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Adaptive nonlinear vibration control of a Cartesian ﬂexible manipulator driven by a ballscrew mechanism Zhi-cheng Qiu n School of Mechanical and Automotive Engineering, South China University of Technology, Guangzhou 510641, PR China

a r t i c l e in f o

abstract

Article history: Received 9 October Received in revised 1 January 2012 Accepted 5 January Available online 26

A ﬂexible Cartesian manipulator is a coupling system with a moving rigid body and ﬂexible structures. Thus, vibration suppression problem must be solved to guarantee the stability and control accuracy. A characteristic model based nonlinear golden section adaptive control (CMNGSAC) algorithm is implemented to suppress the vibration of a ﬂexible Cartesian smart material manipulator driven by a ballscrew mechanism using an AC servomotor. The system modeling is derived to recognize the dynamical characteristics. The closed loop stability is analyzed based on the model. Also, an experimental setup is constructed to verify the adopted method. Experimental comparison studies are conducted for modal frequencies’ identiﬁcation and active vibration control of the ﬂexible manipulator. The active vibration control experiments include set-point vibration control responses, vibration suppression under resonant excitation and simultaneous translating and vibration suppression using different control methods. The experimental results demonstrate that the controller can suppress both the larger and the lower amplitude vibration near the equilibrium point effectively. & 2012 Elsevier Ltd. All rights reserved.

2010 form 2012 January 2012

Keywords: Flexible Cartesian manipulator Active vibration suppression Characteristic model Nonlinear control Adaptive control

1. Introduction Space manipulators should be as light as possible in order to reduce their launching cost [1]. They are usually made of lightweight materials to help launch. Space robots usually have very low damping ratios, high dimensional order and parametric uncertainties in dynamics. Thus, unwanted vibrations of the ﬂexible links will be caused unavoidably [2,3]. In order to meet the increasing demand for high end-position accuracy robotic systems coupled with needs of space application, the issue on modeling and active vibration control for such a ﬂexible Cartesian robot system is rather difﬁcult and challenging. The dynamical modeling and active control of a ﬂexible Cartesian robot have attracted attention of many researchers in the last few decades [5–11]. In most of previous research works, servomotors and piezoelectric patches are used as actuators to achieve desired end-point position while suppressing unwanted vibration. PZT (lead zirconate titanate) as a kind of smart material was effectively implemented to control the mechanical vibration of ﬂexible structures [4]. In the vibration control of ﬂexible robot arms, boundary feedback schemes are employed in order to damp the vibrations and dissipate energy [5]. Hou and Tsui [6] formulated a mathematical model for a ﬂexible robot arm on a moving base with a payload at the tip end using a fourth order partial differential equation with boundary conditions. They show that such

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Z.C. Qiu / Mechanical Systems and Signal Processing 30 (2012) 248–266

249

a system is both controllable and observable in an inﬁnite dimensional Hilbert space, through the state-space formulation. Luo et al. [7] proposed a shear force feedback control method to suppress vibrations arising from structural ﬂexibility of Cartesian type robots. Dadfarnia et al. [8,9] presented theoretical and experimental results of an observer-based control strategy and a Lyapunov-based controller for regulating problem of a ﬂexible Cartesian robot. Ge et al. [10,11] presented a robust distributed controller and an asymptotically stable end-point regulation controller for a single-link Cartesian smart materials robot. Both simulation and experimental results veriﬁed the good performance in suppression of residual vibrations under the environment of disturbances. Although great progress has been achieved in this ﬁeld, the issue is still far from completely solved. The ﬂexible robot system is of inﬁnite dimensionality. Once using the technique of modal truncation, undesired residual vibrations make ﬂexible robots difﬁcult to control with high precision [10]. For ﬂexible Cartesian robot, the ballscrew is a most popular driving mechanism utilized in high-speed and long stroke positioning stages. However, due to the backlash and nonlinear friction force between the ballscrew and nut, it is difﬁcult to obtain sub-micrometer resolution based on a ballscrew mechanism [12]. The non-linear behavior of a single-link ﬂexible visco-elastic Cartesian manipulator was studied by Pratiher and Dwivedy [13]. As to nonlinear control, Yang et al. [14] proposed a feedback nonlinear control law for the endpoint control of a ﬂexible macro–micro manipulator system. Their dynamic behavior presents a coupling between rigid body displacements and ﬂexible modes of vibration. These model-based controllers, originally designed for the demands of high performance, may not be easy to implement due to uncertainties in design models, large variations of loads on the robot’s end-effector, ignored high frequency dynamics and the high order of the designed controllers [7]. In order to design a satisfactory control system, according to traditional control theory, a mathematical model that describes system dynamics must be determined beforehand. However, it is difﬁcult to establish their mathematical models accurately for some plants, as their characteristics and environment may change unpredictably [15]. Even if a precise mathematical model can be established, the order of the model is extremely high and the structure is very complicated. Unfortunately, the high-order controller is very difﬁcult to realize. In general, the controller design in previous researches for ﬂexible structures control depends on the reduced order models by modal truncation. Therefore, the problems of ‘‘observation spillover’’ and ‘‘control spillover’’ are unavoidable due to the ignored high-frequency dynamics [7]. To solve the ‘‘spillover’’ problem in traditional modal truncation methods, a characteristic model (CM) based adaptive control method was proposed by Wu et al. [15,16]. The high order system of ﬂexible structures can be equivalently modeled as a second order time-varying difference equation, named as ‘‘characteristic model’’. It shows a new way to control large ﬂexible structures with low order controllers and avoids the ‘‘spillover’’ problem at the same time. CM is to implement the controller from the engineering point of view. The CM based nonlinear golden section adaptive control (CMNGSAC) algorithm was introduced in Ref. [16,17]. Using this control law, the low amplitude residual vibration near the equilibrium point will be suppressed quickly. In recent years, several researchers have employed the CMNGSAC algorithm for vibration control for ﬂexible structures, spacecraft attitude control and tracking control of robotic manipulators [15–19]. In order to deal with complex nonlinear system, Luo et al. [20] presented a novel neuro-fuzzy dynamic characteristic modeling method by introducing neural network into the fuzzy characteristic modeling control. Among the aforementioned researches, in many cases, only simulation results were obtained. Few experiments are conducted to verify this method. The major contribution of this paper rests on two aspects. The ﬁrst is stability analyses of the CMNGSAC algorithm via equivalent gain adaptive regulating control algorithm, using the model of the Cartesian robot. The second is experimental implementation of the CMNGSAC algorithm to suppress the elastic vibration of a ﬂexible smart Cartesian manipulator. In this regard, a kind of ﬂexible Cartesian smart materials robot using a ballscrew mechanism is designed and constructed for experimental studies. Experimental results are provided to demonstrate the satisfactory control performance and robustness of the CMNGSAC method. The rest of this article is organized as follows. Section 2 introduces the system and the governing equations for a ﬂexible Cartesian smart materials robot. In Section 3, the CMNGSAC algorithm is discussed. The stability of CMNGSAC algorithm for the closed-loop system is analyzed. In Section 4, the experimental setup of the ﬂexible Cartesian smart materials robot is designed and constructed. And experimental comparison studies are conducted using several different control algorithms, including proportional derivative (PD) control, positive position feedback (PPF) control and the CMNGSAC method with respect to the ﬂexible Cartesian robot system. The experimental validation contents include the following cases: set point vibration active control, vibration suppression under resonant excitation and simultaneous translating motion and vibration suppression. The paper ends with conclusions in Section 5. 2. System description and mathematical modeling The schematic diagram of the ﬂexible smart materials Cartesian robot is illustrated in Fig. 1. The ﬂexible manipulator is a translating uniform beam clamped on the translational slider of a ballscrew drive system. An AC servomotor with a rotary encoder is used to drive the ballscrew. PZT patches used as sensors and actuators are stuck on both surfaces of the host beam, close to the clamped side. The vibration of the ﬂexible beam can be measured by the PZT sensor. An acceleration sensor can also be used to measure the vibration, which is mounted close to the tip side of the beam. It is considered as the increase of the centralized mass at the mounted position of the beam. In this work, the measured signal

250

Z.C. Qiu / Mechanical Systems and Signal Processing 30 (2012) 248–266

AC motor Coupling

PZT actuator

Accelerometer

s (t) w (x, t) Slider

PZT sensor

Flexible beam

Ballscrew drive system

Y X Fig. 1. Schematic diagram of the ﬂexible Cartesian smart materials robot.

of the acceleration sensor is not used for feedback control. Acceleration feedback control will be investigated in the future research. Based on the ideas of previous researches [2,7,10,27], a dynamic model of the ﬂexible Cartesian manipulator system with a ballscrew mechanism is derived using Lagrange’s equation. The dynamics model is needed to analyze the stability of closed-loop system as a prerequisite, and it helps to recognize the system’s characteristics. For system modeling, the following assumptions are made [2,11]: (a) the link is thin enough to satisfy the Euler beam assumptions. No axial deformation is considered. So when the translating motion of the slider completely stops, the transverse bending vibration of the beam is the primary motion. (b) PZT sensors and actuators are perfectly bonded to the structure. (c) The ﬂexible manipulator is in the horizontal plane. The base of the ﬂexible manipulator can only move along the Y-axis of the plane. (d) The effects of gravity are neglected which is always true for space applications. It is assumed that the deﬂection of the ﬂexible beam w(x,t) is relatively small compared with the length of the beam L. Using the assumed modes method, the transverse deﬂection of the mid-plane of the beam is expressed as wðx,tÞ ¼

m-1 X

fj ðxÞqj ðtÞ ¼ UðxÞq, j ¼ 1,2,. . .,

ð1Þ

j¼1

where fj and qj denote the jth mode shape function and corresponding generalized coordinate, respectively; UðxÞ ¼ ½f1 ðxÞ. . .fj ðxÞ. . .fm ðxÞ and q ¼ ½q1 . . .qj . . .qm T denote the modal shape function vector and generalized coordinate vector, respectively; m is the modal number. Actually, the boundary conditions used for deﬁning the assumed modes of the ﬂexible moving manipulator are dynamic-free. Moreover, the dynamic boundary condition is dependent on the moving speed produced by the motor’s driving moment. Generally, the speed reduction ratio of the ballscrew system is high, and the motion speed of the slider is comparatively low. Therefore, its high torque-to-force magniﬁcation causes the poor back driving ability. In this case, the boundary conditions are close to those of the clamped-free beam with tip centralized mass. In addition, when the slider is controlled to move with a constant speed, the boundary conditions are also equivalent to the clamped-free case. This can be tested and veriﬁed in Section 4. Since the ﬂexible manipulator is in the horizontal plane, its gravitational energy is not included. Also, the effects of the shear displacements can be neglected because the width of the manipulator under consideration is assumed to be signiﬁcantly larger than its thickness. In this way, the potential energy of the manipulator is related to the bending deformation of the manipulator, and it is expressed as Z L n Z xi2 1 1X Ep ¼ Eb Ib w00 ðx,tÞ2 dx þ Epe Ipe w00 ðx,tÞ2 dx, ð2Þ 2 2 i ¼ 1 xi1 0 3

where Ep is the potential energy; Eb and Ib ¼ bb hb =12 are Young’s modulus and the moment of inertia of the beam, 3 respectively; EbIb is the uniform ﬂexural rigidity; Epe and Ipe ¼ bpe hpe =12 are Young’s modulus and the moment of inertia of

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the PZT patches, respectively; hb and hpe are the thicknesses of the beam and the PZT patches, respectively; bb and bpe are the widths of the beam and the PZT patches, respectively; the primes indicate the partial derivatives with respect to the displacement variable x. The mechanical kinetic energy of the system can be written as Ek ¼

2 1 1 1 ðIr þIc þIbs Þy_ r þ M b s_ ðtÞ2 þ rb Ab 2 2 2

Z

L

2 _ ½s_ ðtÞ þ wðx,tÞ dx

0

n Z xi2 X 1 1 2 2 _ _ ½s_ ðtÞ þ wðx,tÞ dx þ M t ½s_ ðtÞ þ wðl,tÞ , þ rpe Ape 2 2 i ¼ 1 xi1

ð3Þ

where Ek is the kinetic energy; Ir is the inertia of the servomotor rotor; Ic is the inertia of the coupling; Ibs is the inertia of the thread rod; yr is the angular displacement of the servomotor’s rotor; Mb is the mass of the translating slider; s(t) is the displacement of the translational slider, and s(t)¼(yr(t))/(200p) due to the lead of the ballscrew is 10 mm; rb and Ab ¼bbhb are the mass density and the cross-sectional area of the beam, respectively; rpe and Ape ¼bpehpe are the mass density and the cross-sectional area of each PZT patch, respectively; L is the total length of the ﬂexible beam; l is the distance between the acceleration sensor and the clamped side of the beam; (xi1, xi2) denotes the left and the right coordinates of the ith piezoelectric patch; n is the number of PZT patches; Mt is the mass of the acceleration sensor; the overdot indicates the partial derivatives with respect to the time variable t. The non-conservative work done by the AC servomotor torque and piezoelectric actuator can be written as W ¼ ðtðtÞtf ðtÞÞyr þ

n X

_ cV i ðtÞðU0 ðxi2 ÞU0 ðxi1 ÞÞqF1 q,

ð4Þ

i¼1

where t(t) is the control input torque of the AC servomotor; tf(t) is the equivalent friction torque of the drive system in the servomotor side; Vi(t) is the control voltage applied to the ith PZT actuator in the direction of polarization; F1 ¼ ½f 1 ,. . .,f m is the vector accounting for the internal viscous friction in the ﬂexible beam structure; where fi is the internal viscous friction coefﬁcient of the ith mode, f i 4 0, i ¼ 1,. . .,m; c is a constant related to the piezoelectric structures. Friction in mechanical system is a nonlinear phenomenon; friction force can degrade the performance of a motion control system. The equivalent friction torque of the drive system in the AC servomotor side can be expressed as

tf ðtÞ ¼ tf r ðtÞ þ tf l ðtÞ þ

F f s ðtÞ , 200p

ð5Þ

where tfr(t) is the friction torque of the servomotor’s rotor; tﬂ(t) is the thread rod friction torque including the friction torque of screw pair and that of the both sides’ supporting bearings; Ffs(t) is the friction force between the slider and the roller guider. In this research, the friction force tf(t) is mainly viscous damping. It can be approximately calculated as tf ðtÞ ¼ dy Uy_ r where dy is the equivalent damping coefﬁcient in the servomotor side. By substituting the expressions of the kinetic energy of Eqs. (2)–(5) into Lagrange’s equations, the dynamics equation of the single ﬂexible moving manipulator can be derived as Mx€ þ Cx_ þKx ¼ U,

ð6Þ

where x, M, C, K and U are the state vector, mass matrix, centrifugal stiffening and damping matrix, stiffness matrix, control input vector, respectively, and they can be expressed as " q , U ¼ tðtÞ

x ¼ ½yr

#T

n X

T T

0

0

cV i ðtÞðU ðxi2 ÞU ðxi1 ÞÞ

,

i¼1

" MðqÞ ¼

myy

myq

mTyq

mqq

#

" ,

C¼

dy

0

0

dq

#

" ,

K¼

0

0

0

Kq

# ,

where M,C 2 Rðm þ 1Þðm þ 1Þ ; x 2 Rðm þ 1Þ1 ; K 2 Rðm þ 1Þðm þ 1Þ ; dq is the modal damping coefﬁcient vector of the ﬂexible manipulator, dq ¼ diag½dq1 . . .dqi . . .dqm . The variables and vectors can be written as myy ¼ ðIr þIc þIbs Þ þ

myq ¼

rb Ab 200p

Z

M b þ rb Ab L þ nrpe Ape Lpe þM t

L

UðxÞdxþ 0

ð200pÞ2

,

ðmyy 2 R11 Þ:

n Z xi2 rpe Ape X M UðlÞ UðxÞdx þ t , ðmyq 2 R1m Þ: 200p 200p i ¼ 1 xi1

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Z.C. Qiu / Mechanical Systems and Signal Processing 30 (2012) 248–266

The (r,s)th elements of the mass and stiffness matrices are Z L n Z xi2 X rb Ab Uk ¼ r ðxÞUk ¼ s ðxÞdx þ rpe Ape Uk ¼ r ðxÞUk ¼ s ðxÞdxþ Mt UT ðlÞUðlÞ9ðr,sÞ , mqq ðr,sÞ ¼ 0

Kq ðr,sÞ ¼

Z

L 0

Eb Ib U00k ¼ r ðxÞU00k ¼ s ðxÞdx þ

i¼1 n X

Z

i¼1

mqq 2 Rmm ,

xi1

xi2

xi1

Epe Ipe U00k ¼ r ðxÞU00k ¼ s ðxÞdx,

ðKq 2 Rmm Þ:

When the ﬂexible manipulator is deformed, the charge Q(t) generated by the PZT sensor is [21] Z x2 2 @ wðx,tÞ dx, Q ðtÞ ¼ e31 bs r @x2 x1

ð7Þ

where e31 is the piezoelectric stress constant for the ith PZT sensor, corresponding to its piezoelectric strain constant d31 ¼ 166 10 12 m/V, e31 o0; bs is the width, x1 and x2 are the location of the left and right edges of the ith piezoelectric sensor; r is the distance from the middle of the PZT patch to the middle of the beam. The PZT sensor’s signal can be ampliﬁed by a charge ampliﬁer, and the output voltage is V s ðtÞ ¼ kca Q ðtÞ ¼ kca e31 bs r½U0 ðx2 ÞU0 ðx1 Þq,

ð8Þ

where kca is the gain of the charge ampliﬁer. 3. Control algorithms 3.1. PD and PPF controller The PD control algorithm is given as follows upd ¼ K p V s K v V_ s ,

ð9Þ

where K p 4 0 and K v 40 are the proportional and differential gain of PD control law by the AC servomotor actuator, respectively. The control laws of the system include AC servomotor’s position control and active vibration control of the ﬂexible manipulator. The control objective is that the slider achieves its desired position while unwanted vibration of the ﬂexible manipulator is suppressed. Therefore, the controller for the AC servomotor comprises of two components, one is for the rigid motion, and the other is for the ﬂexible motion. The designed controller for simultaneous positioning and vibration control is a kind of composite controller. The realization of the servomotor torque PD controller can be expressed as

t ¼ K p1 ðyd yr Þ þ K v1 ðy_ d y_ r ÞK p V s K v V_ s ,

ð10Þ

where yd and y_ d are the desired angle position and the angle velocity of the servomotor’s rotor, and their units are rad and rad/s, respectively; Kp1 and Kv1 are the proportional and differential gains of PD control law for the servomotor rotor. PPF control requires that the sensor is collocated or nearly collocated with the actuator. In PPF control, structural position information is fed to a compensator. The output of the compensator, magniﬁed by a gain, is fed directly back to the structure. The equations describing PPF operation are given as [22–24] Structure : x€ þ2zox_ þ o2 x ¼ upe , upe ¼ ko2 Z,

ð11aÞ

Compensator : Z€ þ 2zc oc Z_ þ o2c Z ¼ o2c x,

ð11bÞ

where x, z and o are the modal coordinate describing displacement, damping ratio and natural frequency of the ﬂexible structure, respectively; Z, zc and oc are the compensator coordinate, damping ratio and frequency of the compensator, respectively; upe is the control input to the system by PZT actuator; k is a feedback gain, for the closed loop system being stable, the control gain should satisfy 0 oko1. The realization of PPF controller by AC servomotor torque drive can be expressed as

t ¼ K p1 ðyd yÞ þ K v1 ðy_ d y_ Þ þ k1 o2 Z,

ð12Þ

where k1 is a feedback gain using AC servomotor, and it should satisfy 0 ok1 o1. 3.2. Characteristic model based nonlinear golden section adaptive control algorithm In addition to the above mentioned two controllers, a kind of characteristic model based nonlinear golden section adaptive control is implemented. It is seemed necessary to represent the concept and deﬁnition of CM here. The author cited the description from Ref. [15–18] with a little modiﬁcation on nonlinear golden section control law and nonlinear logic integral damping control law. The so-called CM is modeling based on plant dynamics characteristics and control performance requirements, rather than only based on accurate plant dynamic analysis. The CM has the following features: (a) in a dynamic process, for the same input, the output of the CM is approximately equal to that of the practical plant, i.e.,

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253

in the dynamic process the output error can remain within a permitted error limit. In a steady state, the outputs of both the CM and the practical system are equal. (b) The structure of a CM should be simpler, easier and more convenient to implement from the engineering point than the original dynamics equation. Besides plant characteristics, the structure and order of CM mainly rely on control performance requirements. (c) The CM is signiﬁcantly different from the traditional truncation model of a high order system. It compresses all the information of the high dynamics order model into several characteristic parameters, and no information is lost. In general, CM is represented with a slowly time-varying difference equation. Therefore, the aforementioned problems of truncated model based control methods are avoided. If the controlled plant is a minimum-phase system, or a weak non-minimum-phase system, its CM can be given by a second order slowly time-varying difference equation as follows: yðkÞ ¼ f 1 ðkÞyðk1Þ þ f 2 ðkÞyðk2Þ þ g 0 ðkÞuðk1Þ ¼ uT ðk1ÞhðkÞ,

ð13Þ

where the regression vector and the parameter vector of the system are written as

uðk1Þ ¼ ½ yðk1Þ yðk2Þ uðk1Þ T , hðkÞ ¼ ½ f 1 ðkÞ f 2 ðkÞ g 0 ðkÞ T : The coefﬁcients fi(k) (the subscript is i ¼1, 2) and g0(k) in Eq. (13) are slowly time-varying. It is obvious that the range of the coefﬁcients fi(k) can be determined beforehand when no integral item and no multiple-root exist. Consider their maximum range, we can choose f1(k)A(1,2), f2(k)A[ 1,0]; g0(k) meets the condition of g 0 ðkÞ 5 1. For stable plants, it is proved that parameters f1(k), f2(k) and g0(k) belong to the bounded convex set [16]: 8 9 1:4331rf 1 ðkÞ r 1:9974 o2 > > > > > > > > < = 1 o0:9999rf 2 ðkÞ o 0:5134 ð14Þ : Ds ¼ ðf 1 ðkÞ,f 2 ðkÞ,g 0 ðkÞÞ ðkÞ þf ðkÞ r 0:9999o 1:0 0:9196o f > > 1 2 > > > > > > : ; 0 o g r g ðkÞ rg o1 min

0

max

When the static gain of the practical dynamic system is 1, the following can be obtained f 1 ð1Þ þf 2 ð1Þ þ g 0 ð1Þ61:

ð15Þ

Eq. (13) can also be expressed as an estimated parameters’ vector as ð16Þ yðkÞ ¼ f^ 1 ðkÞyðk1Þ þ f^ 2 ðkÞyðk2Þ þ g^ 0 ðkÞuðk1Þ ¼ uT ðk1Þh^ ðkÞ, h iT ^ ^ ^ ^ where hðkÞ ¼ f 1 ðkÞ f 2 ðkÞ g^ 0 ðkÞ ; f i ðkÞ and g^ 0 ðkÞ are deﬁned as the estimated values of the parameters fi(k) and g0(k), respectively. A parameter identiﬁcation of the projection algorithm composes the characteristic parameter estimator [25]:

h^ n ðkÞ ¼ h^ ðk1Þ þ

guðk1Þ ðyðkÞuT ðk1Þh^ ðk1ÞÞ, a þ uT ðk1Þuðk1Þ

h^ ðkÞ ¼ p½h^ n ðkÞ,

ð17aÞ ð17bÞ

where a 4 0 and 0 o g o2; p[x] denotes the orthogonal projection from x to Ds . Thus, ðf^ 1 ðkÞ, f^ 2 ðkÞÞ 2 Ds . The CMNGSAC method includes Maintaining/tracking control law, nonlinear golden section control law, logic integral and logic differential control law. (1) Maintaining/tracking control law u0 ðkÞ ¼

1 ½y ðkÞf^ 1 ðkÞyðkÞf^ 2 ðkÞyðk1Þ, g^ 0 ðkÞ þ l1 r

ð18Þ

where y(k) is the output, yr(k) is the desired output, l1 is a small positive constant. The desired output of vibration suppression is yr(k)¼0, then, Eq. (18) can be written as an adaptive gain regulating proportional and derivative (PD) controller form: u0 ðkÞ ¼ K p4 ðkÞUyðkÞ þK d4 ðkÞU½yðkÞyðk1Þ, ^

^

ð18aÞ

^

f 2 ðkÞ where K p4 ðkÞ ¼ f 1g^ðkÞðkÞþþf 2lðkÞ, K d4 ðkÞ ¼ g^ ðkÞ . þ l1 1 0 0 (2) Nonlinear golden-section adaptive control law. A kind of nonlinear golden-section adaptive control law is [16,17] h ^ i L1 f 1 ðkÞ þ L2 f^ 2 ðkÞ L2 f^ 2 ðkÞ yðkÞ þ L2 f^ 2 ðkÞyðk1Þ Z1 9Z3 UyðkÞ9m þ Z2 , ul ðkÞ ¼ ðg^ 0 ðkÞ þ l1 Þ

ð19Þ

pﬃﬃﬃ ~ wherepZ ﬃﬃﬃ1, Z2, Z3 and m are positive constants selected for nonlinear switch; yðkÞ ¼ yðkÞyr ðkÞ, L1 ¼ ð3 5Þ=2 0:382, L2 ¼ ð 51Þ=2 0:618; when the system is stable, L1 ¼1 and L2 ¼1. According to Eq. (19), Eq. (20) can be also re-written as an adaptive gain regulating PD controller form: ul ðkÞ ¼ K p2 ðk,yðkÞÞyðkÞK d2 ½yðkÞyðk1Þ,

ð20Þ

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Z.C. Qiu / Mechanical Systems and Signal Processing 30 (2012) 248–266

where the control coefﬁcients can be written as K p2 ðk,yðkÞÞ ¼

kp ðkÞ m

Z1 9Z3 UyðkÞ9 þ Z2

,

kp ðkÞ ¼

L1 f^ 1 ðkÞ þ L2 f^ 2 ðkÞ , g^ 0 ðkÞ þ l1

K d2 ðkÞ ¼

L2 f^ 2 ðkÞ : g^ 0 ðkÞ þ l1

Remarks: different from the nonlinear golden-section control law presented in Ref. [16,17], another parameter Z3 was introduced. The advantage is that the nonlinear switching amplitude can be speciﬁed by the parameter Z3 according to the practical control output. The detailed explanation and parameters’ selection of the nonlinear switching law are as following cases: Case 1. When one selects the parameters as Z1 ¼0.0 and Z2 ¼ 1.0, then there is Z19Z3Uy(k)9m þ Z2 ¼1, and Kp2(k,y(k)) ¼kp(k). No switching gain is applied to the controller. Case 2. In the practical application of the algorithm, the parameters Z1, Z2 and m can be selected in the following ranges: Z1,Z2A(0,1), m 4 1. For example, Z1 ¼ Z2 ¼0.5, and m ¼2. The selection of the parameter Z3 can be larger or less than 1.0, according to the amplitude of the practical system’s output. Because the inequality is m 4 1, if the constant m parameter Z3 is given and 9Z3 UyðkÞ94 1, then Z1 9Z3 UyðkÞ9 þ Z2 4 1 and Kp2(k,y(k))okp(k). If 9Z3 UyðkÞ9 ¼ 1, then m Z1 9Z3 UyðkÞ9 þ Z2 ¼ 1 and Kp2(k,y(k))¼ kp(k). If 9Z3 UyðkÞ9 o1, then Z1 9Z3 UyðkÞ9m þ Z2 o 1 and K p2 ðk,yðkÞÞ 4 kp ðkÞ. Therefore, the control gain can be regulated according to the amplitude of the output signal. In this work for vibration control, the control gain is reduced for the larger amplitude vibration while increased for the lower amplitude vibration. The advantages of this case are as follows: (a) for the larger amplitude vibration, the control gain can be reduced to avoid much larger control effect. Thus, the problems of stability and control spillover will not be caused due to very high control gain. (b) For the lower amplitude vibration, the control gain is increased due to the switching control law. This can guarantee that the lower amplitude vibration can be damped out quickly. Case 3. The case of parameters selection as other values is not considered in this paper. (3) Nonlinear logic damping control law. In order to suppress the lower amplitude vibration quickly, a kind of nonlinear logic damping controller is needed before the ﬁnal output. It is given as vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u N u X ð9yðkiÞ9 þ 9yðkiÞ9Þ _ ud ðkÞ ¼ c2 yðkÞUt , ð21Þ m Z 9 Z UyðkÞ9 þ Z2 1 3 i¼0 where c2 is a constant, N is the number of sampling periods considered for logic derivative control, here N¼ 1/ (4DT o1), and DT is the sampling time of the control system. Similarly, Eq. (21) can be also re-written as an adaptive gain regulating proportional controller form: ud ðkÞ ¼ K p3 ðkÞUyðkÞ, where K p3 ðkÞ ¼ c2

ð21aÞ

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ PN ð9yðkiÞ9 þ 9yðkiÞ9 _ Þ i¼0

Z1 9Z3 UyðkÞ9m þ Z2

can be seen as a tuning proportional gain. m

The effect of the nonlinear switching law Z1 9Z3 UyðkÞ9 þ Z2 is the same as mentioned above. The total CMNGSAC law can be expressed as uCM ðkÞ ¼ u0 ðkÞ þ ul ðkÞ þud ðkÞ:

ð22Þ

The detailed proof of the above-mentioned controllers can be referred in previous researches [16–18]. The realization of the CMNGSAC method by AC servomotor actuator can be expressed as

t ¼ K p1 ðyd yr Þ þ K v1 ðy_ d y_ r Þ þ uCM :

ð23Þ

When the CMNGSAC method is applied to the system, the outputs are from the measured signal by PZT sensor and the input is from the control signal uCM. Remark: by now, three control laws are introduced for the AC servomotor, given by Eqs. (10), (12) and (23). All the control laws comprise of two components, one is for the rigid motion, and the other is for the ﬂexible motion. The rigid motion component can control the slider to achieve the desired position. Simultaneously, the ﬂexible motion component is used to suppress the vibration of the ﬂexible beam using the AC servomotor as the actuator. In this work, the proportional and derivative gains Kp1 and Kv1 for the rigid motion are uniform for three controllers. However, the control laws for the ﬂexible motion are different. They are PD controller, PPF controller and CMNGSC algorithm, expressed by Eqs. (10), (12) and (23), respectively. The differences of the three controllers are the control effects for vibration suppression of the ﬂexible beam. Their differences are addressed in detail as follows: PD controller is a constant gain linear controller. To obtain better control effect, the larger proportional and derivative gains will be selected. Thus, the stability will be affected, especially for controlling the larger amplitude vibration; also, the problem of control spillover will be caused. If the selected PD control gains are not much larger, the stability and control

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spillover will not be affected. However, the lower amplitude vibration is difﬁcult to be damped out quickly. In total, the constant PD controller is difﬁcult to achieve a good tradeoff between the larger and the lower amplitude vibration. PPF control is also considered as a second order low-pass ﬁlter; therefore, it rolls off quickly at high frequencies. As compared with PD controller, PPF controller is insensitive to spillover. This algorithm is suitable to control the lower modes of a structure with well-separated modes. However, the natural frequency required in the design of PPF controller should be known exactly, or the performance will be adversely affected. The CMNGSC algorithm is a kind of adaptive nonlinear controller. Its gains can be regulated according to the vibration amplitude. This controller can not only keep the control effect not so high for the larger amplitude vibration, but also keep the considerable control effect for the lower amplitude vibration to overcome the system’s nonlinearity problems, such as dead zone nonlinearity. Therefore, the lower amplitude residual vibration can be suppressed quickly due to much larger control effect, as compared with PD controller. Accordingly, both the larger and the lower amplitude vibrations are effectively suppressed by the CMNGSC algorithm. 3.3. Stability analysis of the CMNGSAC algorithm In this section, stability of the CMMGSAC algorithm is analyzed for the translational manipulator driven by motor, using Lyapunov’s direct method La Salle’s invariance principle. From Eqs. (18a), (20) and (21a), one can know that the combined form Eq. (22) is a kind of gain adaptive regulating PD control algorithm. For vibration of the ﬂexible manipulator, the measured output is the voltage signal of the PZT patch’s sensor. Therefore, the CMNGSAC algorithm can be expressed in the following gain regulating PD controller form: uCM ¼ K~ pN V s K~ vN V_ s , ð24Þ where the gains K~ pN and K~ vN can be regulated with control time according to a nonlinear adaptive law of the CMNGSAC method. The values of the control gains are bounded as following: 0 o K p min o K~ pN o K p max o 1, 0 o K v min o K~ vN o K v max o 1,

ð25aÞ

ð25bÞ where K p min and K p max , K v min and K v max are the strict positive minimum and maximum possible values of the proportional and the derivative gains, respectively. Eq. (24) indicates that the CMNGSAC method is a kind of gain regulating PD controller. The regulating law is nonlinear and adaptive. Then, Eq. (23) can be written as

t ¼ K p1 ðyd yr Þ þ K v1 ðy_ d y_ r ÞK~ pN V s K~ vN V_ s :

ð26Þ

From a practical point of view, the low frequency modes usually play a very important role in the ﬂexible beam structure. Without loss of generality, only the ﬁrst two bending modes are considered in this research. Then, Eq. (8) can be written as V s ðtÞ ¼ kg1 q1 þ kg2 q2 ,

ð27Þ

0 0 kg1 ¼ kca e31 bs r½f1 ðx2 Þf1 ðx1 Þ4 0

0 0 kca e31 bs r½f2 ðx2 Þf2 ðx1 Þ 40

where and kg2 ¼ for the PZT sensor’s signal used here. The dynamic equation considered the ﬁrst two bending modes of vibration driven by motor is myy y€ r þ myq1 q€ 1 þ myq2 q€ 2 þ dy y_ r ¼ t,

ð28Þ

R xi2 rpe Ape Pn r b Ab R L r b Ab R L Mt f1 ðlÞ where the mass parameters are myq1 ¼ 200 i ¼ 1 xi1 f1 ðxÞdx þ 200p , and myq2 ¼ 200p 0 f2 ðxÞdxþ p 0 f1 ðxÞdx þ 200p R rpe Ape Pn xi2 M t f2 ðlÞ i ¼ 1 xi1 f2 ðxÞdx þ 200p . The mass of accelerometer Mt is relatively small, thus, there is myq1 4 0 and myq2 4 0. 200p In general, there is y_ d ¼ 0. Let y~ ¼ ðyd yr Þ. The unique equilibrium point of the closed loop system is ~ ½y y_ r V s V_ s T ¼ 0, i.e., y ¼ y , y_ r ¼ 0, V ¼ 0, V_ s ¼ 0. r

d

s

To study the system’s stability of equilibrium point driven by the AC motor, Lyapunov’s direct method is employed. The Lyapunov function candidate may be written as V¼

2 2 1 1 1 1 1 m y_ þ m q_ 2 þ m q_ 2 þ K p1 y~ þ K~ pN ðkg1 q21 þ kg2 q22 Þ: 2 yy r 2 yq1 1 2 yq2 2 2 2

ð29Þ

Eq. (29) shows that V is a non-negative function. Its total derivative with respect to time is expressed as V_ ¼ y_ r myy y€ r þ q_ 1 myq1 q€ 1 þ q_ 2 myq2 q€ 2 y~ K p1 y_ r þ K~ pN ðq1 kg1 q_ 1 þq2 kg2 q_ 2 Þ:

ð30Þ

Substituting Eqs. (26)–(28) into Eq. (30), one gets 2 V_ ¼ ðK v1 þ dy Þy_ r K~ vN kg1 q_ 21 K~ vN kg2 q_ 22 þ ðq_ 1 y_ r Þðmyq1 q€ 1 þ K~ vN kg1 q_ 1 þ K~ pN kg1 q1 Þ þðq_ y_ r Þðmyq q€ þ K~ vN kg2 q_ þ K~ pN kg2 q Þ: 2

2

2

2

2

ð31Þ

Then, there is 2 V_ r ðK v1 þ dy Þy_ r K~ vN kg1 q_ 21 K~ vN kg2 q_ 22 þ 9ðq_ 1 y_ r Þðmyq1 q€ 1 þ K~ vN kg1 q_ 1 þ K~ pN kg1 q1 Þ9 þ9ðq_ 2 y_ r Þðmyq2 q€ 2 þ K~ vN kg2 q_ 2 þ K~ pN kg2 q2 Þ9:

ð32Þ

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Since all parameters and variables in the expressions 9ðq_ 1 y_ r Þðmyq1 q€ 1 þ K~ vN kg1 q_ 1 þ K~ pN kg1 q1 Þ9 and 9ðq_ 2 y_ r Þðmyq2 q€ 2 þ K~ vN kg2 q_ 2 þ K~ pN kg2 q2 Þ9 are bounded, 9ðq_ 1 y_ r Þðmyq1 q€ 1 þ K~ vN kg1 q_ 1 þ K~ pN kg1 q1 Þ9 and 9ðq_ 2 y_ r Þðmyq2 q€ 2 þ K~ vN kg2 q_ 2 þ K~ pN kg2 q2 Þ9 are also bounded. Therefore, the condition V_ r0 can be held under the condition that the control gain Kv1 is chosen sufﬁciently ‘‘large’’. Remarks: all the coefﬁcients of the functions V and V_ are positive. And stability analyses are explained in the following cases: Case (1). Only the viscous friction is considered in this paper. If y_ r 0, then y€ r 0, there must be t 0. If and only if t 0, y_ r 0 can be held because no driving torque can cause the rotor rotating. Considering the orthogonality of modes, from Eqs. (26) and (27) one can infer that the following equations are satisﬁed: yr ¼ yd, y_ r ¼ 0, q1 ¼q2 ¼0, q_ 1 ¼ q_ 2 ¼ 0. That is to say, the equilibrium point is achieved. Case (2). The initial values of the variables q_ 1 , q€ 1 , q_ 2 and q€ 2 are bounded as the initial excitation is limited, and the scopes of motion variables are also limited. When y_ r a0, if Kv1 is chosen sufﬁciently ‘‘large’’, the condition V_ o 0 can be held at the beginning of the control. Then, V is non-increasing, and the values of the variables q_ 1 , q€ 1 , q_ 2 and q€ 2 will be convergent and bounded in the whole control process. Therefore, the stability of the closed loop system can be achieved if the gain Kv1 is chosen sufﬁciently ‘‘large’’. Case (3). To analyze the global asymptotic stability of the equilibrium, one may explore the use of La Salle’s invariance principle. La Salle’s Theorem may be used in a straightforward way to analyze the global asymptotic stability of the origin [28]. Then, every solution originating in the deﬁnite ﬁeld tends to the set V_ ¼ 0 as time tends to inﬁnity. When V_ 0, from Eqs. (29) and (30) one can deduce the following: yr ¼ yd, y€ r ¼ 0, q1 ¼q2 ¼0 and q_ 1 ¼ q_ 2 ¼ 0, according to the orthogonality of modes. Therefore, the equilibrium point of the closed loop system is asymptotically stable. From a theoretical point of view, asymptotic stability of the nonlinear adaptive control algorithm is analyzed. One may achieve the stable condition if the control gains are chosen reasonably. In the next section, experiments will be conducted to verify the implemented control algorithms. 4. The test-Beds of the ﬂexible manipulator and experimental results 4.1. Introduction of experimental setup In order to verify the implemented control algorithm and analyze the dynamic characteristics of the ﬂexible Cartesian manipulator driven by a ballscrew drive system with an AC servomotor. An experimental setup was constructed for conducting experiments. The photograph of the experimental setup is shown in Fig. 2. In the test-beds, an AC servomotor with a built-in incremental encoder is mounted at the entrance of the ballscrew drive system, connected by a coupling. The ballscrew was made in THK Corporation. Its lead is 10 mm and its stroke is 600 mm. The AC servomotor was made in MITSUBISHI Corporation. The rated power of the servomotor is 400 W. The builtin incremental encoder is used to measure the rotation angle and angular velocity of the servomotor’s rotor. The resolution of the encoder is speciﬁed as 40,000 pulses per revolution or 0.0091 per pulse through a third-axis quadrature encoder counter board, PCL-833, which is used to count the encoder’s pulses. The material of the ﬂexible beam is ﬁberglass colophony. The dimensions of the ﬂexible beam are the length L¼0.68 m, the width b¼0.15 m and the thickness h¼0.002 m. The mass of the acceleration sensor is 50 g, and the mounted position is l ¼0.655 m. One PZT patch is bonded onto the top surface of the host beam and located nearly close to the clamped side

Fig. 2. Photograph of the ﬂexible Cartesian manipulator experimental setup.

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of the beam about X-axis, which is used as a PZT sensor. Another four PZT patches are bonded onto the top and the bottom surfaces of the host beam structure symmetrically close to the clamped end. Their electric circuits are connected in parallel, used as a one-channel actuator. The control system is realized using an industrial personal computer (IPC), the Pentium IV, CPU 2.4 GHz. Considering the hardware conditions and the control algorithms applied for the system, the sampling period was selected as 2 ms. A charge ampliﬁer, YE5850, ampliﬁes the PZT sensor’s signal to the voltage range of 10 V to þ 10 V, and converted into digital data through a PCL-818HD A/D data acquisition card. After scaling, the transverse bending displacement at the tip point of the link is about 8 mm corresponding to 10 V of the PZT sensor’s signal. The outputs of the controller are sent to ampliﬁers of the AC servomotor and PZT actuator, through a PCL-727 D/A control card. The PZT actuator is driven by a high voltage ampliﬁer (APEX PA240CX), which ampliﬁes the low voltage signal in the range 5 V to þ5 V to a high voltage signal in the range 260 V to þ260 V. A signal generator is used to generate the sine signal for experiments on resonant response case. For active vibration control experiments, the PZT sensor is used to measure the vibration of the ﬂexible manipulator. Furthermore, the input is the torque of the AC servomotor. Because the system is minimum phase, thus, the proposed control methods can be applied to the system. 4.2. Experimental identiﬁcation on modal frequencies of the ﬂexible manipulator If the slider is completely stopped by the servomotor’s control action, the ﬂexible manipulator can be approximately treated just as a cantilever beam with centralized mass (payload). In order to identify the modal frequencies of the ﬂexible beam, excitation analyses are carried out for actual beam. The cases of excitation include the external and internal exciting. The external exciting is generated by an impulse force hammer knocking at some certain point of the beam, to generate the vibration of the ﬁrst and the ﬁrst two bending modes, respectively. The internal exciting is generated using the PZT actuator or AC servomotor actuator. The PZT sensor is used to measure the vibration. The frequency response curves can be obtained by employing the fast Fourier transform (FFT) method. After exciting the vibration of the ﬁrst bending mode, the measured time-domain response without control is depicted in Fig. 3(a), and the frequency response shown in Fig. 3(b) is obtained by employing FFT. The curve ﬁtting of the frequency response curve is also shown in Fig. 3(b). The frequency response function of the curve ﬁtting is expressed as GðsÞ ¼

0:635 ðs þ4:5Þ 0:635 ðs þ 4:5Þ : ¼ ðs þ 0:093 þ12:566iÞðs þ 0:09312:566iÞ ðs2 þ 0:186s þ 157:914Þ

ð33Þ

After excitation, the measured time-domain response of the ﬁrst two bending modes without control is depicted in Fig. 4(a). The frequency response can be obtained by employing FFT. The frequency response curve and its curve ﬁtting are shown in Fig. 4(b). The curve ﬁtting’s frequency response function is GðsÞ ¼ ¼

0:33 ðs þ3:3Þðs2 þ12:0s þ 2800:0Þðs þ190:0Þ ðs2 þ 0:162s þ 157:914Þðs2 þ 0:87s þ 6986:1Þðs þ55:0Þ 0:33 ðs þ3:3Þðs þ 6:0 þ 52:574iÞðs þ 6:052:574iÞðs þ 190:0Þ ðsþ 0:081 þ 12:566iÞðs þ 0:08112:566iÞðs þ0:435 þ83:582iÞðs þ 0:43583:582iÞðs þ 55:0Þ

ð34Þ

From Eq. (33), one can know that the gain of the system is k1 ¼ 0.635; the zero is z1 ¼ 4.5, the poles are p1,2 ¼ 0.093 712.566i. Then one can know that the damping ratio and modal frequency of the ﬁrst bending mode are z1 ¼0.0074 and o1 ¼2.0 Hz, respectively.

10

20 10 0 Magnitude (dB)

Voltage (V)

5

0

-5

-10 -20 -30 -40 -50

-10

-60 0

5

10 Time (s)

15

20

100

101 Frequency (Hz)

Fig. 3. Uncontrolled responses of the ﬁrst bending mode: (a) time-domain response; (b) frequency response.

258

Z.C. Qiu / Mechanical Systems and Signal Processing 30 (2012) 248–266

20

10

10 FFT amplitude (dB)

Voltage (V)

5

0

-5

0 -10 -20 -30 -40 -50 -60

-10 0

2

4 6 Time (s)

8

100

10

101 Frequency (Hz)

100

100

50

50

Imaginary axis

Imaginary axis

Fig. 4. Uncontrolled responses of the ﬁrst two bending modes: (a) time-domain response; (b) frequency response.

0

-50

-50

-100 -200

0

-100 -150

-100 -50 Real axis

0

50

-8

-6

-4 -2 Real axis

0

2

Fig. 5. Map of the distributed poles and zeros the ﬁrst two bending modes: (a) all the poles and zeros; (b) zoom in the real axis.

From Eq. (34), one can know that the zeros are z1 ¼ 3.3, z2,3 ¼ 6.0 752.574i, z4 ¼ 190.0; the poles are p1,2 ¼ 0.081 712.566i, p3,4 ¼ 0.435783.582i and p5 ¼ 55.0; the gain of the system is k1 ¼0.33; the damping ratio and modal frequency of the ﬁrst bending mode are z1 ¼0.0064 and o1 ¼2.0 Hz; those of the second bending mode are z2 ¼0.0052 and o2 ¼13.30 Hz, respectively. Remark: on comparing Eq. (33) with Eq. (34), it can be seen that the identiﬁed modal frequency of the ﬁrst bending mode is equal. However, the identiﬁed damping ratio of Eq. (33) is a little larger than that of Eq. (34). This is because that the identiﬁed damping ratio comprises two components: one is the structural damping, and the other is air damping in the environment. The excited vibration peak amplitude of the ﬁrst bending mode in Fig. 3(a) is larger than that in Fig. 4(a). Generally, the air damping is bigger for the larger amplitude vibration. In addition, the structural damping is almost uniform for the two excited cases. Therefore, the identiﬁed damping ratio is a little larger when only the vibration of the ﬁrst bending mode is excited. It is obvious that the damping ratios are comparatively small for free vibration response. Thus, active vibration control action must be applied to suppress the vibration effectively. The map of the distributed poles and zeros in Eq. (34) for the ﬁrst two modes is shown in Fig. 5. In Fig. 5, ‘*’ denotes pole and ‘o’ denotes zero. From Fig. 5, it can be seen that there are also one stable real pole and two stable conjugate zeros and two stable real zeros in the curve ﬁtting model. However, the stable real pole and all the zeros are far from the imaginary axis. Their distances away from the imaginary axis are over ﬁve times greater than those of the poles concerning the ﬁrst two bending modes. So they will not affect the characteristics of the system signiﬁcantly. They have inﬂuence almost solely on the weighted gain along with frequency change. Thus, the poles of the ﬁrst two bending modes dominate the system. From Fig. 5, one can know that the poles of the ﬁrst two modes locate in the left half complex plane, thus the openloop system is stable. However, the damping of ﬂexible beam structure is rather small. From the measured results it can be known that the decay time of the beam vibrations without active control effect is over 50 s. In addition to the external excitation analyses, internal excitation analyses were also used. This is done using a chirp (swept-sine) signal to excite the vibration of the ﬂexible beam. Their responses were measured by the PZT sensor. The swept-sine signal was generated by a signal generator SPF05, which ranges from 0.5–30 Hz, and the exciting time and

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amplitude were speciﬁed as 50 s and 2 V, respectively. The ﬁrst case was excited using the PZT actuator. The generated swept-sine signal was ampliﬁed by a high voltage ampliﬁer APEX PA240CX and then applied to the bonded PZT actuator. Thus, the swept frequency response is obtained. Fig. 6(a) shows the excited chirp signal and the time-domain response of the bending vibration measured by the PZT sensor. One can obtain the frequency response curve from excitation to output as shown in Fig. 6(b) by employing FFT. From Fig. 6(b), one can obtain that the modal frequencies are o1 ¼2.0 Hz and o2 ¼13.30 Hz. The second case was excited using the AC servomotor actuator. In this case, the slider of the ballscrew was controlled at some certain point. The swept-sine signal was applied to the servomotor according to the following control law:

t ¼ K p1 ðyd yÞ þK v1 ðy_ d y_ Þ þ K ex UV ex ,

ð35Þ

3 2 1 0 -1 -2 -3

40 30

0

10

20 30 Time (s)

40

50

Voltage (V)

10 5

FRF Magnitude (dB)

Voltage (V)

where Kex is the selected control gain for exciting analysis using the AC servomotor; Vex is the voltage of the exciting signal. Fig. 7(a) shows the excited chirp signal and the time-domain response excited by AC servomotor using the controller of Eq. (35). Its frequency response was shown in Fig. 7(b) by employing FFT. From the internal excitation analyses’ results illustrated in Figs. 5 and 6, it can be seen that the natural frequencies are the same as those by the external excitation. Comparing Fig. 7(b) with Fig. 6(b), one can know that the ballscrew mechanism will cause noises. From Fig. 7(b), one can

20 10 0 -10 -20

0

-30

-5 -40 -10 0

10

20 30 Time (s)

40

100

50

101 Frequency (Hz)

3 2 1 0 -1 -2

40 30

-3 0

10

20 30 Time (s)

40

50

Voltage (V)

10 5

FRF Magnitude (dB)

Voltage (V)

Fig. 6. Measured swept sine vibration response excited by PZT actuator: (a) time-domain response; (b) frequency response.

20 10 0 -10 -20

0

-30

-5 -40

-10 0

10

20 30 Time (s)

40

50

101

100 Frequency (Hz)

Fig. 7. Measured swept sine vibration response excited by AC servomotor actuator: (a) time-domain response; (b) frequency response.

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Z.C. Qiu / Mechanical Systems and Signal Processing 30 (2012) 248–266

obtain that the modal frequencies are o1 ¼ 2.0 Hz and o2 ¼13.32 Hz. The modal frequencies of Fig. 7(b) and Fig. 6(b) are approximately equal. Thus, the boundary condition of the moving beam driven by ballscrew system is close to that of the cantilever beam.

4.3. Experiments on set-point active vibration control driven by AC servomotor

10

10

5

5 Voltage (V)

Voltage (V)

For set-point vibration control, the PZT sensor is used to measure the elastic vibration. The AC servomotor is used as the actuator. Its driving torque is employed at the entrance of the ballscrew system. The desired angle position of the motor rotor was set as yd ¼const, i.e., the desired position of the slider was at some set-point position, and the desired angle velocity was set as y_ d ¼ 0. In Sections 4.3–4.5, the velocity control mode of the AC motor is employed. The relevant control parameters (gains) are speciﬁed as Kp1 ¼ 2.29, Kv1 ¼0.048, Kp ¼0.057 and Kv ¼0.0019 for the motor side position control. For PPF control, the control gain is k¼ 0.0067 and o1 ¼ 12.57 rad/s. For the CMNGSC algorithm, Z1 ¼ Z2 ¼0.5, Z3 ¼0.45, m ¼2 and a ¼0.97. The initial values of tuning parameters are given as f^ 1 ¼ 1:84, f^ 2 ¼ 0:85 and g^ 0 ¼ 0:01. The controlled time-domain responses under PD and PPF control laws using Eqs. (10) and (12) for the ﬁrst and the ﬁrst two bending modes are depicted alternately in Fig. 8(a) and (b), Fig. 9(a) and (b), respectively. From the experimental results, it can be seen that the vibration of the larger amplitude can be suppressed to the lower amplitude vibration effectively. However, the lower amplitude vibration will last for a period of time to disappear, thus the decay rate is far from satisfactory. By comparison, the decay rate of PPF controller for the lower amplitude vibration is faster than that of the PD controller. This can be explained by the existence of the nonlinear factors, such as nonlinear friction torque, backlash of the ballscrew system, etc. The controlled time-domain responses under CMNGSAC algorithm using Eq. (23) are shown in Fig. 10. Comparing Fig. 10 with Figs. 8 and 9, one can know that the CMNGSAC method can suppress both the larger and the lower amplitude vibrations quickly and effectively. This is due to the fact that this controller has very advantageous adaptive nonlinear damping properties. The advantage of the proposed control method is to achieve satisfactory vibration suppression results.

0

-5

0

-5

-10

-10 0

2

4 6 Time (s)

8

10

0

1

2

3 4 Time (s)

5

6

10

10

5

5 Voltage (V)

Voltage (V)

Fig. 8. Controlled time-domain responses under PD control: (a) the ﬁrst bending mode; (b) the ﬁrst two bending modes.

0

-5

0

-5

-10

-10 0

2

4 6 Time (s)

8

10

0

1

2

3 4 Time (s)

5

6

Fig. 9. Controlled time-domain responses under PPF control: (a) the ﬁrst bending mode; (b) the ﬁrst two bending modes.

10

10

5

5 Voltage (V)

Voltage (V)

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0

261

0

-5

-5

-10

-10 0

2

4 6 Time (s)

8

10

0

1

2

3 4 Time (s)

5

6

Fig. 10. Controlled time-domain responses under CMNGSAC method: (a) the ﬁrst bending mode; (b) the ﬁrst two bending modes.

In addition, different from the PPF control method, the CMNGSAC method does not require providing accurate modal frequencies beforehand. The controller can not only enhance the performance, but also can be easily implemented. 4.4. Experiments on vibration suppression under resonant excitation In this section, the PZT actuator is used to excite vibration of the ﬂexible beam persistently, acting at the structural resonant frequency of the ﬁrst bending mode of the ﬂexible manipulator, i.e., oex ¼2.0 Hz. The resonant frequency is determined from the experimental identiﬁcation results in Section 4.2. The AC servomotor is used to suppress the excited resonant vibration. The control effect of the AC servomotor is applied to the system at the time t ¼5.0 s. Experiments of resonant vibration control under persistent excitation were conducted using PD control, PPF control and the CMNGSAC algorithm. And their experimental results are shown in Fig. 11. The rated power of AC servomotor is 400 W, while the power of the PZT actuator used in this research is less than 2.0 W. Because the PZT actuator is used to excite the resonant vibration persistently, the vibration can be suppressed eventually by AC motor to some smaller steady-state amplitude but not zero. Furthermore, the smaller the controlled vibration amplitude is, the more effective the control algorithm is. The experimental results can verify the ability of the control algorithm for disturbance suppression. Signiﬁcantly, the amount of vibration reduction is greater for the CMNGSAC method than those of PD and PPF control methods. The CMNGSAC method outperformed the PD and PPF control method, i.e. about 93.6% reduction in vibration for CMNGSAC vs. 90.5% for PPF and 87.8% for PD method. Experimental comparison results show the advantage and robustness of the proposed method. Here, the robustness of the proposed control method is its ability to withstand the effects of persistent disturbance. From these data, it can be concluded that an effective damping of the nonlinear adaptive feature can be achieved using the CMNGSAC method, contrary to the classical PD and PPF control. 4.5. Experiments on simultaneous translating and vibration suppression In this section, experiments were conducted on simultaneous control of translating motion and vibration suppression of the ﬂexible Cartesian manipulator. In this case, the desired control performance can be described as the slider translating motion converges to the ﬁnal position, and the elastic vibrations are effectively suppressed. Theoretical analysis was carried out in Section 3.3. The desired speed trajectory is generated using so-called Linear Segments with Parabolic Blends (LSPB) [26]. This type of trajectory has a Trapezoidal Velocity Proﬁle and is appropriate when a constant velocity is desired along a portion of the path. The LSPB trajectory is such that the velocity is initially ‘‘ramped up’’ to its desired value and then ‘‘ramped down’’ when it approaches the global position [26]. The desired speed trajectory is planned just as shown in Fig. 12. The speciﬁc desired constant angular velocity of motor is y_ d ¼ 180 rev=min; the desired angle acceleration is y€ d ¼ 942:48 rad=s2 and y€ d ¼ 942:48 rad=s2 for T-curve velocity trajectory rising and descending moment, respectively. The initial translating motion time is t1 ¼5.0 s, and the stopping time is at t2 ¼11.9 s, as shown in Fig. 12. The rising and the descending time is 0.02 s. For the ﬂexible manipulator side, the desired translating velocity is s_ d ¼ 0:03 m=s, and the desired translating acceleration is s€ d ¼ 1:5 m=s2 and s€ d ¼ 1:5 m=s2 for T-curve velocity trajectory rising and descending moment, respectively. When the slider is driven to move along the desired speed, vibration will be caused due to inertia of the ﬂexible manipulator. The experimental results of translating motion without active vibration control are shown in Fig. 13(a) and Fig. 13(b). In Fig. 13(a), two maps are the translating motion speed and vibration response of the ﬂexible manipulator, measured by the built-in encoder and the PZT sensor. The translating speed is obtained through the rotor rotating speed

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40

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Fig. 11. Active vibration control for persistent resonant excitation of the ﬁrst bending mode: (a) PD control; (b) PPF control; (c) CMNGSAC method; (d) CMNGSAC method for comparatively long time.

measured by the servomotor’s encoder. The relation is s_ ðtÞ ¼

y_ r ðtÞ : 200p

ð36Þ

The frequency response is shown in Fig. 13(b) by employing FFT for the time-domain vibration response shown in Fig. 13(a). From Fig. 13(b), one can know that the frequencies of the ﬁrst two bending modes of vibration are o1 ¼2.0 Hz and o2 ¼13.30 Hz. The modal frequencies in Fig. 13(b) and Fig. 4(b) are approximately equal. Therefore, the boundary condition of the Cartesian ﬂexible manipulator is close to that of the cantilever beam. From the experimental results, one can know that the vibration can be caused when the motor started translating and stopped, due to the beam’s ﬂexibility, translational inertia, etc. Fig. 13(a) shows the result of translating motion and without active vibration control. It can be known that the excited vibration amplitude of the ﬂexible link is larger and will last for a long time without active control. Therefore, active vibration control should be applied to the system, for simultaneous translating and vibration suppression. The experimental results of simultaneous translating motion and vibration control for ﬂexible Cartesian manipulator are shown in Fig. 14(a)–(c), respectively, using PD control, PPF control and CMNGSAC method. In Fig. 14, two maps of each ﬁgure are the translating motion speed and vibration of the ﬂexible manipulator measured by the encoder and the PZT sensor, respectively. Comparing the experimental results shown in Fig. 14(a)–(c) with those in Fig. 13(a), it can be seen that the vibrations are suppressed largely by introducing active vibration control. By comparison, the CMNGSAC method can suppress the excited vibration much more effectively for simultaneous translating motion control. The slider’s trajectory displacement tracking errors of different cases are shown in Fig. 15(a) (d). The small tracking errors are caused during the moving process. This is because only PD control for trajectory motion is employed, without using an integral element. The velocity response curves of Fig. 14(a) (c) are oscillatory compared with those of Fig. 13(a) at the starting and the end of the T-curve velocity. Correspondingly, the displacement trajectory tracking errors of Fig. 15(b) (d) have also oscillatory occurrence compared with those of Fig. 15(a). This is the reason why the control effect is applied to the servomotor for active vibration control. From Fig. 15, it can be seen that the largest oscillatory amplitude of PPF control is larger than those of PD control and the CMNGSAC method. The largest oscillatory amplitude of the CMNGSAC method is approximately equal to that of PD control. However, the rate of decay of vibration is different.

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FRF Magnitude (dB)

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Fig. 12. Motion planning for the driving system tracking a reference trajectory: (a) speed trajectory planning of motor rotating motion; (b) displacement and speed trajectory planning of the slider’s motion.

0 -10 -20 -30 -40

0 -50 -5 -60

-10 0

5

10 Time (s)

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20

100

101 Frequency (Hz)

Fig. 13. Translating and without vibration control: (a) time-domain signal; (b) frequency response.

Therefore, the adaptive nonlinear control method can enhance the damping for the lower amplitude vibration. Experiments conﬁrm that the CMNGSAC method can damp out the vibration in translating motion signiﬁcantly, especially quick vibration suppression at the starting and stopping moment of the slider translating motion. Remarks: in Section 3, the CMNGSAC method is equivalent to a kind of gain regulating PD controller, according to the nonlinear adaptive tuning algorithm of this method. Compared with the constant gain PD and PPF control algorithms, it exhibited the effectiveness due to the adaptive adjustment of the control parameters. Because of the introduction of nonlinear switch adaptive control, the control action is enough but not so large for the large amplitude vibration, while the

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10 Time (s)

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10 5 0 -5 -10

Fig. 14. Simultaneous translating and active vibration control: (a) PD control; (b) PPF control; (c) CMNGSAC method.

control effect can be enhanced in comparison with the constant gain controller. Therefore, the stability will not be affected, and the low amplitude residual vibration can be suppressed quickly, just as shown in Sections 4.3 and 4.5, for the experimental results of the set-point vibration control and simultaneous translating and vibration suppression. For vibration suppression under resonant excitation, as shown in Section 4.4, the CMNGSAC method shows its advantage for disturbance attenuation. In summary, the above-mentioned three experimental cases can show the advantages of the nonlinear adaptive control and the effectiveness of the CMNGSAC method. The author’s future research work will be conducted using the widely used control approaches, such as model-based control, model reference adaptive control, model-based robust control, etc. By comparison, it is expected to show the characteristics and robustness of the all the control methods due to model uncertainties and load variation.

5. Conclusions This paper presents the theoretical analyses and experimental results of the CMNGSAC method for vibration suppression of ﬂexible Cartesian smart material robot. The stability and performance of the CMNGSAC method were analyzed theoretically. Experimental comparison researches of different control methods were conducted, including setpoint vibration control, resonant vibration suppression under persistent excitation and simultaneous control of translating motion and vibration suppression. For the set-point and translating motion cases, experiments show that the adopted algorithm can not only substantially suppress the larger amplitude vibration, but also signiﬁcantly damp out the lower amplitude vibration faster than the traditional control schemes. For resonant vibration suppression under persistent excitation, experiments conﬁrm that the CMNGSAC methods also show good robustness, i.e., the vibration amplitude was attenuated much lower. Experimental results on active vibration control for ﬂexible Cartesian smart beam have shown

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Fig. 15. Trajectory tracking displacement error: (a) without active vibration control; (b) PD control; (c) PPF control; (d) CMNGSAC method.

that the adopted CMNGSAC method is stable and can yield satisfactory vibration suppression effect for the Cartesian manipulator. Experiments veriﬁed the theoretical analyses.

Acknowledgments This work was partially supported by the National Natural Science Foundation of China under Grants 51175181, 60404020 and 90505014, partially supported by the State Key Laboratory of Robotics Foundation (RLO200805) and in part supported by the Fundamental Research Funds for the Central Universities, SCUT (2009ZM0148, 2012ZZ0060). The author gratefully acknowledges these support agencies. The author would like to thank Professor Wu H.X. for his valuable suggestions and discussion. References [1] S.K. Dwivedy, P. Eberhard, Dynamic analysis of ﬂexible manipulators, a literature review, Mech. Mach. Theory 41 (2006) 749–777. [2] D. Sun, J.K. Mills, J.J. Shan, S.K. Tso, A PZT actuator control of a single-link ﬂexible manipulator based on linear velocity feedback and actuator placement, Mechatronics 14 (2004) 381–401. [3] Z.C. Qiu, J.D. Han, X.M. Zhang, Y.C. Wang, Z.W. Wu, Active vibration control of a ﬂexible beam using a non-collocated acceleration sensor and piezoelectric patch actuator, J. Sound Vib. 326 (3–5) (2009) 438–455. [4] S. Hurlebaus, L. Gaul, Smart structure dynamics, Mech. Syst. Signal Process. 20 (2006) 255–281. [5] M.P. Coleman, L.A. McSweeney, Analysis and computation of the vibration spectrum of the Cartesian ﬂexible manipulator, J. Sound Vib. 274 (1–2) (2004) 445–454. [6] X. Hou, S.K. Tsui, A control theory for Cartesian ﬂexible robot arms, J. Math. Anal. Appl. 225 (1998) 265–288. [7] Z.H. Luo, N. Kitamura, B.Z. Guo, Shear force feedback control of ﬂexible robot arms, IEEE Trans. Robotics Autom. 11 (5) (1995) 760–765. [8] M. Dadfarnia, N. Jalili, Z. Liu, D.M. Dawson, An observer-based piezoelectric control of ﬂexible Cartesian robot arms: theory and experiment, Control Eng. Pract. 12 (2004) 1041–1053. [9] M. Dadfarnia, N. Jalili, B. Xian, D.M. Dawson, A Lyapunov-based piezoelectric controller for ﬂexible Cartesian robot manipulators, J. Dyn. Syst. Meas. control 126 (2004) 347–358. [10] S.S. Ge, T.H. Lee, J.Q. Gong, A robust distributed controller of a single-link SCARA/Cartesian smart materials robot, Mechatronics 9 (1999) 65–93. [11] S.S. Ge, T.H. Lee, G. Zhu, Asymptotically stable end-point regulation of a ﬂexible SCARA/Cartesian robot, IEEE/ASME Trans. Mechatronics 3 (2) (1998) 138–144. [12] J. Lin, C.H. Chen, Positioning and tracking of a linear motion stage with friction compensation by fuzzy logic approach, ISA Trans. 46 (2007) 327–342.

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