PDE model-based state-feedback control of constrained moving vehicle-mounted flexible manipulator with prescribed performance

Accepted Manuscript PDE model-based state-feedback control of constrained moving vehicle-mounted flexible manipulator with prescribed performance Xueyan Xing, Jinkun Liu PII:

S0022-460X(18)30696-5

DOI:

https://doi.org/10.1016/j.jsv.2018.10.023

Reference:

YJSVI 14439

To appear in:

Journal of Sound and Vibration

Received Date: 23 October 2017 Revised Date:

24 August 2018

Accepted Date: 14 October 2018

Please cite this article as: X. Xing, J. Liu, PDE model-based state-feedback control of constrained moving vehicle-mounted flexible manipulator with prescribed performance, Journal of Sound and Vibration (2018), doi: https://doi.org/10.1016/j.jsv.2018.10.023. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT PDE Model-Based State-Feedback Control of Constrained Moving Vehicle-Mounted Flexible Manipulator with Prescribed Performance Xueyan Xinga and Jinkun Liua* a

School of Automation Science and Electrical Engineering, Beihang University (Beijing

University of Aeronautics and Astronautics), Beijing, People’s Republic of China *

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Corresponding author: Jinkun Liu, School of Automation Science and Electrical Engineering,

Beihang University (Beijing University of Aeronautics and Astronautics), Beijing 100191, People’s Republic of China. E-mail: [email protected]. Abstract

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In this study, we investigate the state-feedback control problem of a moving vehicle-mounted flexible manipulator with output constraints. Both position regulation and vibration suppression can be accurately carried out based on the partial differential equation (PDE) model we

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established. The dynamics of piezoelectric actuators and sensors are considered for compensating the gravity effect of the flexible manipulator in the modeling and prescribed performance can be guaranteed with the aid of the performance functions. Further, the tracking error can converge to an arbitrarily small residual set with convergence rate no less than a pre-specified value. State variables of the closed-loop system are proven to be asymptotically stable by using Lyapunov’s direct method and the extended LaSalle’s Invariance Principle. Simulations are included to

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validate the effectiveness of the proposed control scheme. Keywords

Moving vehicle-mounted flexible manipulator, output constraints, prescribed performance, position regulation, vibration suppression, state-feedback control.

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1. Introduction

Flexible structures have become increasingly prevalent in factories and industrial plants

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throughout the developed world. Their light weight, long service life, and flexible operation represent the forefront of modern robotics technology [1-5]. However, the structural flexibility of the flexible system is likely to cause vibrations, which impact control accuracy and stability. Therefore, vibration suppression is a vital and popular research topic relevant to flexible systems. In [6], a nonlinear model for giant magnetostrictive materials is adopted to predict the active vibration suppression process of a cantilever laminated composite plate. In [7], active damping strategies are developed to eliminate the structural vibrations of a space manipulator with flexible links during on-orbit operations. In [8], a state feedback controller is proposed to damp out the tip vibrations and regulate the endpoint of a flexible robot. A flexible link vibration control scheme as-attached to a rigid robot is investigated based on an impedance control technique in [9]. The active vibration control problem of piezoelectric smart structures is discussed with distributed 1

ACCEPTED MANUSCRIPT sensors and actuators in [10] via a finite element modeling. However, the extant research described by [6-10] centers around ordinary differential equation (ODE) models, which restrict the system to a few critical modes and greatly influence the control performance, especially for nonlinear systems. Consequently, researchers have instead attempted to establish partial differential equation models which can exactly preserve the higher order modes of original flexible systems. In [11], a coupled PDE-ODE model is established for the

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vibration adaptive boundary control of a Timoshenko beam with external uncertain distributed disturbance and time-varying boundary disturbance. In [12], a PDE model is applied and PD nonlinear strain feedback controllers are developed for improving the tip regulation performance of a single-link flexible manipulator. A boundary control technique is presented for flexible

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articulated wings on a robotic aircraft as-described by PDEs, where the output of the system is given by a spatial integral of weighted functions of states in [13]. In [14], adaptive boundary control laws are proposed for a flexible link robot arm with a payload mass at the end of the link

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to compensate for the parametric uncertainty of the system.

The flexible manipulator has become an important research direction in the flexible systems field. Various control methods have been proposed for single-link and multi-link flexible manipulators. In [15], a neural network control is proposed to suppress the vibration of a flexible robotic manipulator system and eliminate the effects of input deadzone in the actuators. In [16], combined PD feedback and distributed piezoelectric polymer are used to control a single-link

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flexible manipulator via Lyapunov approach. In [17], the trajectory control problem in a flexible two-link manipulator is investigated based on a non-linear PDE observer which aims to estimate distributed positions and velocities along the flexible arms. An energy-based robust control scheme is presented in [18], which guarantees the closed-loop stability of multi-link flexible performance.

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robots, and gains of the additional control terms can be automatically tuned to improve the control

The vehicle-mounted manipulator is commonly employed in search, repair, rescue missions,

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surveillance, remote monitoring, and cooperative transportation applications, among others [19]. In [20], the coordinated control of unmanned aerial vehicles equipped with robotic manipulators is addressed by establishing a behavioral control technique. In [21], a motion coordination algorithm for an autonomous underwater vehicle-manipulator system is developed for ocean exploration applications. An extended Jacobian transpose control algorithm is established for mobile manipulators including vehicle suspension characteristics in [22]. The nonholonomic mechanical structure of space robots on free-flying vehicles and corresponding path planning are investigated in [23]. The vehicular manipulators discussed by [19-23] are all rigid, however, which reduces the flexibility of the manipulators. Vibration suppression and trajectory tracking are two major control objectives in flexible 2

ACCEPTED MANUSCRIPT manipulator. The end-point trajectory control of a single-link flexible arm is presented in [24] by an inverse dynamic solution. A robust tracking control scheme is developed to stabilize the vibration of a two-link flexible manipulator in [25]. A trajectory tracking control problem is considered for a spatial three-link articulated manipulator in [26]. In [27], a control approach for a single-link flexible manipulator is proposed to achieve precise end-point positioning with effective vibration suppression. Despite these achievements, almost all of these researches in [24-27]

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neglect the movement of the flexible manipulator base.

Real control systems are rife with output constraints which, when violated, cause serious threats to the system. Thus it is crucial to design proper controllers for systems with output constraints. In [28], a vibration control scheme is designed for a flexible beam with boundary

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output constraints via Barrier Lyapunov Function. In [29], both the steady state and the transient performance of a neural networks enhanced telerobot control system can be ensured to satisfy a prescribed performance. A boundary control method is considered for an Euler-Bernoulli beam

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with input and output restrictions in [30].

Compared to using Barrier Lapunov Function which restricts system state variable within a given range without considering the convergence rate, prescribed performance control guarantees not only the convergence of the output tracking error to a pre-defined arbitrarily small residual set, but also the convergence rate no less than a pre-specified value [31-34]. However, most researches on the prescribed performance control are based on ODE models.

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In this study, we establish a novel PDE model and design flexible manipulator control laws for location regulation and vibration suppression. The proposed model is built under Hamilton’s principle for a flexible manipulator attached to a moving vehicular base. In the original PDE model which contains the dynamics of piezoelectric actuators and sensors, all modes including the

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higher order modes of the system are considered in order to improve the control accuracy. By contrast to previous studies on flexible manipulator systems, we take into account the vehicle movement in the vehicle-mounted flexible manipulator system so as to extend the operating range

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of the flexible manipulator. The results presented here may, to this effect, have significant practical value. Under the proposed state-feedback control laws, both position regulation and vibration suppression can be achieved with prescribed performance. Tracking errors and their convergence rates can be regulated within certain pre-specified values. To the best of our knowledge, it is the first time to consider the vehicle motion in such a vehicle-mounted flexible manipulator control scheme based on a PDE model. The rest of this paper is organized as follows. Section 2 gives the PDE model of the vehicle-mounted flexible manipulator system under Hamilton’s principle. Several assumptions and lemmas which are essential to the proof of the theorem are also provided. In Section 3, state-feedback control laws which guarantee the prescribed performance are discussed and the 3

ACCEPTED MANUSCRIPT asymptotic stability analyses of the closed-loop system are strictly proven by LaSalle’s Invariance Principle. Simulation results are given in Section 4 which validates the proposed control method. Section 5 gives a brief summary and conclusion. 2. Problem Description and Preliminaries 2.1 Dynamic Model

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In this paper, we study a vehicle-mounted flexible manipulator system in the presence of output constraints which moves in the vertical plane and the elongation of the flexible link is small enough to be neglected. The gravity of the link and payload is considered in this paper. The system is shown in Fig. 1.

Y2

Y1

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Y

P ( x,

um ( t )

O

r (t )

H ( x, t )

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D ( x, t )

τ (t ) Ih

O1 ( O2 )

t)

θ (t )

uL ( t ) M

w ( x, X1

t)

X2 X

m Z1 ( Z 2 )

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Z

Fig. 1. A vehicle-mounted flexible manipulator.

In Fig. 1, OXYZ represents the global inertial coordinate system and Oi X iYi Z i is a

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body-fixed coordinate system which is attached to the mass of the vehicle where i = 1, 2 . Since the distributed control input will be employed in this paper for compensating the gravity effect of

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the flexible link, the dynamics of piezoelectric actuators and sensors is taken into account in the paper. As shown in Fig. 2, the flexible arm under study is enveloped by two plates of piezoelectric materials, which can be regarded as actuators or sensors for better controller performance. In Fig. 2, the piezoelectric materials can convert electrical to mechanic energy to suppress the vibration of the manipulator by regulating the supply voltage.

4

ACCEPTED MANUSCRIPT Y1 b c1

c2

X1 O :Piezoelectric Materials

Z1

:Pure Manipulator

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a

Fig. 2. Structure of the flexible link robot enveloped with piezoelectric materials.

The system parameters and state variables are presented as follows. L is the length of the

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flexible arm; ρ is the mass per unit span of the piezoelectric smart materials link; EI is the uniform flexural rigidity of the piezoelectric smart materials link; m and M denote the mass of

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vehicle and payload, respectively; I h is the hub inertia of the flexible arm; θ ( t ) is the angular position of the shoulder motor; the elastic deflection is defined as w ( x , t ) with respect to frame O2 X 2Y2 Z 2 at position x for time t ; the displacement of the vehicle is defined as r ( t ) in

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OXYZ ; u L ( t ) is the control force generated by boundary actuator of the flexible link; um ( t ) is the control force produced by actuator of the vehicle; τ ( t ) is the control torque at the shoulder motor; v ( x, t ) is the input voltage applied to the piezoelectric actuator; D y ( x , t ) is the electrical

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displacement along Y1 axis, which is perpendicular to the link; H ( x, t ) is the magnetic field intensity; a is the thicknesses of the flexible arm and b is the width of both the arm and the

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piezoelectric materials; the thicknesses of upper and lower surfaces of piezoelectric materials plate are denoted by c1 and c2 , respectively; hL is the coupling parameter per unit length of the smart materials link; µ L is the permeability per unit length of the smart materials manipulator;

β L is the impermittivity per unit length of the smart materials link. The position vector of the flexible link with respect to frame XOY is denoted by P ( x, t ) and P ( x, t ) can be expressed as  Px ( x, t )   x cos θ ( t ) − w ( x, t ) sin θ ( t ) + r ( t )  =  x sin θ ( t ) + w ( x, t ) cos θ ( t )   Py ( x, t )  

P ( x, t ) = 

5

(1)

ACCEPTED MANUSCRIPT where Px ( x, t ) and Py ( x, t ) are the axial components of X axis and Y axis respectively. Similarly, P ( L, t ) can be written as  Px ( L, t )   L cos θ ( t ) − w ( L, t ) sin θ ( t ) + r ( t )  = .  Py ( L, t )   L sin θ ( t ) + w ( L, t ) cos θ ( t ) 

P ( L, t ) = 

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(2)

By calculation and simplification, following equations can be obtained as

P& T ( x, t ) P& ( x, t ) = x 2θ& 2 ( t ) +θ& 2 ( t ) w2 ( x, t ) + w& 2 ( x, t ) + r& 2 ( t ) + 2 xθ& ( t ) w& ( x, t )

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−2 x sin θ ( t ) θ& ( t ) r& ( t ) − 2 cos θ ( t ) θ& ( t ) w ( x, t ) r& ( t ) − 2sin θ ( t ) w& ( x, t ) r& ( t )

and

P& T ( L, t ) P& ( L, t ) = L2θ& 2 ( t ) +θ& 2 ( t ) w2 ( L, t ) + w& 2 ( L, t ) + r& 2 ( t ) + 2 Lθ& ( t ) w& ( L, t )

(4)

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−2 L sin θ ( t ) θ& ( t ) r& ( t ) − 2 cos θ ( t ) θ& ( t ) w ( L, t ) r& ( t ) − 2sin θ ( t ) w& ( L, t ) r& ( t ) .

(3)

To derive the PDE model of this system, kinetic energy E k ( t ) , potential energy E p ( t ) and non-conservative work W ( t ) of the system can be represented as follows [35]

1 L &T 1 ρ ∫ P ( x, t ) P& ( x, t ) dx + MP& T ( L, t ) P& ( L, t ) 0 2 2

L 1 1 + mr& 2 ( t ) + µ L ∫ 0 2 2

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1 2

Ek ( t ) = I hθ& 2 ( t ) +

(∫

x

0

)

2

D& y (ξ , t )dξ dx

L EI L 2 wxx ( x, t ) dx +g ρ ∫ Py ( x, t ) dx + MgPy ( L, t ) ∫ 0 0 2 L L 1 + β L ∫ Dy2 ( x, t )dx + hL ∫ Dy ( x, t )wxx ( x, t ) dx 0 0 2

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E p (t ) =

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W ( t ) = τ ( t )θ ( t ) + uL ( t ) w( L,t ) + um ( t ) r ( t ) + ∫ bv( x,t ) Dy ( x, t )dx

(5)

(6)

L

0

(7)

where g is the acceleration of gravity. Since a motor acts on the free end of the flexible manipulator for generating boundary control

signals, we make v ( L, t ) = 0 . The Hamilton’s principle is given as

∫ (δ E ( t ) − δ E ( t ) + δ W ( t ) )dt = 0 t2

t1

k

(8)

p

where δ (.) represents the variation of (.) ; t1 and t1 are two time constants. 6

ACCEPTED MANUSCRIPT By using Hamilton’s principle, we further obtain the following structure dynamics of the system with the governing equations as

( M + m + ρ L ) &&r ( t ) −  ρ

1 2  L + ML  cos θ ( t ) θ& 2 ( t ) 2 

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

(9)

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1 3  2  && &  I h + ρ L + ML  θ ( t ) + 2 M θ ( t ) w ( L, t ) w& ( L, t ) 3   1   −  ML + ρ L2  sin θ ( t ) && r ( t ) − M cos θ ( t ) w ( L, t ) && r (t ) 2   1 && ( L, t ) + g ρ L2 cos θ ( t ) + MgL cos θ ( t ) − Mgw ( L, t ) sin θ ( t ) + M θ&& ( t ) w 2 ( L, t ) + MLw 2 2 & & ( x, t ) θ ( t ) + w ( x, t ) θ&& ( t ) + xw && ( x, t )  L  2 w ( x, t ) w +ρ ∫  dx − τ ( t ) =0 0   − cos θ ( t ) w ( x, t ) r&& ( t ) − gw ( x, t ) sin θ ( t )

 1  −  ρ L2 + ML  sin θ ( t ) θ&&( t ) + M sin θ ( t ) θ& 2 ( t ) w ( L, t ) − M cos θ ( t ) θ&&( t ) w ( L, t )  2  & && ( L, t ) −2 M cos θ ( t ) θ ( t ) w& ( L, t ) − M sin θ ( t ) w

(10)



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&2 && L sin θ ( t ) θ ( t ) w ( x, t ) − cos θ ( t ) θ ( t ) w ( x, t )  +ρ ∫  dx − um ( t ) = 0 0 && ( x, t )   −2 cos θ ( t ) θ& ( t ) w& ( x, t ) − sin θ ( t ) w

&& ( x, t ) + ρ xθ&&( t ) − ρ sin θ ( t ) && ρw r ( t ) +  EI − 

Dy ( x , t ) = −

hL

βL

wxx ( x, t ) +

vxx ( x, t ) = 0

b

βL

v ( x, t )

(11)

(12)

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∀ ( x, t ) ∈ [ 0, L ) × [ 0, ∞ ) .

βL

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− ρθ&2 ( t ) w ( x, t ) + g ρ cosθ ( t ) +

bhL

hL2   w ( x, t ) β L  xxxx

The corresponding boundary conditions are derived as

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&& ( L, t ) + MLθ&&( t ) − M sin θ ( t ) r&&( t ) − M θ& 2 ( t ) w ( L, t ) + Mw  h2  bh −  EI − L  wxxx ( L, t ) + Mg cos θ ( t ) − L vx ( L, t ) − u L ( t ) = 0 β βL  L 

(13)

wxx ( L, t ) = 0

(14)

w( 0, t ) = wx ( 0, t ) = 0

(15)

∀t ∈ [ 0, ∞ ) .

2.2 Preliminaries 7

ACCEPTED MANUSCRIPT Several important assumptions and lemmas are given in this section, which will be required in the following design and analysis of the control scheme. Assumption 1. The kinetic and potential energy of the system are bounded for ∀t ∈[ 0, ∞) and all

parameters of the system are known.

r& ( t ) , w& ( L, t ) and w& ( x, t ) are bounded for ∀( x, t ) ∈[ 0, L] ×[ 0, ∞) .

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Lemma 1 [36]. If the kinetic energy Ek ( t ) in Eq. (5) is bounded for ∀t ∈[ 0, ∞) , then θ& ( t ) ,

Lemma 2 [36]. If the potential energy Ep ( t ) in Eq. (6) is bounded for ∀t ∈[ 0, ∞) , then

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wxx ( x, t ) is bounded for ∀( x, t ) ∈[ 0, L] ×[ 0, ∞) .

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We define the tracking errors of joint angle and vehicle as e1 ( t ) = θ ( t ) −θd , e2 ( t ) = r ( t ) − rd , respectively, where θ d is the desired joint angle and rd is the desired position of the vehicle. Similarly, e3 ( t ) = w ( L, t ) denotes the error of the deflection at the end of the manipulator. As a result, we have e&1 ( t ) = θ& ( t ) , e&2 ( t ) = r& ( t ) and e&3 ( t ) = w& ( L, t ) .

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The following control objectives will be achieved in this paper.

(1) Design control laws to regulate the positions and speeds of both the joint angle and vehicle. We also manage to suppress elastic vibration while ensuring that the output signals of the closed-loop system are constrained within the given bounds. It means that θ ( t ) →θd , θ& ( t ) → 0 ,

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r ( t ) → rd , r& ( t ) → 0 , w( L, t ) → 0 , w& ( L, t ) → 0 , w( x, t ) →0 , w& ( x, t ) →0 are expected to be realized for t → ∞ .

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(2) Prescribed performance bounds on the position tracking error ei ( t ) and w ( x , t ) can

also be guaranteed for t > 0 where i = 1, 2,3 .

2.3 Prescribed Performance Function Strictly positive and decreasing function λi ( t ) , i = 1, 2,3 , is given as a performance function as follows to guarantee that the error can converge to a small bounded neighborhood of zero with a constrained convergence rate. λi ( t ) should have the following properties [37]: (1) λi ( t ) : ℜ + → ℜ + − {0} is a smooth function; 8

ACCEPTED MANUSCRIPT (2) λi ( t ) is decreasing; (3) lim λi ( t ) = λi∞ > 0 . t →∞

Based on the former description, λi ( t ) is selected as an exponentially decaying

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performance function in the following math expression:

λi ( t ) = ( λi 0 − λi∞ ) e−l t + λi∞

(16)

i

where λi 0 , λi∞ , and li are appropriate positive constants and λi 0 > λi∞ .

the transient and steady performances of tracking errors.

(17)

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−λi ( t ) < ei ( t ) < λi ( t ) , i = 1, 2,3 , ∀t > 0 .

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As a result, the following prescribed constraint conditions in inequality (17) can guarantee

From Eqs. (16) and (17), we can know that λi∞ is the maximum value of the error ei ( t ) in the stable state, and the maximum overshoot of ei ( t ) is specified less than λi ( 0 ) . Moreover, the

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decline rate of λi ( t ) also places a lower bound on the required convergence speed of ei ( t ) .

Assumption 2. Suppose that the initial tracking errors of joint and vehicle are respectively satisfied e1 ( 0 ) < λ1 ( 0 ) and e2 ( 0 ) < λ2 ( 0 ) . Besides, the initial elastic deformation of the end

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point of the flexible arm meets e3 ( 0 ) = w ( L, 0) < λ3 ( 0 ) .

3. Controller Design

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In this section, controllers of a constrained moving vehicle-mounted flexible manipulator with described performance is developed. To achieve control goals mentioned above, the error transformation is introduced. Define

ei ( t ) = λi ( t ) Si ( ε i )

(18)

where i = 1, 2,3 , ε i is the transformed error and Si ( ε i ) is a smooth, strictly increased function. Si ( ε i ) should have the following properties:

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ACCEPTED MANUSCRIPT (1) Si ( ε i ) is a monotonically increasing function; (2) −1 < Si ( ε i ) < 1 ; (3) lim Si ( ε i ) = −1 and lim Si ( ε i ) = 1 . ε i →∞

To satisfy the above three properties, Si ( ε i ) is designed as Si ( ε i ) = tanh ( ε i ) =

eε i − e − ε i . eε i + e −ε i

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εi →−∞

(19)

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Therefore, from Eq. (18) and the properties of Si ( ε i ) , we can get

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−λi ( t ) < ei ( t ) < λi ( t )

where i = 1, 2,3 .

(20)

From Eq. (18), we can obtain the inverse function of Si ( ε i ) as

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e (t ) 1+ i   e t λ ( ) 1 1 + Si 1 1 λ ( t ) + ei ( t ) i (t ) = ln = ln i ε i ( t ) = Si−1  i  = ln . ei ( t ) 2 λi ( t ) − ei ( t ) 2  λi ( t )  2 1 − Si 1− λi ( t )

(21)

Differentiating Eq. (21) with respect to time yields

where i = 1, 2,3 .

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1  λ&i ( t ) + e&i ( t ) λ&i ( t ) − e&i ( t )  λi ( t ) e&i ( t ) − λ&i ( t ) ei ( t ) − = λi2 ( t ) − ei2 ( t ) 2  λi ( t ) + ei ( t ) λi ( t ) − ei ( t ) 

ε&i ( t ) = 

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If ε i ( t ) can be made bounded for t ≥ 0

(22)

( i = 1,2,3) , inequality (20) will hold under the

properties of the error transformation in Eq. (18) so that the transient and steady performance of the system can be guaranteed. To ensure that tracking errors e1 ( t ) , e2 ( t ) and the deflection error at the end of the flexible

manipulator e3 ( t ) converge to zero with the prescribed performance, control laws of the system are proposed as

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ACCEPTED MANUSCRIPT 1 g ρ L2 cos θ ( t ) + MgL cos θ ( t ) − Mgw ( L , t ) sin θ ( t ) 2 L 3 3 1 − ρ ∫ gw ( x , t ) sin θ ( t )d x + e12 ( t ) ε 1 ( t ) − λ1 ( t ) e1 ( t ) − λ12 ( t ) ε 1 ( t ) 0 2 2 2

τ ( t ) = − k p1e1 ( t ) − k d 1e&1 ( t ) +

(23)

(24)

3 u L ( t ) = − k p 3 e3 ( t ) − k d 3 e&3 ( t ) + Mg cos θ ( t ) + e32 ( t ) ε 3 ( t ) 2 3 1 2 − λ3 ( t ) e3 ( t ) − λ3 ( t ) ε 3 ( t ) 2 2

(25)

x

0

βL hL b

χ

x

d4 1 0

0

and c2 =

χ

1

0

x 2 cos θ ( t ) + c2 wxx ( x, t )

(26)

hL ; k p1 , kd 1 , k p 2 , k d 2 , k p 3 , kd 3 , k p 4 and k d 4 are positive control b

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where c1 =

g ρc ∫ w (ς , t )dς dχ + k c ∫ ∫ w& (ς , t )dς dχ − 2

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v ( x, t ) = k p 4 c1 ∫

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3 3 1 um ( t ) = − k p 2 e2 ( t ) − kd 2 e&2 ( t ) + e22 ( t ) ε 2 ( t ) − λ2 ( t ) e2 ( t ) − λ22 ( t ) ε 2 ( t ) 2 2 2

coefficients; τ ( t ) , uL ( t ) , um ( t ) , v ( x, t ) are four control inputs which act on the joint, payload, vehicle, flexible manipulator, respectively, ∀ ( x, t ) ∈ [ 0, L ) × [ 0, ∞ ) .

Remark 1. Since the piezoelectric material has the advantages of fast response and stable performance, piezoelectric smart structure is widely used in industry. In practice, all signals in

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control inputs (23)-(26) can be achieved by different kinds of sensors and the backward difference algorithm. For instance, θ ( t ) can be measured by a photoelectrical encoder and θ& ( t ) can be obtained by a tachometer. r ( t ) and w( L, t ) can be sensed by laser displacement sensors. r& ( t )

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& ( L, t ) can be received with speed sensors. w ( x , t ) can be realized by a piezoelectric patch and w

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actuator via a piezoelectric sensor [38-41]. w& ( x , t ) and wxx ( x, t ) can be obtained by a backward difference algorithm. The boundary control input u L ( t ) can be implemented with a flap-based effector [42] or a micro-trailing edge effector [43]. All the input signals in Eqs. (23)-(26) are feasibly measurable.

Theorem 1. Under the aforementioned assumptions and lemmas, the closed-loop system described by Eqs. (9)-(15) satisfies the following properties. (1) Both position regulation and vibration suppression can be realized under the proposed control

& ( L, t ) → 0 , laws, that is, θ ( t ) →θd , θ& ( t ) → 0 , r ( t ) → rd , r& ( t ) → 0 , w( L, t ) → 0 , w 11

ACCEPTED MANUSCRIPT w( x, t ) →0 , w& ( x, t ) →0 are guaranteed for t → ∞ . (2) If the initial boundary conditions e1 ( 0) < λ1 ( 0) , e2 ( 0 ) < λ2 ( 0 ) and e3 ( 0 ) < λ3 ( 0 ) hold,

{

}

then the tracking errors remain in the region of Ωei = ei ( t ) ∈ℜ : ei ( t ) < λi ( 0 )

where i = 1, 2

{

}

Ωe3 = e3 ( t ) ∈ℜ : e3 ( t ) < λ3 ( 0 )

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and the deformation error of the end of the flexible link is also within the set of with a bounded speed. That is to say, the prescribed performance

bounds of both tracking errors and deflection error can be ensured with the given control laws.

Proof: To realize the control goals, a Lyapunov candidate function is proposed to be

1 &2 1 L 1 & ( L, t ) I hθ ( t ) + ρ ∫ P& T ( x, t ) P& ( x, t ) dx + MP& T ( L, t ) P 0 2 2 2

1 EI + mr& 2 ( t ) + 2 2

V2 ( t ) =

∫

L

0

2 xx

w

( x, t )dx

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V1 ( t ) =

SC

V ( t ) = V1 ( t ) + V2 ( t ) + V3 ( t )

L 1 1 1 1 k p1e12 ( t ) + k p 2 e22 ( t ) + k p 3 e32 ( t ) + k p 4 ∫ w2 ( x, t )dx 0 2 2 2 2

1 2 ( λ1 ( t ) − e12 ( t )) e1 ( t ) ε1 (t ) + 12 ( λ22 ( t ) − e22 (t ) ) e2 ( t ) ε 2 ( t ) 2 1 1 1 1 + ( λ32 ( t ) − e32 ( t ) ) e3 ( t ) ε 3 ( t ) + e12 ( t ) λ1 ( t ) + e22 ( t ) λ2 ( t ) + e32 ( t ) λ3 ( t ) 2 2 2 2

TE D

V3 ( t ) =

(27)

(28)

(29)

(30)

where V1 ( t ) is a term of system energy; V2 ( t ) is a term of tracking errors; V3 ( t ) is a term to

EP

guarantee the described performance of the system.

AC C

The derivative of Eq. (27) with respect to time is given as

12

ACCEPTED MANUSCRIPT

SC

RI PT

 2 x 2θ& ( t ) θ&&( t ) +2w ( x, t ) w& ( x, t ) θ& 2 ( t ) + 2 w2 ( x, t ) θ& ( t ) θ&&( t )    && & && ( x, t ) + 2r& ( t ) && && ( x, t )  r ( t ) + 2 xθ ( t ) w& ( x, t ) + 2 xθ ( t ) w  +2w& ( x, t ) w   2  −2 x cos θ ( t ) θ& ( t ) r& ( t ) − 2 x sin θ ( t ) θ&&( t ) r& ( t )  L 1  2 & && & & V& ( t ) = I hθ ( t )θ ( t ) + ρ ∫  −2 x sin θ ( t ) θ ( t ) && r ( t ) + 2sin θ ( t ) θ ( t ) w ( x, t ) r& ( t )  dx 2 0  && & & & &  −2 cos θ ( t ) θ ( t ) w ( x, t ) r ( t ) − 4 cos θ ( t ) θ ( t ) w ( x, t ) r ( t )   −2 cos θ ( t ) θ& ( t ) w ( x, t ) &&  && ( x, t ) r& ( t ) r ( t ) − 2 sin θ ( t ) w    −2sin θ ( t ) w& ( x, t ) &&  r (t )    2 L2θ& ( t ) θ&&( t ) + 2w ( L, t ) w& ( L, t ) θ& 2 ( t ) + 2w2 ( L, t ) θ& ( t ) θ&&( t ) + 2w& ( L, t ) w && ( L, t )    && ( L, t ) − 2 L cos θ ( t ) θ& 2 ( t ) r& ( t )  +2r& ( t ) r&&( t ) + 2 Lθ&&( t ) w& ( L, t ) + 2 Lθ& ( t ) w   1  2 + M  −2 L sin θ ( t ) θ&&( t ) r& ( t ) − 2 L sin θ ( t ) θ& ( t ) && r ( t ) + 2 sin θ ( t ) θ& ( t ) w ( L, t ) r& ( t )  2   && &  −2 cos θ ( t ) θ ( t ) w ( L, t ) r& ( t ) − 4 cos θ ( t ) θ ( t ) w& ( L, t ) r& ( t )   −2 cos θ ( t ) θ& ( t ) w ( L, t ) r&&( t ) − 2sin θ ( t ) w && & & && ( L, t ) r ( t ) − 2 sin θ ( t ) w ( L, t ) r ( t )   r ( t ) + EI ∫ wxx ( x, t )w& xx ( x, t ) dx + k p1e1 ( t ) e&1 ( t ) + mr& ( t ) && L

0

+ k p 2 e2 ( t ) e&2 ( t ) + k p 3 e3 ( t ) e&3 ( t ) + k p 4 ∫ w ( x, t ) w& ( x, t )dx L

0

(

)

(

)

)

(31)

TE D

(

M AN U

1 + λ1 ( t ) λ&1 ( t ) − e1 ( t ) e&1 ( t ) e1 ( t ) ε1 ( t ) + ( λ12 ( t ) − e12 ( t ) ) e&1 ( t ) ε1 ( t ) 2 1 2 + ( λ1 ( t ) − e12 ( t ) ) e1 ( t ) ε&1 ( t ) + λ2 ( t ) λ&2 ( t ) − e2 ( t ) e&2 ( t ) e2 ( t ) ε 2 ( t ) 2 1 2 1 + ( λ2 ( t ) − e22 ( t ) ) e&2 ( t ) ε 2 ( t ) + ( λ22 ( t ) − e22 ( t ) ) e2 ( t ) ε&2 ( t ) 2 2 1 + λ3 ( t ) λ&3 ( t ) − e3 ( t ) e&3 ( t ) e3 ( t ) ε 3 ( t ) + ( λ32 ( t ) − e32 ( t ) ) e&3 ( t ) ε 3 ( t ) 2 1 1 2 + ( λ3 ( t ) − e32 ( t ) ) e3 ( t ) ε&3 ( t ) + e1 ( t ) e&1 ( t ) λ1 ( t ) + e12 ( t ) λ&1 ( t ) + e2 ( t ) e&2 ( t ) λ2 ( t ) 2 2 1 2 1 2 & & + e2 ( t ) λ2 ( t ) + e3 ( t ) e&3 ( t ) λ3 ( t ) + e3 ( t ) λ3 ( t ) . 2 2

& ( L, t ) , we can get the Multiplying Eq. (9) by θ& ( t ) , Eq. (10) by r& ( t ) and Eq. (13) by w

EP

following equations by using

1   3   1   −  ML + ρ L2  sin θ ( t ) r&&( t ) θ& ( t ) − M cos θ ( t ) w ( L, t ) && r ( t ) θ& ( t ) 2   1 && ( L, t ) θ& ( t ) + g ρ L2 cos θ ( t ) θ& ( t ) + M θ&&( t ) w2 ( L, t ) θ& ( t ) + MLw 2 & & + MgL cos θ ( t )θ ( t ) − Mgw ( L, t ) sin θ ( t ) θ ( t )

AC C

θ& ( t )τ ( t ) =  I h + ρ L3 + ML2  θ&&( t ) θ& ( t ) + 2 M θ& 2 ( t ) w ( L, t ) w& ( L, t )

 & ( x, t ) θ& 2 ( t ) + w2 ( x, t ) θ&&( t ) θ& ( t ) L  2 w ( x, t ) w +ρ ∫  dx 0 && ( x, t ) θ& ( t ) − cos θ ( t ) w ( x, t ) && r ( t ) θ& ( t ) − gw ( x, t ) sin θ ( t ) θ& ( t )   + xw

13

(32)

ACCEPTED MANUSCRIPT  1  r& ( t ) um ( t ) = ( M + m + ρ L ) && r ( t ) r& ( t ) −  ρ L2 + ML  cos θ ( t )θ& 2 ( t ) r& ( t )  2   1  −  ρ L2 + ML  sin θ ( t ) θ&&( t ) r& ( t ) + M sin θ ( t ) θ& 2 ( t ) w ( L, t ) r& ( t )  2  && − M cos θ ( t )θ ( t ) w ( L, t ) r& ( t ) − 2M cos θ ( t ) θ& ( t ) w& ( L, t ) r& ( t )

(33)

&2 & ( t ) − cos θ ( t ) θ&&( t ) w ( x, t ) r& ( t )  L sin θ ( t ) θ ( t ) w ( x, t ) r +ρ ∫  dx 0 & && ( x, t ) r& ( t )   −2 cos θ ( t ) θ ( t ) w& ( x, t ) r& ( t ) − sin θ ( t ) w

RI PT

&& ( L, t ) r& ( t ) − M sin θ ( t ) w

&& ( L , t ) w& ( L , t ) + MLθ&& ( t ) w& ( L , t ) w& ( L , t ) u L ( t ) = − M θ& 2 ( t ) w ( L , t ) w& ( L , t ) + Mw

(34)

SC

 h2  − M sin θ ( t ) && r ( t ) w& ( L , t ) −  EI − L  wxxx ( L , t ) w& ( L , t ) βL   bh + M g cos θ ( t ) w& ( L , t ) − L v x ( L , t ) w& ( L , t ) .

βL

−

bhL

βL

∫ w& ( x, t ) v ( x, t ) dx = ∫ L

xx

0

L

0

M AN U

Multiplying Eq. (11) by w& ( x , t ) and integrating it respect to x from 0 to L , we can get && ( x, t ) w& ( x, t )dx + ∫ ρ xθ&&( t ) w& ( x, t ) dx ρw L

0

L L h2  r ( t ) w& ( x, t ) dx + ∫  EI − L  wxxxx ( x, t ) w& ( x, t ) dx −∫ ρ sin θ ( t ) && 0 0 βL   L L − ρθ&2 ( t ) w( x, t ) w& ( x, t ) dx + g ρ cosθ ( t ) w& ( x, t ) dx.

∫

(35)

∫

0

0

TE D

Substituting Eqs. (32)-(35) into Eq. (31) gives bh V& ( t ) = θ& ( t )τ ( t ) + r& ( t ) um ( t ) + w& ( L, t ) u L ( t ) − L

βL

1 ∫ w& ( x, t ) v ( x, t ) dx − 2 g ρ L L

2

xx

0

cos θ ( t ) θ& ( t )

L − MgL cos θ ( t ) θ& ( t ) + Mgw ( L, t ) sin θ ( t ) θ& ( t ) + ρ ∫ gw ( x, t ) sin θ ( t ) θ& ( t )dx

0

L L h2 + EI ∫ wxx ( x, t )w& xx ( x, t ) dx − ∫  EI − L 0 0 βL 

AC C

0

 h2  g ρ cos θ ( t ) w& ( x, t ) dx +  EI − L  wxxx ( L, t ) w& ( L, t ) βL  

EP

− Mg cos θ ( t ) w& ( L, t ) − ∫

L

 bhL vx ( L, t ) w& ( L, t )  wxxxx ( x, t ) w& ( x, t )dx + βL 

+ k p1e1 ( t ) e&1 ( t ) + k p 2 e2 ( t ) e&2 ( t ) + k p 3 e3 ( t ) e&3 ( t ) + k p 4 ∫ w ( x, t ) w& ( x, t )dx

(

L

0

)

1 + λ1 ( t ) λ&1 ( t ) − e1 ( t ) e&1 ( t ) e1 ( t ) ε1 ( t ) + ( λ12 ( t ) − e12 ( t ) ) e&1 ( t ) ε1 ( t ) 2 1 2 + ( λ1 ( t ) − e12 ( t ) ) e1 ( t ) ε&1 ( t ) + λ2 ( t ) λ&2 ( t ) − e2 ( t ) e&2 ( t ) e2 ( t ) ε 2 ( t ) 2 1 2 1 + ( λ2 ( t ) − e22 ( t ) ) e&2 ( t ) ε 2 ( t ) + ( λ22 ( t ) − e22 ( t ) ) e2 ( t ) ε&2 ( t ) 2 2 1 + λ3 ( t ) λ&3 ( t ) − e3 ( t ) e&3 ( t ) e3 ( t ) ε 3 ( t ) + ( λ32 ( t ) − e32 ( t ) ) e&3 ( t ) ε 3 ( t ) 2 1 2 1 + ( λ3 ( t ) − e32 ( t ) ) e3 ( t ) ε&3 ( t ) + e1 ( t ) e&1 ( t ) λ1 ( t ) + e12 ( t ) λ&1 ( t ) + e2 ( t ) e&2 ( t ) λ2 ( t ) 2 2 1 2 1 + e2 ( t ) λ&2 ( t ) + e3 ( t ) e&3 ( t ) λ3 ( t ) + e32 ( t ) λ&3 ( t ) . 2 2

(

(

)

)

14

(36)

ACCEPTED MANUSCRIPT Using integration by parts of the middle terms in Eq. (36), we get − ∫ EIwxxxx ( x, t ) w& ( x, t )dx + EIwxxx ( L, t ) w& ( L, t ) + EI ∫ wxx ( x, t )w& xx ( x, t ) dx L

L

0

0

= − ∫ EIwxxxx ( x, t ) w& ( x, t )dx + EIwxxx ( L, t ) w& ( L, t ) L

0

+ EIwxx ( x, t ) w& x ( x, t )

L 0

− EI ∫ wxxx ( x, t )w& x ( x, t ) dx L

0

= − ∫ EIwxxxx ( x, t ) w& ( x, t )dx + EIwxxx ( L, t ) w& ( L, t ) + EIwxx ( L, t ) w& x ( L, t ) L

L − EIwxx ( 0, t ) w& x ( 0, t ) −  EIwxxx ( x, t ) w& ( x, t ) 0L − EI ∫ wxxxx ( x, t )w& ( x, t ) dx  0   = EIwxxx ( L, t ) w& ( L, t ) + EIwxx ( L, t ) w& x ( L, t ) − EIwxx ( 0, t ) w& x ( 0, t )

− ( EIwxxx ( L, t ) w& ( L, t ) − EIwxxx ( 0, t ) w& ( 0, t ) )

= EIwxx ( L, t ) w& x ( L, t ) − EIwxx ( 0, t ) w& x ( 0, t ) + EIwxxx ( 0, t ) w& ( 0, t )

SC

= 0.

(37)

RI PT

0

Since the derivative of the boundary input voltage is adjustable, the derivative of the

vx ( L, t ) =

M AN U

boundary piezoelectric input voltage is chosen as [44] hL wxxx ( L , t ) . b

(38)

Substituting Eqs. (37) and (38) into Eq. (36), we can obtain bh V& ( t ) = θ& ( t )τ ( t ) + r& ( t ) um ( t ) + w& ( L, t ) u L ( t ) − L

βL

hL2

βL

∫

L

0

wxxxx ( x, t ) w& ( x, t )dx −

L

xx

0

1 g ρ L2 cos θ ( t ) θ& ( t ) − MgL cos θ ( t ) θ& ( t ) 2

TE D

+

∫ w& ( x, t ) v ( x, t ) dx

L + Mgw ( L, t ) sin θ ( t ) θ& ( t ) + ρ ∫ gw ( x, t ) sin θ ( t ) θ& ( t )dx 0

− Mg cos θ ( t ) w& ( L, t ) − ∫ g ρ cos θ ( t ) w& ( x, t ) dx + k p1e1 ( t ) e&1 ( t ) + k p 2 e2 ( t ) e&2 ( t ) + k p 3 e3 ( t ) e&3 ( t ) L

0

(

)

+ k p 4 ∫ w ( x, t ) w& ( x, t )dx + λ1 ( t ) λ&1 ( t ) − e1 ( t ) e&1 ( t ) e1 ( t ) ε1 ( t ) L

0

AC C

(

EP

1 2 1 λ1 ( t ) − e12 ( t ) ) e&1 ( t ) ε1 ( t ) + ( λ12 ( t ) − e12 ( t ) ) e1 ( t ) ε&1 ( t ) ( 2 2 1 & + λ2 ( t ) λ2 ( t ) − e2 ( t ) e&2 ( t ) e2 ( t ) ε 2 ( t ) + ( λ22 ( t ) − e22 ( t ) ) e&2 ( t ) ε 2 ( t ) 2 1 2 2 + ( λ2 ( t ) − e2 ( t ) ) e2 ( t ) ε&2 ( t ) + λ3 ( t ) λ&3 ( t ) − e3 ( t ) e&3 ( t ) e3 ( t ) ε 3 ( t ) 2 1 2 1 + ( λ3 ( t ) − e32 ( t ) ) e&3 ( t ) ε 3 ( t ) + ( λ32 ( t ) − e32 ( t ) ) e3 ( t ) ε&3 ( t ) + e1 ( t ) e&1 ( t ) λ1 ( t ) 2 2 1 2 1 1 + e1 ( t ) λ&1 ( t ) + e2 ( t ) e&2 ( t ) λ2 ( t ) + e22 ( t ) λ&2 ( t ) + e3 ( t ) e&3 ( t ) λ3 ( t ) + e32 ( t ) λ&3 ( t ) . 2 2 2 +

)

(

)

Using control laws in Eqs. (23)-(26) to Eq. (39), we have

15

(39)

ACCEPTED MANUSCRIPT

βL

+

2 L

h

0

∫

βL

RI PT

1   2  − k p1e1 ( t ) − kd 1e&1 ( t ) + 2 g ρ L cos θ ( t ) + MgL cos θ ( t ) − Mgw ( L, t ) sin θ ( t )  & & V (t ) = θ (t )    − ρ L gw x, t sin θ t dx + 3 e 2 t ε t − 3 λ t e t − 1 λ 2 t ε t  ( ) () ( ) ( ) ( ) ( ) ( ) ( )  ∫0  1 1 1 1 1 1 2 2 2   3 3 1   + r& ( t )  − k p 2 e2 ( t ) − kd 2 e&2 ( t ) + e22 ( t ) ε 2 ( t ) − λ2 ( t ) e2 ( t ) − λ22 ( t ) ε 2 ( t )  2 2 2   3 2    − k p 3 e3 ( t ) − kd 3 e&3 ( t ) + Mg cos θ ( t ) + 2 e3 ( t ) ε 3 ( t )  + w& ( L, t )    − 3 λ t e t − 1 λ2 t ε t    3( ) 3 ( ) 3 ( ) 3( )  2 2  bh L − L ∫ w& ( x, t ) ( k p 4 c1 w ( x, t ) + kd 4 c1 w& ( x, t ) − g ρ c1 cos θ ( t ) + c2 wxxxx ( x, t ) ) dx L

0

wxxxx ( x, t ) w& ( x, t )dx −

1 g ρ L2 cos θ ( t ) θ& ( t ) − MgL cos θ ( t )θ& ( t ) + Mgw ( L, t ) sin θ ( t ) θ& ( t ) 2

+ ρ ∫ gw ( x, t ) sin θ ( t ) θ& ( t )dx − Mg cos θ ( t ) w& ( L, t ) − ∫ g ρ cos θ ( t ) w& ( x, t ) dx L

L

0

0

(

SC

+ k p1e1 ( t ) e&1 ( t ) + k p 2 e2 ( t ) e&2 ( t ) + k p 3 e3 ( t ) e&3 ( t )

)

+ k p 4 ∫ w ( x, t ) w& ( x, t )dx + λ1 ( t ) λ&1 ( t ) − e1 ( t ) e&1 ( t ) e1 ( t ) ε 1 ( t ) L

0

1 2 ( λ1 (t ) − e12 (t ) ) e&1 (t ) ε1 ( t ) + 12 ( λ12 (t ) − e12 ( t ) ) e1 (t ) ε&1 ( t ) 2 1 + λ2 ( t ) λ&2 ( t ) − e2 ( t ) e&2 ( t ) e2 ( t ) ε 2 ( t ) + ( λ22 ( t ) − e22 ( t ) ) e&2 ( t ) ε 2 ( t ) 2 1 2 2 + ( λ2 ( t ) − e2 ( t ) ) e2 ( t ) ε&2 ( t ) + λ3 ( t ) λ&3 ( t ) − e3 ( t ) e&3 ( t ) e3 ( t ) ε 3 ( t ) 2 1 2 1 + ( λ3 ( t ) − e32 ( t ) ) e&3 ( t ) ε 3 ( t ) + ( λ32 ( t ) − e32 ( t ) ) e3 ( t ) ε&3 ( t ) + e1 ( t ) e&1 ( t ) λ1 ( t ) 2 2 1 2 1 1 + e1 ( t ) λ&1 ( t ) + e2 ( t ) e&2 ( t ) λ2 ( t ) + e22 ( t ) λ&2 ( t ) + e3 ( t ) e&3 ( t ) λ3 ( t ) + e32 ( t ) λ&3 ( t ) 2 2 2 3 3 1   = e&1 ( t )  − kd 1e&1 ( t ) + e12 ( t ) ε 1 ( t ) − λ1 ( t ) e1 ( t ) − λ12 ( t ) ε 1 ( t )  2 2 2   3 3 1   + e&2 ( t )  − kd 2 e&2 ( t ) + e22 ( t ) ε 2 ( t ) − λ2 ( t ) e2 ( t ) − λ22 ( t ) ε 2 ( t )  2 2 2  

(

M AN U

+

)

)

TE D

(

3 3 1   + e&3 ( t )  − kd 3 w& ( L, t ) + e32 ( t ) ε 3 ( t ) − λ3 ( t ) e3 ( t ) − λ32 ( t ) ε 3 ( t )  2 2 2   L 2 & − k w& ( x, t )dx + λ ( t ) λ ( t ) − e ( t ) e& ( t ) e ( t ) ε ( t ) 0

d4

(

EP

∫

1

1

1

1

)

1

1

1 2 ( λ1 (t ) − e12 (t ) ) e&1 (t ) ε1 ( t ) + 12 ( λ12 (t ) − e12 ( t ) ) e1 (t ) ε&1 ( t ) 2 1 + λ2 ( t ) λ&2 ( t ) − e2 ( t ) e&2 ( t ) e2 ( t ) ε 2 ( t ) + ( λ22 ( t ) − e22 ( t ) ) e&2 ( t ) ε 2 ( t ) 2 1 2 + ( λ2 ( t ) − e22 ( t ) ) e2 ( t ) ε&2 ( t ) + λ3 ( t ) λ&3 ( t ) − e3 ( t ) e&3 ( t ) e3 ( t ) ε 3 ( t ) 2 1 2 1 1 + ( λ3 ( t ) − e32 ( t ) ) e&3 ( t ) ε 3 ( t ) + ( λ32 ( t ) − e32 ( t ) ) e3 ( t ) ε&3 ( t ) + e1 ( t ) e&1 ( t ) λ1 ( t ) + e12 ( t ) λ&1 ( t ) 2 2 2 1 2 1 2 & & + e2 ( t ) e&2 ( t ) λ2 ( t ) + e2 ( t ) λ2 ( t ) + e3 ( t ) e&3 ( t ) λ3 ( t ) + e3 ( t ) λ3 ( t ) . 2 2 +

AC C

(

)

(

)

Now, it follows from Eqs. (22) and (40) that

16

(40)

ACCEPTED MANUSCRIPT 3 3 1 V& ( t ) = − kd 1e&12 ( t ) + e12 ( t ) ε1 ( t ) e&1 ( t ) − λ1 ( t ) e1 ( t ) e&1 ( t ) − λ12 ( t ) ε1 ( t ) e&1 ( t ) 2 2 2 3 3 1 −kd 2 e&22 ( t ) + e22 ( t ) ε 2 ( t ) e&2 ( t ) − λ2 ( t ) e2 ( t ) e&2 ( t ) − λ22 ( t ) ε 2 ( t ) e&2 ( t ) 2 2 2 3 3 1 −kd 3 e&32 ( t ) + e32 ( t ) ε 3 ( t ) e&3 ( t ) − λ3 ( t ) e3 ( t ) e&3 ( t ) − λ32 ( t ) ε 3 ( t ) e&3 ( t ) 2 2 2 L 1 2 −kd 4 ∫ w& ( x, t ) dx + λ1 ( t ) λ&1 ( t ) − e1 ( t ) e&1 ( t ) e1 ( t ) ε1 ( t ) + ( λ12 ( t ) − e12 ( t ) ) e&1 ( t ) ε1 ( t ) 0 2 & λ ( t ) e&1 ( t ) − λ1 ( t ) e1 ( t ) 1 + ( λ12 ( t ) − e12 ( t ) ) e1 ( t ) 1 + λ2 ( t ) λ&2 ( t ) − e2 ( t ) e&2 ( t ) e2 ( t ) ε 2 ( t ) 2 λ12 ( t ) − e12 ( t ) λ ( t ) e&2 ( t ) − λ&2 ( t ) e2 ( t ) 1 1 + ( λ22 ( t ) − e22 ( t ) ) e&2 ( t ) ε 2 ( t ) + ( λ22 ( t ) − e22 ( t ) ) e2 ( t ) 2 2 2 λ22 ( t ) − e22 ( t )

(

)

(

)

RI PT

(

)

(41)

1 2 ( λ3 ( t ) − e32 ( t ) ) e&3 (t ) ε 3 ( t ) 2 λ ( t ) e&3 ( t ) − λ&3 ( t ) e3 ( t ) 1 1 + ( λ32 ( t ) − e32 ( t ) ) e3 ( t ) 3 + e1 ( t ) e&1 ( t ) λ1 ( t ) + e12 ( t ) λ&1 ( t ) 2 2 λ3 ( t ) − e3 ( t ) 2 2

SC

+ λ3 ( t ) λ&3 ( t ) − e3 ( t ) e&3 ( t ) e3 ( t ) ε 3 ( t ) +

d1 1

d2 2

d3 3

M AN U

1 1 +e2 ( t ) e&2 ( t ) λ2 ( t ) + e22 ( t ) λ&2 ( t ) + e3 ( t ) e&3 ( t ) λ3 ( t ) + e32 ( t ) λ&3 ( t ) 2 2 L 2 2 2 2 = −k e& ( t ) − k e& ( t ) − k e& ( t ) − k w& ( x, t )dx + λ ( t ) λ& ( t ) e ( t ) ε

∫

d4 0

1

1

1

1

(t )

+λ2 ( t ) λ&2 ( t ) e2 ( t ) ε 2 ( t ) + λ3 ( t ) λ&3 ( t ) e&3 ( t ) ε 3 ( t ) .

Since

(42)

TE D

λi ( t ) λ&i ( t ) ei ( t ) εi ( t ) ≤ 0 , i = 1, 2,3

can be obtained from Eqs. (16) and (21), we can further obtain V& ( t ) ≤ 0 on the basis of

EP

inequality (42), which means that V ( t ) is negative semi-definite. In this paper, the extended LaSalle’s Invariance Principle is applied to prove the asymptotic stability of the system.

AC C

If V& ( t ) ≡ 0 then from Eq. (41), we get

θ& ( t ) = r& ( t ) = w& ( L, t ) = w& ( x, t ) = e1 ( t ) ε1 ( t ) = e2 ( t ) ε 2 ( t ) = e3 ( t ) ε 3 ( t ) ≡ 0

(43)

which means

&& ( L, t ) = w && ( x, t ) ≡ 0 . θ&&( t ) = && r (t ) = w

(44)

Applying Eqs. (43) and (44) into Eqs. (9)-(13) yields L 1 g ρ L2 cos θ ( t ) + MgL cos θ ( t ) − Mgw ( L, t ) sin θ ( t ) − ρ ∫ gw ( x, t ) sin θ ( t )dx − τ ( t ) =0 0 2

17

(45)

ACCEPTED MANUSCRIPT (46)

um ( t ) =0

(47)

 hL2  bhL vxx ( x, t ) = 0 .  EI −  wxxxx ( x, t ) + ρ g cos θ ( t ) + βL  βL 

(48)

From Eqs. (43) and (48), it is clear that hL2

βL

wxxxx ( x, t ) − ρ g cos θ ( t ) −

bhL

βL

vxx ( x, t ) = −k p 4 w ( x, t ) .

(49)

SC

EIwxxxx ( x, t ) =

RI PT

− EIwxxx ( L, t ) + Mg cos θ ( t ) − u L ( t ) =0

By using the technique of separation of variables [45], we separate w ( x , t ) as (50)

M AN U

w ( x, t ) = Φ ( x ) ⋅ q ( t )

where Φ ( x ) and q ( t ) are two undetermined functions with respect to space and time respectively.

Φ xxxx ( x ) Φ ( x)

=−

k p4 EI

2 αx 2

EI

(51)

and solving Eq. (51), we obtain

  − 2 2 α x + b2 sin α x  + e  b1 cos 2 2  

2 αx 2

  2 2 α x + b4 sin α x   b3 cos 2 2  

(52)

AC C

Φ ( x) = e

k p4

EP

Setting α 4 =

.

TE D

Substituting Eq. (50) into Eq. (49), we can get

where bi ∈ ℜ, i = 1, 2, 3, 4 are four constants to be determined. From Eqs. (25), (43) and (46), we know that the following equation holds for all time

3 1 k p 3 w ( L, t ) + λ3 ( t ) e3 ( t ) + λ32 ( t ) ε 3 ( t ) =EIwxxx ( L, t ) . 2 2

(53)

Hence, we can obtain that wxxx ( L, t ) = 0 and w ( L, t ) = 0 . Using

boundary

conditions

(13)-(15)

and

wxxx ( L, t ) = 0

,

we

have

Φ ( 0 ) = Φ x ( 0 ) = Φ xx ( L ) = Φ xxx ( L ) = 0 . Then from Eq. (52), the following equations can be 18

ACCEPTED MANUSCRIPT obtained as follows b1 + b3 = 0

(54)

b1 + b2 − b3 + b4 = 0

(55)

  − 2 2 α L − b1 sin α L  + e  b2 cos 2 2  

2 αL 2

  2 2 α L + b3 sin α L  = 0  −b4 cos 2 2  

  2 2 α L + ( b2 − b1 ) cos α L   ( −b2 − b1 ) sin 2 2   2  − αL  2 2 α L + ( b4 +b3 ) cos α L  = 0. +e 2  ( −b3 +b4 ) sin 2 2   αL

(57)

SC

2

e2

(56)

RI PT

2 αL 2

e

By resolving Eqs. (54)-(57), we can derive that bi = 0, i = 1, 2, 3, 4 , is the unique solution.

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Therefore, Φ ( x ) = 0 and w ( x, t ) = 0 can be obtained.

Substituting control laws in Eqs. (23)-(26) into Eqs. (45)-(48), we can get e1 ( t ) = θ ( t ) − θ d = 0

e3 ( t ) = w ( L , t ) = 0 w ( x, t ) = 0 .

(59)

(60)

TE D

e2 ( t ) = r ( t ) − rd = 0

(58)

(61)

EP

On the basis of the above analysis, the asymptotical stability of the closed-loop system is verified and θ ( t ) → θ d , θ& ( t ) → 0 ,

AC C

w ( x, t ) → 0 , w& ( x, t ) → 0

when t → ∞

r ( t ) → rd ,

r& ( t ) → 0 ,

w ( L, t ) → 0 ,

w& ( L, t ) → 0 ,

can be proven by applying extended LaSalle’s

Invariance Principle. Furthermore, under the assumptions and control inputs proposed in Eqs. (23)-(26), the transient performance of the system is adjustable and output errors e1 ( t ) , e2 ( t ) , e3 ( t ) can be constrained within prescribed performance bounds for −λi ( t ) < ei ( t ) < λi ( t ) , i = 1, 2,3 . Therefore, Theorem 1 has been proven.

4. Simulations In this part, we run several simulation experiments on the proposed controller of the moving vehicle-mounted flexible manipulator system to validate both position regulation and vibration 19

ACCEPTED MANUSCRIPT suppression control performances with guaranteed transient and steady state error bounds. We use the finite difference method [46] to discretize the system and perform the simulations. In order to obtain the approximate solution of the nonlinear PDE system, we choose a 1 temporal step of ∆ t = 2 × 10 -4 s and spatial step size of ∆ x = 6 × 10 -2 m , which satisfies ∆t ≤ ∆x 2

[47]. The accuracy of the system solution can be adjusted by choosing proper temporal and spatial

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step sizes. The initial conditions of the system are set as w ( x,0 ) = 0 , w& ( x,0 ) = 0 , θ ( 0 ) = 0 ,

θ& ( 0) = 0 , r ( 0 ) = 0 , r& ( 0 ) = 0 in order to satisfy the rigid Assumption 2. The desired positions of the joint and vehicle are given as θ d = 0.2 rad and rd = 1 m . The final signals of the closed-loop

SC

system are expected to be θ ( t ) → θ d , θ& ( t ) → 0 , r ( t ) → rd , r& ( t ) → 0 , w ( L, t ) → 0 , w& ( L, t ) → 0 , w ( x, t ) → 0 , w& ( x, t ) → 0 for t → ∞ . The parameters of the system and λi ( t ) in

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Eq. (16) are given in Table 1 and Table 2, respectively, where i = 1, 2,3 .

Table 1. Parameters of a moving vehicle-mounted flexible manipulator. Parameter

Description

Value

Length of the flexible link

0.6 m

ρ

Mass per unit span of the piezoelectric smart materials link

0.08 kgm-1

M

Mass of the tip payload

0.03 kg

EI

Flexural rigidity of the piezoelectric smart materials link

0.2 Nm2

Ih

Moment of inertia of the flexible link

0.2 kgm2

m

Mass of the vehicle

0.5 kg

Thicknesses of the beam

0.08 m

Width of both the beam and the piezoelectric materials

0.1 m

c1

Thicknesses of upper surface of piezoelectric materials

0.008 m

c2

Thicknesses of lower surface of piezoelectric materials

0.004 m

hL

Coupling parameter per unit length of the smart materials

2 Vm 2

βL

Impermittivity per unit length of the smart materials link

50 m 3 F −1

AC C

b

EP

a

TE D

L

20

ACCEPTED MANUSCRIPT Table 2. Parameters of described performance functions. Parameter

Value

λ10

0.4

λ1∞

0.002

l1

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1.6

λ20

1.2

λ2 ∞

0.003

l2

1.8

λ30

SC

0.002

λ3∞

2 ×10−5

l3

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2

The control gains of the system are set in Table 3.

Table 3. Control gains of the system. Parameter

k p1

kp2 kd 2

k p3

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kd1

Value 40 10 40 20 40 15

k p4

10

kd 4

20

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EP

kd 3

To validate the control performance and demonstrate the advantages of the proposed control

over the traditional control scheme, the dynamic responses of the system are simulated with the above parameters in the following three cases. Case 1: Without control: τ ( t ) =0 , um ( t ) =0 , u L ( t ) =0 , v ( x, t ) =0 ;

Case 2: With an improved PID control with gravity compensation: τ ( t ) = − k1e1 ( t ) − k 2 e&1 ( t ) − k 3 ∫ e1 (σ )d σ + t

0

1 g ρ L2 cos θ ( t ) 2

+ MgL cos θ ( t ) − Mgw ( L , t ) sin θ ( t ) − ρ ∫ gw ( x , t ) sin θ ( t )d x L

0

21

(62)

ACCEPTED MANUSCRIPT um ( t ) = −k4 e2 ( t ) − k5e&2 ( t ) − k6 ∫ e2 (σ )dσ

(63)

uL ( t ) = −k7 e3 ( t ) − k8e&3 ( t ) − k9 ∫ e3 (σ )dσ + Mg cosθ ( t )

(64)

t

0

t

0

x

+k12 ∫

t

0

χ

χ

∫ w (ς , t )dς dχ + k ∫ ∫ w& (ς , t )dς dχ x

11 0

0

x

∫∫

0 0

χ

0

0

w& (ς , σ )dς dχ dσ +

(65)

g ρ c1 2 x cos θ ( t ) 2

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v ( x, t ) = k10 ∫

where k1 = 20 , k 2 = 10 , k3 = 10 , k 4 = 20 , k5 = 20 , k 6 = 2 , k 7 = 20 , k 8 = 15 , k9 = 10 , k10 = 10 , k11 = 20 , k12 = 10 ;

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Case 3: With the proposed control in Eqs. (23)-(26).

Simulation results for Cases 1-3 are shown in Figs. 3-18. The dynamic responses of the system without control inputs in Case 1 are shown in Figs. 3-6 to clearly display the control

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effects. It is apparent that the system is unstable and the vibration of the manipulator cannot converge within 4 s.

Elastic deflection of the flexible manipulator

0.4

0

-0.2

EP

-0.4 0.6

TE D

Deflection (m)

0.2

4

AC C

0.4

x (m)

3 2

0.2 1 0

0

Time (s)

Fig. 3. The elastic deflection of the flexible manipulator without control.

22

ACCEPTED MANUSCRIPT Speed of elastic deflection of the flexible manipulator

Speed of deflection (m s-1)

2

1

-1

-2 0.6

RI PT

0

4

0.4

3

2

0.2 0

x (m)

SC

1 0

Time (s)

Angle tracking (rad)

2

-2

0.5

1

1.5

2 Time (s)

2.5

TE D

0

3

3.5

4

4 2 0 -2

EP

Angle speed tracking (rad s -1)

Angle tracking Desired angle

0

-4

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Fig. 4. The speed of the elastic deflection of the flexible manipulator without control.

-4

0

0.5

1

1.5

Angle speed tracking Desired angle speed 2 Time (s)

2.5

3

AC C

Fig. 5. The angular position and speed of the joint without control.

23

3.5

4

0.5

0

0.5

1

1.5

2 Time (s)

2.5

3

0.2

4

Vehicle speed tracking Desired vehicle speed

0.1 0 -0.1 -0.2

3.5

RI PT

0

Vehicle speed tracking (m s -1)

Position tracking Desired position

1

0

0.5

1

1.5

2 Time (s)

2.5

3.5

4

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Fig.6. The position and speed of the vehicle without control.

3

SC

Vehicle tracking (m)

ACCEPTED MANUSCRIPT

The performance of the system with control input for Cases 2 and 3 is shown in Figs. 7-11. Fig. 7 gives the position of the joint and its speed. Fig. 8 shows both the position and speed of the vehicle. Fig. 9 shows the elastic deflection and speed of the elastic deflection at the free end of the flexible manipulator. The elastic vibration of the flexible manipulator with control inputs is shown in Fig. 10, and the speed of the elastic deflection in the closed-loop system is shown in Fig. 11.

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Figs. 7(b)-11(b) indicate that under the proposed control, the joint and vehicle are positioned to the desired set points and the speeds of the joint and vehicle converge to zero over time. The vibration of the flexible manipulator is also significantly suppressed under the proposed control within 4 s, which indicates that the proposed controller has very good control effect. Compared with Figs. 7(b)-11(b), Figs. 7(a)-11(a) show that the control performance is poorer and the convergence rate

AC C

EP

is slower with the PID control.

24

ACCEPTED MANUSCRIPT

Angle tracking Desired angle

0.1

0.5

1

1.5

2 Time (s)

2.5

3

3.5

0.05

4

Angle speed tracking Desired angle speed

0.4 0.2 0

0

0.5

1

1.5

2 Time (s)

2.5

3

3.5

4

Angle tracking Desired angle

0.1

0 0

0.6

-0.2

0.2 0.15

0

0.5

1

1.5

2 Time (s)

2.5

3

3.5

4

0.8 Angle speed tracking Desired angle speed

0.6 0.4 0.2 0 0

0.5

1

1.5

2 Time (s)

2.5

3

3.5

4

(b)

SC

(a)

RI PT

0.2

0

Angle speed tracking (rad s -1)

Angle tracking (rad)

0.3

Angle speed tracking (rad s -1)

Angle tracking (rad)

0.4

Fig. 7. The angular position and speed of the joint. (a) With the PID control for Case 2. (b) With

0.5

Position tracking Desired position 1

1.5

2 Time (s)

2.5

3

3.5

Vehicle speed tracking Desired vehicle speed

1

0.5

0

0.5

1

1.5

2 Time (s)

2.5

3

3.5

4

(a)

Position tracking Desired position

0.5

4

Vehicle speed tracking (m s -1)

0.5

1.5

0

1

0

0

EP

Vehicle speed tracking (m s -1)

0

Vehicle tracking (m)

1

TE D

Vehicle tracking (m)

1.5

M AN U

the proposed control for Case 3.

0

0.5

1

1.5

2 Time (s)

2.5

3

3.5

2.5

Vehicle speed tracking Desired vehicle speed

2 1.5 1 0.5 0 0

0.5

1

1.5

2 Time (s)

2.5

3

3.5

(b)

Fig. 8. The position and speed of the vehicle. (a) With the PID control for Case 2. (b) With the

AC C

proposed control for Case 3.

25

4

4

ACCEPTED MANUSCRIPT

Boundary deflection (m)

x 10

Desired boundary deflection Boundary deflection

-5

-10

0

0.5

1

1.5

2 Time (s)

2.5

3

3.5

4

0.01

0 Desired speed of boundary deflection Speed of boundary deflection

-0.01

-0.02

0

0.5

1

1.5

2 Time (s)

2.5

3

3.5

4

5

x 10

-4

0 Desired boundary deflection Boundary deflection

-5

-10

0

0.5

1

1.5

2 Time (s)

2.5

3

3.5

4

0.01

RI PT

0

Speed of boundary deflection (m s-1)

Speed of boundary deflection (m s-1)

Boundary deflection (m)

-4

5

0 -0.01

Desired speed of boundary deflection Speed of boundary deflection

-0.02 -0.03

0

0.5

1.5

2 Time (s)

2.5

3

3.5

4

(b)

SC

(a)

1

Fig. 9. The elastic deflection and speed of the elastic deflection at the end of the flexible

Elastic deflection of the flexible manipulator

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manipulator. (a) With the PID control for Case 2. (b) With the proposed control for Case 3.

Elastic deflection of the flexible manipulator

-3

x 10

-4

x 10

2

2

-1

-2 0.6

TE D

0

4

0.4

3

Deflection (m)

1 0

-1 -2 -3 -4 0.6 4

0.4

3

2

0.2 0

Time (s)

1 x (m)

EP

0

(a)

2

0.2

1

x (m)

0

0

Time (s)

(b)

Fig. 10. The elastic deflection of the flexible manipulator. (a) With the PID control for Case 2. (b)

AC C

Deflection (m)

1

With the proposed control for Case 3.

26

ACCEPTED MANUSCRIPT Speed of elastic deflection of the flexible manipulator

Speed of elastic deflection of the flexible manipulator

x 10

-3

3 Speed of deflection (m s-1)

0.05 0 -0.05 -0.1 -0.15 0.6

2

1

0

-1 0.6 4

0.4

4

0.4

3

3

2

0.2 0

2

0.2

1 0

x (m)

RI PT

Speed of deflection (m s-1)

0.1

1

Time (s)

0

x (m)

(a)

0

Time (s)

SC

(b)

Fig. 11. The speed of the elastic deflection of the flexible manipulator. (a) With the PID control

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for Case 2. (b) With the proposed control for Case 3.

Tracking errors of the joint and vehicle are shown in Figs. 12 and 13, respectively, with the proposed described performance function curves. The error of boundary deflection of the flexible manipulator is shown in Fig. 14. As per our theoretical analysis, system error ei ( t ) , i = 1, 2,3 , can converge to the prescribed performance bounds without transgression of the prescribed speed only under the proposed control in Figs. 12(b)-14(b). As compared with the PID control in Figs

TE D

12(a)-14(a), it is obvious that the prescribed performance can always be satisfied under the proposed control.

0.4

0.4

EP

λ 1(t)

0.1 0 -0.1 -0.2 -0.3 -0.4

0

0.5

1

1.5

2 Time (s)

λ 1(t) 0.3

e1(t)

0.2

AC C

Tacking error of joint (rad)

0.2

e1(t)

Tacking error of joint (rad)

0.3

0.1 0 -0.1

-3

6 4 2 0 -2

-0.2 -0.3

x 10

3.92

2.5

3

3.5

4

-0.4

(a)

0

0.5

1

1.5

2 Time (s)

2.5

3.94 3

3.96 3.5

4

(b)

Fig. 12. The tracking error of the joint. (a) With the PID control for Case 2. (b) With the proposed control for Case 3.

27

ACCEPTED MANUSCRIPT 1.5

1.5

λ 2(t)

e2(t)

0

-0.5

0.5

0

-3

x 10

-0.5

RI PT

Tacking error of vehicle (m)

1

0.5

5 0

-1

-1

-5

3.963.965 3.97 3.975

-1.5

-1.5

0

0.5

1

1.5

2 Time (s)

2.5

3

3.5

4

0

0.5

1

1.5

(a)

2 Time (s)

2.5

3

3.5

4

SC

(b)

Fig. 13. The tracking error of the vehicle. (a) With the PID control for Case 2. (b) With the

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proposed control for Case 3.

-3

-3

2

x 10

2

x 10

λ 3(t)

λ 3(t) 1.5

Boundary deflection error (m)

1 0.5

TE D

0 -0.5 -1 -1.5

0

0.5

1

1.5

2 Time (s)

2.5

3

3.5

w(L,t)

1

0.5

0

-0.5

-5

x 10

-1

2

-1.5

-2

0

4

EP

-2

1.5

w(L,t)

3.9 -2

0

(a)

0.5

1

1.5

2 Time (s)

2.5

3.95 3

3.5

4

(b)

Fig. 14. The error of the boundary deflection of the flexible manipulator. (a) With the PID control for Case 2. (b) With the proposed control for Case 3.

AC C

Boundary deflection error (m)

Tacking error of vehicle (m)

λ 2(t)

e2(t)

1

Four control inputs τ ( t ) , um ( t ) , uL ( t ) , and v ( x, t ) which can be achieved by different

sensors are shown in Figs. 15-18 as follows.

28

4.5

9

4

8

3.5

7 6

2.5 2 1.5

5 4 3 2

1

1 0.5 0

0 0

0.5

1

1.5

2 Time (s)

2.5

3

3.5

-1

4

0

0.5

1

1.5

2 Time (s)

2.5

3

3.5

4

(b)

SC

(a)

RI PT

3

Control input τ(t) (N)

Control input τ(t) (N)

ACCEPTED MANUSCRIPT

Fig. 15. Control input τ ( t ) . (a) The PID control for Case 2. (b) The proposed control for Case 3.

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25

45 40

20

30

Control input um(t) (N)

15

10

0

0

0.5

1

1.5

2 Time (s)

(a)

2.5

3

3.5

4

20 15 10 5 0 -5

0

0.5

1

1.5

2 Time (s)

2.5

3

3.5

4

(b)

Fig. 16. Control input um ( t ) . (a) The PID control for Case 2. (b) The proposed control for Case 3.

EP

-5

TE D

5

25

AC C

Control input um(t) (N)

35

29

ACCEPTED MANUSCRIPT 0.7

0.9 0.8

0.4

0.3

0.2

0.1

0

0.7 0.6 0.5 0.4 0.3 0.2 0.1

0

0.5

1

1.5

2 Time (s)

2.5

3

3.5

0

4

0

0.5

1

1.5

(a)

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Boundary control input uL(t) (N)

0.5

2 Time (s)

2.5

3

3.5

4

(b)

SC

Fig. 17. Boundary control input uL ( t ) . (a) The PID control for Case 2. (b) The proposed control

-190 -192 -194 -196

-200 0.6

TE D

-198

Distributed control voltage v(x,t) (V)

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for Case 3.

4

0.4

3

-150 -160 -170 -180 -190 -200 0.6

4

0.4

3

2

0.2

1

0

x (m)

0

Time (s)

2

0.2 1 x (m)

0

Time (s)

(b)

EP

(a)

0

Fig. 18. Distributed control voltage v ( x, t ) . (a) The PID control for Case 2. (b) The proposed control for Case 3.

AC C

Distributed control voltage v(x,t) (V)

Boundary control input uL(t) (N)

0.6

The proposed control laws are indeed effective according to the simulation results shown in

Figs. 3-18. Under the proposed control scheme, positions of the joint and vehicle can be regulated to desired values while vibration of the flexible manipulator is suppressed. When t → ∞ , θ ( t ) → θ d , θ& ( t ) → 0 ,

r ( t ) → rd ,

r& ( t ) → 0 ,

w ( L, t ) → 0 ,

w& ( L, t ) → 0 ,

w ( x, t ) → 0 ,

w& ( x, t ) → 0 can be realized in the closed-loop system. Furthermore, the prescribed performance

can be guaranteed under the proposed controller. Our results also indicate that compared to the traditional ODE model with modal truncation [48], there is no control spillover problem existing 30

ACCEPTED MANUSCRIPT in the proposed control method based on the infinite dimensional PDE model. Thus, no additional approach is required in this paper to eliminate the control and observation spillover [49-51].

5. Conclusions This paper addresses a control problem of a moving vehicle-mounted flexible manipulator with output constraints under a prescribed performance method. A PDE model is established to

RI PT

describe the system dynamics with considering the dynamics of piezoelectric actuators and sensors. Based on the model, state-feedback control laws are proposed to not only regulate joint and vehicle positions, but also eliminate flexible link vibration with prescribed performance. With the proposed control scheme, the output errors converge to predefined arbitrarily small residual sets and the convergence rates are no less than pre-specified values. Simulations are conducted to

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validate the control scheme. In the future, we plan to study the vibration control of vehicle-mounted flexible manipulators based on the ODE model systemically to make further comparison between the PDE control and ODE control. Moreover, disturbance observer and

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robust adaptive control for the moving vehicle-mounted flexible manipulator system will also be developed to compensate the influence of uncertainties in the future work.

Acknowledgments

This work was supported by the Research Fund for the National Natural Science Foundation of China [grant number 61374048].

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References

[1] W. He, S. S. Ge, Cooperative Control of a Nonuniform Gantry Crane with Constrained Tension, Automatica 66(4) (2006) 146-154.

[2] P. Williams, D. Sgarioto, P. Trivailo, Optimal control of an aircraft-towed flexible cable system,

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Journal of Guidance, Control, and Dynamics 29(2) (2006) 401-410. [3] W. He, T. Meng, X. He, S. S. Ge, Unified Iterative Learning Control for Flexible Structures

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with Input Constraints, Automatica 86 (2018) 326-336. [4] V. A. Spector, H. Flashner, Modelling and design implications of noncollocated control in flexible systems, Journal of Dynamic Systems, Measurement, and Control 112(2) (1990) 186-193. [5] Z. Liu, J. Liu, W. He, Dynamic modeling and vibration control for a nonlinear three-dimensional flexible manipulator, International Journal of Robust and Nonlinear Control 28(13) (2018) 3927-3945. [6] Y. Zhang, H, Zhou, Y. Zhou, Vibration Suppression of Cantilever Laminated Composite Plate with Nonlinear Giant Magnetostrictive Material Layers, Acta Mechanica Solida Sinica 28(1) (2015) 50-61. [7] M. Sabatini, P. Gasbarri, R. Monti, G. B. Palmerini, Vibration control of a flexible space 31

ACCEPTED MANUSCRIPT manipulator during on orbit operations, Acta Astronautica 73 (2012) 109-121. [8] S. K. Tso, T. W. Yang, W. L. Xu, Z. Q. Sun, Vibration control for a fexible-link robot arm with deflection feedback, International Journal of Non-Linear Mechanics 38(1) (2003) 51-62. [9] S. Kalayctoglu, M. Giray, H. Asmer, Vibration control of flexible manipulators using smart structures, Proceedings of the 12th IEEE International Symposium on Intelligent Control, Istanbul,

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Turkey, 1997, pp. 415-420. [10] S. Narayanan, V. Balamuruga, Finite element modelling of piezolaminated smart structures for active vibration control with distributed sensors and actuators, Journal of Sound and Vibration 262(3) (2003) 529-562.

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[11] S. S. Ge, S. Zhang, W. He, Vibration Control of a Flexible Timoshenko Beam under Unknown External Disturbances, Proceedings of the 30th Chinese Control Conference, Yantai, China, 2011, pp. 1031-1036.

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[12] S. S. Ge, T. H. Lee, G. Zhu, Improving Regulation of a Single-Link Flexible Manipulator with Strain Feedback, IEEE Transactions on Robotics and Automation 14(1) (1998) 179-185. [13] A. A. Paranjape, J. Guan, S.-J. Chung, M. Krstic, PDE Boundary Control for Flexible Articulated Wings on a Robotic Aircraft, IEEE Transactions on Robotics 29(3) (2013) 625-640. [14] M. S. Queiroz, D. M. Dawson, M. Agarwal, F. Zhang, Adaptive Nonlinear Boundary Control

779-787.

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of a Flexible Link Robot Arm, IEEE Transactions on Robotics and Automation 15(4) (1999)

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Appendix

The proving process of sufficient conditions of LaSalle’s Invariance Principle is provided as

follows.

We define

q = [ q1 q2

q3

q4

q5

q6

q7

q8 ]

T

= e1 ( t ) e&1 ( t ) e2 ( t ) e&2 ( t ) w ( x, t ) w& ( x, t ) w ( L, t ) w& ( L, t )  . T

The PDE system described by Eqs. (9)-(15) can be compactly written as

35

(A1)

ACCEPTED MANUSCRIPT q& = A% q = Aq + f ( q) , q ( 0) ∈ H .

(A2)

In (A2), the infinite-dimensional linear operator A is

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q2     − k p1q1 − kd 1q2     1 3 2 I h + ρ L + ML   3   q4     −k p 2 q3 − kd 2 q4    , ∀q ∈ D ( A ) M + m + ρL Aq =    q6    −k p 4 q5 − kd 4 q6 − EIq5 xxxx    ρ     q8    −k p 3 q7 − kd 3 q8 + EIq7 xxx      M

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(A3)

and the infinite-dimensional nonlinear operator f ( q ) is given as f ( q ) = [ 0 b1 0 b2

0 b3

0 b4 ]

where

T

(A4)

(A5)

 1 2    1 2  2   ρ L + ML  cos ( q1 + θ d ) q2 +  ρ L + ML  sin ( q1 + θ d ) q&2    2   2   − M sin ( q1 + θ d ) q22 q7 + M cos ( q1 + θ d ) q&2 q7 + 2M cos ( q1 + θ d ) q2 q8    1 2  &2 q5   L  sin ( q1 + θ d ) q2 q5 − cos ( q1 + θ d ) q b2 = M + m + ρ L  + M sin ( q1 + θ d ) q&8 − ρ ∫0  dx    −2 cos ( q1 + θ d ) q2 q6 − sin ( q1 + θ d ) q&6      + 3 q 3 ln λ2 ( t ) + q3 − 3 λ t q − 1 λ 2 t ln λ2 ( t ) + q3   4 3 λ (t ) − q 2 2 ( ) 3 4 2 ( ) λ (t ) − q  2 3 2 3  

(A6)

b3 = sin ( q1 + θ d ) q& 4 + q22 q5 − xq& 2

(A7)

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 1 2    −2Mq2 q7 q8 +  ML + ρ L  sin ( q1 + θ d ) q&4  2      + M cos ( q1 + θ d ) q7 q&4 − Mq&2 q72 − MLq&8  1   b1 = L 2   1 3   & & & q q q q q xq q q q x 2 + + − cos + d θ 2 −ρ ( ) 5 2 6 1 d 5 4 I h + ρ L + ML  ∫0  5 6 2  3  3 λ ( t ) + q1 3 λ ( t ) + q1  1  + q12 ln 1  − λ1 ( t ) q1 − λ12 ( t ) ln 1  4 4 λ1 ( t ) − q1 2 λ1 ( t ) − q1  

36

ACCEPTED MANUSCRIPT b4 = q22 q7 + sin ( q1 + θ d ) q&4 − Lq& 2 +

3 2 λ3 ( t ) + q7 q7 ln λ3 ( t ) − q7 4M

(A8)

λ ( t ) + q7 3 1 2 − . λ3 ( t ) q7 − λ3 ( t ) ln 3 2M 4M λ3 ( t ) − q7 The spaces mentioned in Eqs. (A2) and (A3) are defined as

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H = ℜ 4 × H 2 × L2 × ℜ 2

D ( A ) = {q ∈ℜ4 × H 4 × H 2 ×ℜ2 | q5 ( 0 ) = 0, q5 x ( 0 ) = 0, q5 xx ( L ) = 0, q8 = q6 ( L ) , q7 = q5 ( L )}

where

{

}

2

Ω

{

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L2 ( Ω ) = f | ∫ f ( x ) dx < ∞

}

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H k ( Ω ) = f | f , f ' ,..., f ( k ) ∈ L2 ( Ω ) , Ω = [ 0, L ] .

We define an inner-product as

q, z

H

L 1   =  I h + ρ L3 + ML2  q2 z2 + ( M + m + ρ L ) q4 z4 + Mq8 z8 + ∫ ρ q6 z6 dx 0 3   L

(A9)

L

+ ∫ EIq5 xx z5 xx dx + k p1q1 z1 + k p 2 q3 z3 + k p 3 q7 z7 + ∫ k p 4 q5 z5 dx 0

0

E (t ) =

(H,

1 q, q 2

H

⋅, ⋅ =

H

)

1 q 2

z3

z4

z5

z6

z7

z8 ] ∈ H . T

is a Hilbert space [52], we describe the energy of Eq. (A9) as 2

.

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Since

z2

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where q ∈ H and z = [ z1

H

(A10)

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Differentiating the Lyapunov function E ( t ) along the solution trajectories of q& = A q with respect to time, we can obtain L E& ( t ) = −kd 1θ&2 ( t ) − kd 2 r& 2 ( t ) − kd 3 w& 2 ( L, t ) − kd 4 ∫ w& 2 ( x, t )dx ≤ 0 . 0

Therefore, it can be seen that A

h = [ h1 h2

h3

h4

h5

h6

h7

(A11)

is a dissipative operator. Then we define vector

h8 ] ∈ H . Consider the equation T

Aq = h

(A12)

where 37

ACCEPTED MANUSCRIPT h1 = q 2

(A13)

− k p1 q1 − k d 1 q2 1 I h + ρ L3 + ML2 3

h2 =

(A14)

h3 = q 4

−k p 2 q3 − kd 2 q4

(A16)

M + m + ρL

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h4 =

(A15)

h5 = q 6

(A17)

−k p 4 q5 − kd 4 q6 − EIq5 xxxx

h6 =

(A18)

ρ

−k p 3 q7 − kd 3 q8 + EIq7 xxx

M

.

(A20)

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h8 =

(A19)

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h7 = q8

By solving Eqs. (A13)-(A20) with boundary conditions (13)-(15), we can get the uniquely solution of q . Hence Eq. (A12) has a unique solution q ∈ D ( A ) which means that A − 1 exists and maps H into ℜ4 × H 4 × H 2 × ℜ2 . Furthermore, since A − 1 maps every bounded set of H into bounded set of ℜ4 × H 4 × H 2 × ℜ2 and the embedding of the latter space into H is compact, it follows that A − 1 is a compact operator. Then if we choose λ different from the eigenvalues

− A ) is a compact operator [53]. For the operator A defined in Eq. (A3), it is clear −1

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( λI

of A ,

that 0 ∈ R ( A ) . Consequently, the solution trajectories of system q& = A q are precompact in H for t ≥ 0 [54].

− A ) q = A ( λ A −1 − I ) q = g

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(λI

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Given λ > 0 and g ∈ H , let

where

I

(A21)

is a unit operator.

It proves that for 0 < λ < A −1

−1

, Eq. (A21) has a unique solution q ∈ D ( A ) by contraction

mapping theorem in [55]. Hence the operator λ I − A is onto H for 0 < λ < A −1

−1

. Moreover,

the operator λ I − A is onto H for all λ > 0 . Therefore, we can have that D ( A ) is dense in

H [56]. Based on the Lumer-Phillips theorem [57], operator A generates a C0 -semigroup of

( )

contraction T ( t ) on H . Moreover, since f ( q ) is a bounded operator and 0 ∈ R A% , we can

38

ACCEPTED MANUSCRIPT know that operator A% in Eq. (A2) generates a C0 -semigroup of contraction T ( t ) on H . Thus all the solution trajectories of system in Eq. (A2) are precompact in H for t ≥ 0 .

From above proving, all sufficient conditions of extended LaSalle’s Invariance Principle have been received. By these conditions, the extended LaSalle’s Invariance Principle can be applied to

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the system.

39