystem is determined as [21](37)uf=-Kfη=-R-1BfTPη,where Kf is the feedback gain matrix and P is the solutions of the Riccati equation as follows:(38)PAf+AfTP-PBfR-1BfTP+Q=0.


## 4.1. Slow Subsystem Controller Design

Assuming the desired trajectory asθd(t), the trajectory tracking error e(t) and speed tracking error e˙t can be written as [22](26)et=θdt-θt,e˙t=θ˙dt-θ˙t.Define a sliding surface s as(27)s=e˙t+λet,where λ is positive proportionality coefficient.Obviously, if the slow subsystem states stay on the sliding mode surface, it ensures thatθ(t)→θd(t) and θ˙(t)→θ˙d(t) and can realize the trajectory tracking control. The exponential reaching law function is shown as(28)s˙=-ψsgn⁡s-δs,ψ>0,δ>0.Taking the derivative of the sliding surface s˙ and (22) which are substituted into it, then (29) is obtained as(29)s˙=e¨t+λe˙t=θ¨dt-θ¨t+λe˙t=θ¨dt-M-θθ-1τs+gθUs-h-θθθ˙+λe˙t.Equation (28) is substituted into (29); the control law is expressed as(30)τs=M-θθθ¨dt+λθ˙dt-θ˙t+ψsgn⁡s+δs-gθUs+h-θθθ˙.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 Figure5. In the sliding mode controller, the switching variable ψ is selected as a time-varying value which is denoted as Ψ and realized by fuzzy controller. Based on the two-dimensional fuzzy controller, the sliding surface s and its derivative s˙ are selected as the inputs of fuzzy controller, and the switching variable ψ is selected as the output of fuzzy controller. Linguistic variables correspond to seven different linguistic terms {NB,NM,NS,ZO,PS,PM,PB}, 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 S, DS, and Ψ represent sliding surface s, derivative of the sliding surface s˙, and switching variable ψ in the fuzzy set [24]. The universes of discourses of S, DS, and Ψ are from −1 to 1, −1 to 1, and 0 to 1, respectively. Then [24](31)S=k1s,where k1=1/smax and smax is the maximum value of the sliding surface s.(32)DS=k2s˙,where k2=1/s˙max and s˙max is the maximum value of derivative of the sliding surface s˙.(33)ψ=k3Ψ,where k3=ψmax/1 and ψmax is the maximum value allowed for switching variable ψ.Figure 5
Adaptive fuzzy sliding mode controller.s m a x, s˙max, and ψmax 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.Figure 6
Membership functions of fuzzy controller ((a) linguistic variableS; (b) linguistic variable DS; (c) linguistic variable Ψ; (d) schematic of fuzzy inference).
(a) (b) (c) (d)Table1 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 s and its derivative which are obtained by simulations.Table 1
Fuzzy inference rules of fuzzy controller.
Ψ DS S NB NM NS ZO PS PM PB NB PB PM PS ZO PS PM PB NM PM PS ZO NS ZO PS PM NS PS ZO NS NM NS ZO PS ZO ZO NS NM NB NM NS ZO PS PS ZO NS NM NS ZO PS PM PM PS ZO NS ZO PS PM PM PB PM PS ZO PS PM PBFigure 7
Surface of the fuzzy rules.


## 4.2. Fast Subsystem Controller Design

Define boundary layer variables asη=η1,η2T; (25a) and (25b) are rewritten as(34)η˙=Afη+Bfuf,where(35)Af=0I-N-qqK~qq0,Bf=00N-qθN-qθgθ+N-qqgq,uf=τf,UfT.It 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 as(36)J=12∫0∞ηTQη+ufTRufdt,where Q and R are weighting matrix for state variables and control effort, respectively, and uf represents the input or control variable of the fast subsystem system. Then, the optimum control input for fast subsystem is determined as [21](37)uf=-Kfη=-R-1BfTPη,where Kf is the feedback gain matrix and P is the solutions of the Riccati equation as follows:(38)PAf+AfTP-PBfR-1BfTP+Q=0.


## 5. Optimal Placement and Composite Control of the SFM

To verify the composite control strategy on the trajectory tracking and vibration suppression of the SFM, numerical simulations are conducted in this section. Owing to the lower order modes play leading role in the vibration of the SFM, only the first two order modes are considered, and the active control simulations are performed using the independent mode space control method. The structural parameters of the SFM and actuators are shown in Tables2 and 3, respectively.Table 2
Structural parameters of SFM.
Parameter Value Unit LengthLb 0.400 m Widthbb 0.035 m Thicknesstb 0.0027 m Densityρb 2.03 × 103 kg/m3 Elastic modulusEb 25.24 GPa Rotational inertiaJh 2.5 × 10−4 kg⋅m2 Hub radiusr 0.0175 m Tip massmL 0.03 kgTable 3
Structural parameters of actuators.
Parameter Value Unit LengthLp 0.04 m Widthbp 0.01 m Thicknesstp 0.0008 m Densityρp 7.65 × 103 kg/m3 Elastic modulusEp 57.3 GPa


### 5.1. Optimal Placement of Actuators

The three-dimensional model of the SFM with surface bonded actuators is established by ANSYS Workbench software, and it is shown as Figure9. Then, the position of the piezoelectric actuator is set as input parameter, and the first-order natural frequency and second-order natural frequency are set as output parameters. Based on the response surface method, the changing trends of natural frequencies along with the position of piezoelectric actuator are shown as Figure 10. From Figure 10(a), the maximum value of the first-order natural frequency is got when the piezoelectric actuator is bonded in the root of the SFM. The maximum value of the second-order natural frequency is in root, and the second maximum value is in the position of x=0.255 m, as shown in Figure 10(b). Hence, the first pair of piezoelectric actuators should be bonded in the root of the SFM and the second pair of piezoelectric actuators should be bonded in the position of x=0.255 mm. That is, a11=0 m and a12=0.04 m; a21=0.255 m and a22=0.295 m.Figure 9
Three-dimensional model of the SFM with surface bonded actuators.Figure 10
Changing trends of natural frequencies along with the position of piezoelectric actuator ((a) first-order natural frequency; (b) second-order natural frequency).
(a) (b)


### 5.2. Numerical Simulations

The angular displacement trajectory of the SFM is shown as(39)θdt=6tte5-15tte4+10tte3θe-θ0+θ0.The control parameters are chosen as follows.(  1) Adaptive Fuzzy Sliding Mode Controller. λ = 50, δ=0.1, k1=5, k2=0.025, k3=3, and ψ is obtained by fuzzy controller.(  2) LQR Controller (40) Q = 350000 0 0 0 0 10000 0 0 0 0 1000 0 0 0 0 100 , R = 2 0 0 3 .The simulation curves of composite control for trajectory tracking and vibration suppression of the SFM are shown in Figure11. Figure 11(a) shows the relation curve of the switching variable ψ with time, which is adjusted by fuzzy controller. Figures 11(b) and 11(c) show the output control of the speed reducer which is driven by a motor and the control voltage of actuators, respectively. Figure 11(d) shows the tip angle of the SFM, and it demonstrates that the composite controller successfully implements the trajectory tracking and vibration suppression of the SFM. Figures 11(e) and 11(f) show the first-order modal coordinate and second-order modal coordinate, respectively. It is obtained that the first-order modal coordinate is much larger than the second-order modal coordinate and illustrates that the lower order modes play leading role in the vibration of the SFM. The tip deflection of the SFM in a time-domain is obtained by the modal coordinates and shown in Figure 11(g). Meanwhile, it is evident from Figure 11(g) that the composite controller can successfully suppress the simultaneous elastic vibration in movement of the SFM. Figure 11(h) shows the structural strain history along the length of SFM for the first 3 s of the manipulator motion. It can be observed that the highest strain experienced occurs in the root of the SFM and the strain is reduced gradually in the control of composite controller.Figure 11
Simulation curves of composite control for trajectory tracking and vibration suppression of the SFM ((a) time-varying control variable; (b) control torque; (c) control voltage; (d) tip angle; (e) first-order modal coordinate; (f) second-order modal coordinate; (g) tip deflection; (h) strain time history).
(a) (b) (c) (d) (e) (f)  (g) (h)


### 5.3. Control Experiments

In order to better demonstrate the effectiveness of the composite controller, the experiments on composite control of the SFM are constructed. The overall experimental setup and the flowchart of composite control experiments are depicted in Figures12 and 13, respectively.Figure 12
Photograph of the experimental apparatus.Figure 13
Flowchart of the composite control experiments.The comparison between the simulated and experimental results of the SFM is shown in Figure14. It can be seen that the results of the experimental results are in good agreement with the simulation result in Figure 11. Figure 14(d) shows the tip angle of the SFM, and it demonstrates that the composite controller successfully achieved the trajectory tracking of the SFM. Meanwhile, the elastic vibration in movement of the SFM is suppressed successfully by the composite controller and shown in Figure 14(e). Figure 14(f) shows that the strain trends of the SFM are got by strain gauges which are bonded below the PZTs, respectively. In addition, it should be noted that the results exhibit a certain error between the simulated and experimental results, and this is mainly caused by the difference between experiment system and dynamic model, such as viscous damping, air resistance, the machining precision of structure, and the precision of sensors.Figure 14
Comparison between the simulated and experimental results of the SFM ((a) time-varying control variable; (b) control torque; (c) control voltage; (d) tip angle; (e) tip deflection; (f) strain trend).
(a) (b) (c) (d) (e) (f)


## 5.1. Optimal Placement of Actuators

The three-dimensional model of the SFM with surface bonded actuators is established by ANSYS Workbench software, and it is shown as Figure9. Then, the position of the piezoelectric actuator is set as input parameter, and the first-order natural frequency and second-order natural frequency are set as output parameters. Based on the response surface method, the changing trends of natural frequencies along with the position of piezoelectric actuator are shown as Figure 10. From Figure 10(a), the maximum value of the first-order natural frequency is got when the piezoelectric actuator is bonded in the root of the SFM. The maximum value of the second-order natural frequency is in root, and the second maximum value is in the position of x=0.255 m, as shown in Figure 10(b). Hence, the first pair of piezoelectric actuators should be bonded in the root of the SFM and the second pair of piezoelectric actuators should be bonded in the position of x=0.255 mm. That is, a11=0 m and a12=0.04 m; a21=0.255 m and a22=0.295 m.Figure 9
Three-dimensional model of the SFM with surface bonded actuators.Figure 10
Changing trends of natural frequencies along with the position of piezoelectric actuator ((a) first-order natural frequency; (b) second-order natural frequency).
(a) (b)


## 5.2. Numerical Simulations

The angular displacement trajectory of the SFM is shown as(39)θdt=6tte5-15tte4+10tte3θe-θ0+θ0.The control parameters are chosen as follows.(  1) Adaptive Fuzzy Sliding Mode Controller. λ = 50, δ=0.1, k1=5, k2=0.025, k3=3, and ψ is obtained by fuzzy controller.(  2) LQR Controller (40) Q = 350000 0 0 0 0 10000 0 0 0 0 1000 0 0 0 0 100 , R = 2 0 0 3 .The simulation curves of composite control for trajectory tracking and vibration suppression of the SFM are shown in Figure11. Figure 11(a) shows the relation curve of the switching variable ψ with time, which is adjusted by fuzzy controller. Figures 11(b) and 11(c) show the output control of the speed reducer which is driven by a motor and the control voltage of actuators, respectively. Figure 11(d) shows the tip angle of the SFM, and it demonstrates that the composite controller successfully implements the trajectory tracking and vibration suppression of the SFM. Figures 11(e) and 11(f) show the first-order modal coordinate and second-order modal coordinate, respectively. It is obtained that the first-order modal coordinate is much larger than the second-order modal coordinate and illustrates that the lower order modes play leading role in the vibration of the SFM. The tip deflection of the SFM in a time-domain is obtained by the modal coordinates and shown in Figure 11(g). Meanwhile, it is evident from Figure 11(g) that the composite controller can successfully suppress the simultaneous elastic vibration in movement of the SFM. Figure 11(h) shows the structural strain history along the length of SFM for the first 3 s of the manipulator motion. It can be observed that the highest strain experienced occurs in the root of the SFM and the strain is reduced gradually in the control of composite controller.Figure 11
Simulation curves of composite control for trajectory tracking and vibration suppression of the SFM ((a) time-varying control variable; (b) control torque; (c) control voltage; (d) tip angle; (e) first-order modal coordinate; (f) second-order modal coordinate; (g) tip deflection; (h) strain time history).
(a) (b) (c) (d) (e) (f)  (g) (h)


## 5.3. Control Experiments

In order to better demonstrate the effectiveness of the composite controller, the experiments on composite control of the SFM are constructed. The overall experimental setup and the flowchart of composite control experiments are depicted in Figures12 and 13, respectively.Figure 12
Photograph of the experimental apparatus.Figure 13
Flowchart of the composite control experiments.The comparison between the simulated and experimental results of the SFM is shown in Figure14. It can be seen that the results of the experimental results are in good agreement with the simulation result in Figure 11. Figure 14(d) shows the tip angle of the SFM, and it demonstrates that the composite controller successfully achieved the trajectory tracking of the SFM. Meanwhile, the elastic vibration in movement of the SFM is suppressed successfully by the composite controller and shown in Figure 14(e). Figure 14(f) shows that the strain trends of the SFM are got by strain gauges which are bonded below the PZTs, respectively. In addition, it should be noted that the results exhibit a certain error between the simulated and experimental results, and this is mainly caused by the difference between experiment system and dynamic model, such as viscous damping, air resistance, the machining precision of structure, and the precision of sensors.Figure 14
Comparison between the simulated and experimental results of the SFM ((a) time-varying control variable; (b) control torque; (c) control voltage; (d) tip angle; (e) tip deflection; (f) strain trend).
(a) (b) (c) (d) (e) (f)


## 6. Conclusions

This paper mainly discusses the modelling and composite control of the SFM with piezoelectric actuators. The dynamic model of the SFM with surface bonded actuators is described by means of the assumed mode method, piezoelectric coupling model, and Hamilton’s principle. Based on the singular perturbation method, the coupled dynamic equation is decomposed into slow (rigid) and fast (flexible) subsystems. Then, the composite controller for the slow subsystem and fast subsystem is achieved using the fuzzy sliding mode control and LQR technique. Furthermore, the optimal placement of actuators is determined by the index of natural frequencies of SFM.By the analysis of numerical simulation, the optimal placement of actuators is feasible based on the maximal natural frequency. Meanwhile, the composite controller presented in this paper can not only realize the trajectory tracking of the SFM and can successfully suppress the simultaneous elastic vibration in movement of the SFM, and the closed-loop stability of the SFM is achieved. In addition, the structural strain history along the length of SFM demonstrates that the highest strain experienced occurs in the root of the SFM in the early stage and the strain is reduced gradually in the control of composite controller. Finally, the experiments demonstrate that the control effect of the composite controller reaches the expectation and the experiment results are consistent with simulation results, which further demonstrated the feasibility of the proposed composite control method.



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*Source: 2689178-2016-10-09.xml*