Experimental study on active vibration control for a kind of two-link flexible manipulator

Mechanical Systems and Signal Processing 118 (2019) 623–644

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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp

Experimental study on active vibration control for a kind of two-link flexible manipulator Zhi-cheng Qiu ⇑, Cheng Li, Xian-min Zhang Guangdong Province Key Laboratory of Precision and Manufacturing Technology, School of Mechanical and Automotive Engineering, South China University of Technology, Guangzhou 510641, PR China

a r t i c l e

i n f o

Article history: Received 16 March 2018 Received in revised form 28 August 2018 Accepted 1 September 2018

Keywords: Two-link flexible manipulator Active vibration control Generalized minimum variance self-tuning control T-S model based fuzzy neural network controller

a b s t r a c t This paper presents experimental investigations on active vibration control of a two-link flexible manipulator (TLFM), utilizing a generalized minimum variance self-tuning control (GMVSTC) and Takagi-Sugeno model based fuzzy neural network control (TS-FNN) schemes. The GMVSTC algorithm consists of an on-line identifier in the form of controlled autoregressive moving average model and a vibration control signal generator, and the TSFNN control algorithm generates control actions taking full advantages of fuzzy logic controller and a neural controller. Experimental setup of the two-link flexible manipulator is constructed. Experimental comparison research on vibration attenuation is conducted during and after the motor motion, to verify the designed controllers. The effectiveness of the designed controllers is evaluated in terms of vibration suppression as compared to that of the classical PD control. The experimental results demonstrate that the designed controller can damp out both the large and the small amplitude vibration of a two-link flexible manipulator more quickly than that the traditional linear PD controller, especially for the small amplitude residual vibration. Ó 2018 Elsevier Ltd. All rights reserved.

1. Introduction The most common construction of a flexible manipulator consists of flexible beams attached to rigid actuator such as motors. Compared with the conventional heavy and bulky rigid robots, flexible link manipulators have the potential advantage of lower cost, higher operational speed, greater payload-to-manipulator-weight ratio, smaller actuators, lower energy consumption, better maneuverability, better transportability and safer operation due to reduced inertia [1,2]. However, the greatest disadvantage of these manipulators is the vibration problem due to low stiffness [2]. The oscillatory behavior of the flexible manipulator needs to be considered especially during the operation due to the flexibility of their links. Due to the flexible nature of the system, the dynamics are highly nonlinear and complex. The complexity of the problem increases dramatically for a two-link flexible manipulator where several other factors such as coupling between both links have to be considered [3]. Numerous theoretical analyses were carried out in various ways for modeling the two-link flexible manipulator (TLFM) system. Generally, three methods are mainly used for mathematical modeling of flexible manipulators. These are finite element method (FEM), assumed modes method (AMM) and lumped parameter methods used for truncation of the system [4]. The most widely used method for modeling of flexible manipulators is AMM. Book et al. [5] studied two methods of mod⇑ Corresponding author. E-mail address: [email protected] (Z.-c. Qiu). https://doi.org/10.1016/j.ymssp.2018.09.001 0888-3270/Ó 2018 Elsevier Ltd. All rights reserved.

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eling a TLFM in time domain and frequency space, incorporating transfer matrices and numerical techniques. A dynamic model of a planar two-link flexible manipulator moving in a horizontal plane is developed using the AMM [6]. Also, experiments have been performed for validation of the dynamic model. For dynamic behavior, Abdollahpouri et al. [7] implemented the moving horizon estimation algorithm to observe the dynamic states and parameter variations of an active cantilever beam in real time. The practical behavior of this algorithm has been investigated in various experimental scenarios. It is difficult to design high-performance controllers for flexible manipulators, due to the coupling dynamics between the rigid and the flexible modes [8]. The control strategies for flexible manipulator systems are mainly classified into two categories: feedforward and feedback control schemes. Feedforward control involves developing the control input through consideration of the physical and vibrational properties of the system. Feedback control techniques use measurements and estimates of the system states and changes the actuator input accordingly for control of rigid body motion and vibration suppression of the system [9]. Generally, motors were used as the actuators for driving the joint, while the piezoelectric patches were used as vibration actuators to reduce the vibration amplitudes of flexible manipulators. Shin and Choi established a nonlinear model including inertial effects using Lagrange’s equation, and designed a sliding mode controller (SMC) for the position control of a two-link flexible manipulator with PZT actuators and sensors [10]. Mirzaee et al. [11] investigated active vibration control of a two-link manipulator with PZT actuator and sensor, using a hybrid variable structure and Lyupunov based controller for maneuver tracking. Input preshaping controller is also applied for a two-link flexible manipulator [12]. An improved recursive least square-based adaptive input shaping is investigated for zero residual vibration control of flexible system [13]. A hybrid control scheme was designed to simultaneously suppress the excited and the residual vibration during and after the motor motion, for a macro-micro manipulator [14]. The hybrid control strategy is composed of a trajectory planning approach and an adaptive variable structure control. Experiments on an inner/outer loop controller for a two-link flexible manipulator are conducted [15]. Sometimes the model of a two-link flexible manipulator system is hard to formulate properly or is not known accurately. In addition, even if a relatively accurate model of the flexible robot can be developed, it is often too complex to be used in controller development [16]. Thus, non-model based intelligent control algorithms are considered for vibration suppression of two-link flexible manipulators. Moudgal et al. [16] proposed a rule-based fuzzy model reference learning controller for the tip position control and vibration suppression of a TLFM. Although FLC was successful used in control of FLM, the drawback, such as the establishment of fuzzy control rules is highly dependent on the experience and to achieve a smooth control response. Therefore, the number of rule bases rises exponentially, which increases the design difficulty. To deal with these problems, the fuzzy neural network (FNN) controller is proposed. FNN inherits the reasoning ability of FLC and the learning ability of artificial neural networks [17]. It is able to deal with nonlinearities and uncertainties of the control systems in the control fields. New fuzzy and neuro-fuzzy approaches to tip position regulation of a flexible link manipulator are presented in [18], and the proposed adaptive neuro-fuzzy controller (NFC) can tune the input and output scale parameters of the fuzzy controller on-line. Caswara and Unbehauen [19] used the neuro-fuzzy controller as a nonlinear compensator for a flexible four-link manipulator and achieved a tradeoff between the tracking accuracy and manipulator link vibration control. Fuzzy logic and artificial neural network have some similar properties. Both of them may be able to map the typical nonlinear relation of the input-output without a precise mathematical formula or model between the input and output variables [20]. An improved Takagi-Sugeno (T-S) fuzzy neural network has a compact structure, high training speed, good simulation precision, and generalization ability [20,21]. For limiting the transient amplitude of payload deflection transferred by crane and minimizing the residual vibration in the presence of system’s parameters variation, Smoczek [22] investigated a generalized predictive control and linear parameter varying based approaches, with the recursive least squares method and P1-TS interpolator of model’s parameters. Niu et al. [23] enhanced the classical filtered-x least mean square control algorithm to adaptively suppress the vibration of time-varying structures, such as the vertical tail of aircraft. During operation of two-link flexible manipulators, the configuration changes and the analyses become very complicated, due to the flexibility of each link [4]. Self-tuning is an important branch of control. The objective of self-tuning is to control systems with unknown constant or slowly varying parameters, as a method for controlling time-varying or nonlinear plant over a range of operating points [24]. A nonlinear adaptive model predictive control is designed for tip position control of a flexible manipulator. And experiments are conducted, compared the effectiveness with that of a self-tuning control and a nonlinear adaptive controller [25]. A self-tuning controller is designed for the case in which a two-link flexible manipulator carries an unknown payload at the end point of the second link [26]. In consideration of the aforementioned advantages of the T-S FNN control and the self-tuning control, two adaptive control algorithms are developed controller for the vibration control of a two-link flexible manipulator. A two-link flexible manipulator experimental setup is constructed, using two AC servo motors with high-ratio speed reducer as actuators. PZT patch sensors are bonded on two flexible links to detect the vibration. Experiments are conducted, utilizing the designed T-S fuzzy model with neural networks and self-tuning controller. The GMVSTC control algorithm is made up by an online identifier in the form of controlled autoregressive moving average model and a vibration control signal generator. Also, the parameters identification is implemented by applying the recursive augmented least square estimation (RELS). The rest of this article is organized as follows. System description is presented in Section 2. Controller design including the T-S FNN algorithm and the GMVSTC is described in Section 3. Section 4 presents the experimental results on vibration suppression of the two-link flexible manipulator using the T-S FNN algorithm and the GMVSTC strategies. Finally, the concluding remarks are presented in Section 5.

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2. The description of the two-link flexible manipulator system 2.1. Dynamics of the two-link flexible manipulator The structure of a planar two-link flexible manipulator system is shown in Fig. 1. The two links are cascaded in a serial fashion and actuated by individual AC servo motors with speed reducer at the hub of the flexible manipulator. X 0 Y 0 is the ^iY ^ i are the rigid body coordinate frame attached with the ith link and the moving coorinertial coordinate frame. X i Y i and X dinate frame of the ith link, respectively. hi is the joint angular of the i th link. wi ðxi ; t Þ is the transverse component of the displacement of the i th link. According to Ref. [28], following assumptions were considered in the development of a dynamic model of the flexible manipulator: (a) Transverse shear and the rotary inertia effects are negligible. (b) Each link is assumed to be an EulerBernoulli beam, moving in the horizontal plane. (c) The deformation of each link is assumed in horizontal direction only. (d) The deflection of each link is assumed to be very small.
 x1 It is assumed that 1 r 1 ðx1 Þ ¼ is the position vector that describes arbitrary point of link-1 with the rigid coorw1 ðx1 ; tÞ dinate frame X 1 Y 1 , and the same point in coordinate X 0 Y 0 is

r1 ¼ A11 r 1

ð1Þ




 cosh1 sinh1 where A1 ¼ . sinh1 cosh1 The kinetic energy of link-1 is

T l1 ¼

q1 2

Z

l1

Ab1 r_ 1 r_ 1 dx1 T

0

ð2Þ

where q1 is the density of link-1; the over dot represents the derivative with respect to time.
 x2 Similarly, a point along link-2 with the local coordinate frame X 2 Y 2 is 2 r 2 ðx2 Þ ¼ , and the same point in coorw2 ðx2 ; t Þ dinate X 0 Y 0 is

r2 ðx2 Þ ¼ A1 "



1



r 1 ðl1 Þ þ A2 

2

r 2 ðx2 Þ



ð3Þ

#

cos h2 sinh2 , here,  h2 ¼ h2 þ w1 ðx1 ; tÞ.   sinh2 cos h2 The kinetic energy of link-2 is

where A2 ¼

T l2 ¼

q2 2

Z 0

l2

T Ab2 r_ 2 r_ 2 dx2

ð4Þ

where q2 is the density of link-2; The kinetic energy due to the ith joint is

T hi ¼

1 1 T J h2 þ M hi p_ i p_ i 2 hi i 2

ð5Þ

where M hi and J hi is the mass and the inertia of the ith joint. So the total kinetic energy of TFLM is

Y0

w2 x2 , t

Y2

Xˆ 2

Link-2

Yˆ2

2

Yˆ1

X2

2

w1 x1 , t

Link-1 1

Xˆ 1

X0

1

Fig. 1. Schematic diagram of a planar two-link flexible manipulator.

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T ¼ T l1 þ T l2 þ T h1 þ T h2

ð6Þ

The deflection wi ðxi ; tÞ can be decomposed into a combination of mode-shapes and time dependent generalized coordinate as

wi ðxi ; tÞ ¼

n X

uij ðxÞqij ðtÞ

ð7Þ

j¼1

where uij ðxÞ and qij ðt Þ are the mode shape and the corresponding generalized coordinate of the j th mode of the ith link; and n is the number of assumed modes. The total potential energy due to the deformation of the ith link is

U¼

2 Z X 0

i¼1

li

ðEIÞi

! 2 d ðwi ðxi ; tÞÞ dxi dx2i

ð8Þ

where ðEIÞi is the flexural rigidity of the ith link. The generalized coordinate vector is written as g ¼ ½h1 ; h2 ; q11 ; q12 ; q21 ; q22 T . The driving torque vector is written as

s ¼ ½s1 ; s2 ; 0; 0; 0; 0T ; here, si is the torque applied to the ith link. Then, the dynamic equations of the TFLM can be derived by using Euler-Lagrange’s equations

  @ @L @L  ¼s @t @ g_ @g

ð9Þ

where L ¼ T  U. Thus, the equation of motion of the TFLM can be obtain as [3]

# " # "


 _ q; qÞ €h _ 0 0 s c 1 ðh; h; þD þK ¼ þ _ q; qÞ _ € q_ q 0 c 2 ðh; h; q

Mðh; qÞ

ð10Þ

where M is the mass matrix; q ¼ ½q11 ; q12 ; q21 ; q22 T is the generalized modal coordinate vector; c1 and c2 are the vectors containing of Coriolis and Centrifugal forces; K is the stiffness matrix; D is the damping matrix. 2.2. Introduction of the two-link flexible manipulator experimental platform Fig. 2 shows the schematic diagram of the overall two-link flexible manipulator experimental facility which is constructed for vibration suppression. The hardware components are composed of the following main parts: a two-link flexible manipulator, measuring and control parts. Two flexible links are made up of rectangular epoxy resin beam. And either beam is bonded with a piezoelectric (PZT) patch sensor for detecting the strain signals. The mechanical and geometric parameters of two links and PZT patches are detailed listed in Table 1. Two Mitsubishi electric servo motors and two speed reducers used as actuators for two joints. Their parameters are listed in Table 2. One end of link-2 (elbow) is free; and the other end of link-2 is clamped onto the hub which is driven by Mitsubishi electric servo motor-2 through a planetary speed reducer of 64:1 reduction ratio. One end of link-1 (shoulder) carries the driver of link-2; the other end of link-1 is driven by Mitsubishi electric servo motor-1 through a planetary speed reducer of 64:1 reduction ratio. Fig. 3 shows photographs of the two-link flexible manipulator experimental setup. The built-in shaft motor’s encoders and PZT patch sensors bonded on the links are used to measure the joint angular displacement and the vibration of two links, respectively. The resolution of these encoders is specified as 40,000 pulses per revolution for the measurement of the motor rotation. The two-link flexible manipulator is mounted on the slider of a ballscrew driving unit. In this experiment, the ballscrew unit is not used to drive. It is only considered as the base of the two-link flexible manipulator. A GALIL motion control card (DMC-1846) with 4 axes controlling ability is used as data acquisition and control board, to control these two motors. Two charge amplifiers (YE5850) are used to amplify the vibration signals measured by the PZT sensors to the range from 10 V to 10 V, and these analog signals are also converted to digital signals by the GALIL motion control card. A personal computer with Intel core i5-4590 processor working at 3.3 GHz is used as the processor in experiments. The motion control card (DMC-1846) is also used as a data acquisition card, namely, A/D and D/A converter. The output of the processor is converted to analog signals by the motion control card. The interconnected module (PICM2900) produced by GALIL company provides a direct connect platform the sensors, actuators and the processor. In order to apply the control schemes and watch the controlling effect conveniently, an MFC interactive interface is programmed to connecting the PC and TFLM system, and these algorithm is implemented by C++ to produce the control signal in real-time. 3. Controllers design The designed control schemes include feedback control based on proportional and derivative (PD) controller, the T-S model fuzzy neural network (TS-FNN) controller and the general minimum variance self-tuning controller (GMVSTC), for

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PZT sensor-2

Link-2

Link-1

PZT sensor-1

Motor-2

Motor-1 Encoder-1

Encoder-2

Charge amplifier-2

Servo amplifier-2

Servo amplifier-1

Charge amplifier-1

DMC-Interconnection module

DMC-1846 motion control card

Personal computer Fig. 2. Schematic diagram of the experimental setup showing hardware and signal transmission between each part.

Table 1 Mechanical and geometric parameters of two links and PZT patches. Property

Symbols

Shoulder link

Elbow link

PZT patches

Unit

Length Cross-section Density Young’s modulus Poisson’s ratio PZT strain constant

l A

0.25 3  80 1840 34.64 0.33 –

0.64 2  100 1840 34.64 0.33 –

0.05 1  15 7650 63 0.3 166  1012

m mm2 kg/m3 GPa – m/V

q E

l d31

Table 2 Mechanical parameters of the two joints. Property

Shoulder joint

Elbow joint

Unit

Motor power Motor model Encoder resolution Reduction ratio

400 HC-KFS43 40,000 64:1

100 HC-KFS13 40,000 64:1

W – Pulses per revolution –

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Fig. 3. Photographs of the two-link flexible manipulator experimental setup.

vibration control of the two-link flexible manipulator. The feedback signals are detected by the PZT patch sensors bonded on the flexible links. For the control of joint motions of the two-link flexible manipulator, another PD control algorithm is utilized. 3.1. Design of joint positions controller The block-diagram of the joints position and vibration suppression controller is illustrated in Fig. 4. The driving mode of two AC servo motors is set at speed control mode. A PD feedback controller is applied to control the joint motion. The PD control is 0

0

u0i ðt Þ ¼ kpi ei ðtÞ þ kdi e_ i ðt Þ; i ¼ 1; 2 where

ð11Þ

ei ðtÞ ¼ hri ðtÞ  hi ðtÞ and e_ i ðtÞ are the angular position error and the angular velocity error of the ith link, respectively;

hri ðtÞ and h_ ri ðt Þ are the planned position and angular velocity of the ith motor joint, respectively; kpi and kdi are the ith joint position control proportional and derivative gains, respectively. 0

0

3.2. Vibration controller design 3.2.1. PD controller The vibration signal ypi is the measured vibration signal of the ith link by using the PZT sensor, and the AC servomotors are used as actuators for the vibration suppression. The strain feedback PD control laws for the vibration suppression applied to the AC servo motors are

ui ðt Þ ¼ kpi ei ðt Þ þ kdi e_ i ðtÞ; i ¼ 1; 2

ð12Þ

where ei ðt Þ and e_ i ðt Þ are the error signal and its derivation of the ith link’s vibration, respectively; kpi and kdi are the proportional and derivative control gains for the ith link, respectively. A set of gains for better control performance is tuned experimentally. Control performance, such as increasing the vibration suppression speed, will be improved by increasing the PD control gains theoretically. However, too large gains will result in unpredictable instability of the closed-loop in practical application. In addition, the PD control with constant gains

Vibration controllers for the flexible links

di

,

.

upi

.

di

+

e i,e i

PD controllers for the motors

u

ypi

i

TLFM i i

,

+ rpi

PZT senrsor

. i

Encoder Fig. 4. The block-diagram of the joints position and vibration suppression controller.

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is unable to perform excellently in system parameters variation. Therefore, the T-S fuzzy model with neural networks and a generalized minimum variance self-tuning control schemes are designed to suppress the vibration effectively. 3.2.2. T-S model based fuzzy neural networks (FNN) controller The architecture of an FNN controller is shown in Fig. 5, which consists of an antecedent network and a consequence network. The basic configuration of the FNN, the procedures of the fuzzification, fuzzy inference and defuzzification are implemented in such an FNN by performing mapping form the input vector to the output. The FNN is a kind of an essentially multiplayer forward neural network. The learning algorithm for parameters adjustment can be designed by using the error back propagation method in the BP network. The signal propagation and the basic function in each layer of the FNN controller are subsequently provided. The antecedent network consists of four layers, and the output of the antecedent network corresponds to the fitness value of a fuzzy rule [17–21]: Layer 1 (Input layer): The nodes in the input layer pass the inputs to the next layer. In this layer, the input and output of each node is described as

Net1i ðkÞ ¼ x1i ;

1 y1i ðkÞ ¼ f i Net1i ðkÞ ¼ Net1i ðkÞ;

i ¼ 1; 2

ð13Þ

1 where x11 ¼ e,x_ 2 ¼ e_ are the measured error signal and its derivative with respect time by using the PZT sensor bonded on the flexible link, respectively; k denotes the kth sampling time; and y1i is the output of this layer. Layer 2 (Membership layer): Every node in this layer represents a linguistic variable which can be referred as the fuzzifiction operation. The Gauss function is implemented to calculate the membership of each input variable. The node input and output are expressed as

Net2ij ðkÞ ¼ 

ðxi  cij Þ2

r2ij

;

y2ij ðkÞ ¼ expðNet 2ij ðkÞÞ;

i ¼ 1; 2; j ¼ 1; 2; :::; mi

y1

pij

x0

ð14Þ

consequence network

y2 y3

x1

u

ym

x2 1 1

a1

a1

2 1

a2

a2

m2 1 1 2

am1

1

am1

1

2 2

m1 2

Input layer

Membership layer

am Rule layer

am

Normalization layer

Output layer

antecedent network

Fig. 5. The structure of fuzzy neural networks controller based on Takagi-Sugeno model.

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where cij and rij are the mean and standard deviation of the Gaussian function in the jth term of the ith input variable to the node of this layer, respectively; mi is the number of fuzzy segmentation in this layer for each input; expðÞ is the exponential function, and y2ij is the output of layer 2. Layer 3 (Rule layer): Each node represents a fuzzy rule, which is used to match the antecedent network of the fuzzy rule and get the fitness for every rule. The output of each node in this layer is determined by using fuzzy AND operation. Each node is denoted by P to multiply the input signals. The input and the output of this layer is the same, which can be expressed as

Net3j ðkÞ ¼ y21i ðkÞ  y22i ðkÞ;

y3j ðkÞ ¼ Net 3j ðkÞ;

i ¼ 1; 2; :::; mi ;

j ¼ 1; 2; :::; m

ð15Þ

where m ¼ m1  m2 , and the subsequent variables are also m ¼ m1  m2 . Layer 4 (Normalization layer): This layer is used to normalize the output of layer 3. The number of node in this layer is the same as the upper layer unit. The normalization procedure is expressed as

y3j ðkÞ ; Net4j ðkÞ ¼ Pm 3 i¼1 yi ðkÞ

y4j ðkÞ ¼ Net4j ðkÞ;

j ¼ 1; 2; :::; m

ð16Þ

The first layer of the consequence network is input layer which transmit the input of variable to the second layer. The value x0 ¼ 1 in the input layer node is utilized to provide the constant in the fuzzy rule of the consequence networks. There are m nodes in the second layer of the consequences network. Each node represents a fuzzy rule, and this layer is utilized to calculate the consequence of every fuzzy rule. The output of this layer is written as

Net5j ðkÞ ¼ pj0 þ pj1 x1 þ pj2 x2 ;

y5j ðkÞ ¼ Net 5j ðkÞ;

j ¼ 1; 2; :::; m

ð17Þ

The third layer of the consequences network is also the output layer of the FNN. The output node computes the output as the summation of all input signals with the following type

uðkÞ ¼

m X

y5j ðkÞy4j ðkÞ

ð18Þ

j¼1

To obtain the online learning algorithm of the T-S model FNN controller, it is assumed that the fuzzy division number is predetermined. The parameters supposed to be adjusted are the connecting weight values pji ði ¼ 0; 1; 2; j ¼ 1; 2; :::; mÞ, used in the second layer of the consequent networks, the central value and the width of the network membership function in antecedent networks. The supervised gradient decent method in BP networks also can be used here. To describe the leaning algorithms, an energy function is adopted as

EðkÞ ¼

1 1 2 ðy ðkÞ  yðkÞÞ ¼ e2 ðkÞ 2 r 2

ð19Þ

where eðkÞ represents the error in the learning process of the TS-FNN controller for each discrete time k; yr ðkÞand yðkÞ are the desired output and the practical output of the system at the kth sampling time. The learning process for updating parameters can be deduced in the consequence network. The weight pji is updated by

Dpji ¼ g

@E @eðkÞ @uðkÞ @eðkÞ 4 y ðkÞxi ðkÞ ¼ geðkÞ ¼ geðkÞ @pji @uðkÞ @pji @uðkÞ j

ð20Þ

where g is the learning rate; uðkÞ is the output of the FNN controller. @eðkÞ of the system accurately due to unknown dynamics. To overcome this problem, It is difficult to calculate the Jacobian @u ðkÞ h i @eðkÞ ðk1Þ ; the resulting computational inaccuracy can be a delta adaptation law and sign function are adopted as @u ¼ sgn ueððkkÞe ðkÞ Þuðk1Þ

compensated by tuning the learning rate g. The connective weight pji is updated according to the following equation

pji ðk þ 1Þ ¼ pji ðkÞ þ Dpji ;

i ¼ 0; 1; 2;

j ¼ 1; 2; :::; m

ð21Þ

When the parameters rij ; cij are updated, the connective weight pji is fixed. The error term to be propagated in the output layer of the antecedent network is given by

d5 ¼ 

@EðkÞ @eðkÞ ¼  eðkÞ @uðkÞ @uðkÞ

ð22Þ

The error terms propagated in layer 4 and layer 3 of the antecedent network are given by

d4j ¼  and

@EðkÞ @u ¼ d5 y5j ; @uðkÞ @y4j

j ¼ 1; 2; :::; m

ð23Þ

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P d4j mi¼1 y3i ðkÞ 4 3 @y ð k Þ @Net ð k Þ @E ð k Þ @E ð k Þ j j i–j d3j ¼  ¼ ¼ Pm 2 3 @Net 3j ðkÞ @Net 4j ðkÞ @y3j ðkÞ @Net3j ðkÞ i¼1 y ðkÞ

ð24Þ

i

The error terms to be propagated in layer 2 of the antecedent network are given by

d2ij ¼ 

m X

@EðkÞ @EðkÞ ¼ 3 @Net2ij ðkÞ @Net k ðkÞ k¼1

@Net 3k ðkÞ @y2ij ðkÞ

@y2ij ðkÞ @Net 2ij ðkÞ

¼

m X

0 

B d3k sij exp@

k¼1

xi  c2ij

2 1

r2ij

Since the rule layer in antecedent networks is implemented by multiplication, sij ¼

C A; dNet3n dy2ij

i ¼ 1; 2;

¼

Q

j ¼ 1; 2; :::; mi

ð25Þ

y2ij if Net2ij is the input of the

kth (k ¼ 1; 2; :::; m) node in layer 4 of antecedent network, else sij ¼ 0. The update laws of the mean and the width of the Gaussian function cij and dij are

8 > @E @E > < Dcij ¼ a @cij ¼ a @Net2

@Net 2ij

> > @E : Drij ¼ b @@E rij ¼ b @Net2

@Net 2ij

@cij

ij

@ rij

ij

2ðxi cij Þ

¼ ad2ij ¼ bd2ij

r2ij 2ðxi cij Þ

2

ð26Þ

; i ¼ 1; 2; j ¼ 1; 2; :::; mi

r3ij

where a and b are the learning rate of cij and rij . cij and rij of the membership function are updated according to



rij ðk þ 1Þ ¼ rij ðkÞ þ Drij cij ðk þ 1Þ ¼ cij ðkÞ þ Dcij

; i ¼ 1; 2;

j ¼ 1; 2; :::; mi

ð27Þ

In order to attenuate the vibration of the TFLM quickly, the inputs of the FNN controller are chosen as x0 ¼ 1, x1 ¼ eðtÞ and _ here, ei ðt Þ and e_ i ðt Þ are the measured error signal and its derivative with respect to time, respectively, of the ith link x2 ¼ eðtÞ; detected by the PZT sensors. 3.2.3. Generalized minimum variance self-tuning controller A multivariable self-tuning controller (STC) has three main elements such as [25]: 1) a control law generator in terms of multivariable difference equation; 2) an online parameters estimator that uses measured system output and input values; and 3) an algorithm that relates the estimated parameters and control parameters. Instead of determining explicit and complicated expressions for the model, it can be assumed that the model can be represented by a controlled autoregressive moving average (CRAMA) model, in which these parameters are estimated at each sampling instance on the basis of the available measurements. Fig. 6 shows the structure of GMVSTC vibration controller for the two-link flexible manipulator. According to Refs. [25–27], the self-tuning control is designed for the two-link flexible manipulator. The system input/output relation will be described by using the CRAMA model for the controller design as

Y ðkÞ ¼

 1  d  1 z B z uðkÞ þ C z1 nðkÞ Aðz1 Þ

ð28Þ

RELS Controller parameters design

rpi

di

,

. di

+

-

Vibration controller with variable parameters

.

e i,e i

Joint movtion controller

CRAMA

upi u

ypi TLFM

i

i

,

PZT senrsor

. i

Encoder Fig. 6. The structure of vibration suppression for the TFLM with GMVSTC controller.

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  where the polynomials A z1 ¼ 1 þ a1 z1 þ a2 z2 þ ::: þ ana zna , B z1 ¼ b0 þ b1 z1 þ b2 z2 þ ::: þ bnb znb ,  1 ¼ 1 þ c1 z1 þ c2 z2 þ ::: þ cnc znc ; k refers to the sampling instant; d represents the maximum delay; z1 is a backward C z shift operator; Y ðkÞ is the measured output detected by the PZT sensor attached on the ith flexible link. The measured output and control input of the TLFM system at the kth sampling instant and before are expressed as n o k Y ; U k ¼ fyðkÞ; yðk  1Þ; :::; uðkÞ; uðk  1Þ; :::g. The prediction output for the ith flexible link at the ðk þ dÞth sampling instant n o   ^ðk þ djkÞ. It is assumed that the polynomials A z1 and C z1 satisfy the following based on Y k ; U k can be described as y Diophantine equations

(     C z1 ¼ A z1 E z1 þ zd G z1  1  1  1 F z ¼B z E z

where the polynomials are  F z1 ¼ f 0 þ f 1 z1 þ ::: þ f nf znf .

written

ð29Þ as

 E z1 ¼ 1 þ e1 z1 þ ::: þ ene zne ,

 G z1 ¼ g0 þ g 1 z1 þ ::: þ g ng zng

and

The dth step output prediction model is

yðk þ dÞ ¼ Enðk þ dÞ þ

F G uðkÞ þ yðkÞ C C

ð30Þ

The forward dth step optimal prediction output y ðk þ djkÞ must satisfy the following equation

   C z1 y ðk þ djkÞ ¼ G z1 yðkÞ þ F z1 uðkÞ

ð31Þ

The controller is designed to make the output yðk þ dÞ follow the desired output yr ðk þ dÞ, and the weight term of control value is added to the performance index, to restrict the changes of the control action dramatically. Thus, the performance index function can be described as

n o 2 2 J ¼ E Pðz1 Þyðk þ dÞ  Rðz1 Þyr ðk þ dÞ þ ½Q ðz1 ÞuðkÞ

ð32Þ

where yðk þ dÞ and yr ðk þ dÞ are the actual output and the desired output of the system at the ðk þ dÞth sampling instance,  is the weighted polynomial of the control value; respectively; Q z1 ¼ 1 þ q1 z1 þ q2 z2 þ :::: þ qnp znp  1  1 1 2 np and R z ¼ 1 þ p1 z þ p2 z þ :::: þ pnp z ¼ r 0 þ r 1 z1 þ r2 z2 þ :::: þ prp zrp are the weighted polynomials of the P z actual output and the desired output, respectively; and uðkÞ is the controlled value at the kth sampling instance. According to Eqs. (30) and (31), the system output at the ðk þ dÞth sampling time can be written as

yðk þ dÞ ¼ Enðk þ dÞ þ y ðk þ djkÞ

ð33Þ

where Enðk þ dÞ is the linear combination of the random disturbance nðk þ dÞ; it is independent of

n

Y k; Uk

o

and fyr ðkÞg;

therefore, their cross-correlation function is zero. Then, Eq. (32) can be rewritten as

n o 2  2 2 J ¼ E Pðz1 ÞEnðk þ dÞ þ Pðz1 Þy ðk þ djkÞ  Rðz1 Þyr ðk þ dÞ þ ½Q ðz1 ÞuðkÞ

Let the partial derivative

@J @uðkÞ

ð34Þ

be zero, the necessary condition of minimizing the performance index can be obtained.

There is

 1  @Pðz1 Þy ðk þ djkÞ @Q ðz1 ÞuðkÞ þ 2½Q ðz1 ÞuðkÞ ¼0 Pðz Þy ðk þ djkÞ  Rðz1 Þyr ðk þ dÞ @uðkÞ @uðkÞ

ð35Þ

 1  Pðz Þy ðk þ djkÞ  Rðz1 Þyr ðk þ dÞ b0 þ ½Qðz1 ÞuðkÞq0 ¼ 0

ð36Þ

  According to Q Z 1 and Eq. (31), Eq. (35) can be written as

where

@Pðz1 Þy ðkþdjkÞ @uðkÞ

¼ b0 and

@Q ðz1 ÞuðkÞ @uðkÞ

¼ q0 .

Then, the generalized minimum variance control law can be given as

uðkÞ ¼

Cðz1 Þyr ðk þ dÞ  Gðz1 ÞPðz1 ÞyðkÞ q0 Cðz1 ÞQ ðz1 Þ þ Fðz1 ÞPðz1 Þ b0

ð37Þ

Eq. (31) can be written as the following vector representation

y ðk þ djkÞ ¼ hT uðkÞ where

ð38Þ

Z.-c. Qiu et al. / Mechanical Systems and Signal Processing 118 (2019) 623–644

8   < uT ðkÞ ¼ ½yðkÞ; :::; y k  ng ; uðkÞ; :::; u k  nf ; y  ðk þ d  1jk  1Þ; :::; y  ðk þ d  1jk  nc Þ h i : hT ¼ g 0 ; :::; g ng ; f 0 ; :::; f n ; c1 ; :::; cnc ¼ ½g; f ; c  f

633

ð39Þ

h iT ^; ^ The estimated parameter vector ^ h¼ g f; ^ c can be obtained using a recursive augmented least squares (RELS) algorithm as

8 T ^ ^ ^ > > < hðkÞ ¼ hðk  1Þ þ KðkÞ½yðkÞ  u ðk  dÞhðk  1Þ Pðk1ÞuðkdÞ KðkÞ ¼ 1þuT ðkdÞPðk1ÞuðkdÞ > > : PðkÞ ¼ ½I  KðkÞuT ðk  dÞPðk  1Þ

ð40Þ

where P ðkÞ is the covariance matrix, which is proportional to the covariance matrix of the estimates. Substituting the estimated parameters into Eq. (37), one can obtain the GMVSTC vibration control value uðkÞ. The procedures applied the GMVSTC algorithm to the vibration control are listed as Step1: Choosing the polynomials Pi ðz1 Þ, Q i ðz1 Þ and Ri ðz1 Þ for the ith link. Step2: Initializing the value of ^ hð0Þ and P ð0Þ used in Eq. (39) for the ith link. Step3: Conducting data acquisition of the system output yðkÞ and giving the desired outputyr ðk þ dÞ. Step4: Constructing the observation data vector uðkÞ. ^ðkÞ at the kth samStep5: Applying RELS algorithm defined in Eq. (40) to estimate the real-time controller parameters h pling time online for the ith link. Step6: The self-tuning control law uðkÞ is generated by solving Eq. (37). 4. Experimental results To verify the validity of the vibration suppression of the designed two-link flexible manipulator and the adopted control algorithms, experiments are conducted. 4.1. Experimental vibration identification In experiments, the two Mitsubishi electric servo motors are both specified as the velocity working mode. The ability to suppress the residual vibration of the two flexible links is considered as the main assess the performance of the designed control strategies. The velocity planning of each joint is the same in vibration control experiments. The motors follow the trapezoidal angular velocity profile given in Fig. 7. The motion control units produce motion control signals according to the profile to drive the motors with the proposed PD feedback control. Considering the structural size and carrying capacity of the flexible arm of the experimental system, trapezoidal wave is used to plan the motor running speed. Furthermore, the motor speed should be specified within a certain range. The acceleration values of the acceleration and deceleration sections of the trapezoidal wave velocity are determined through experiments, so as not to exceed the carrying capacity of the flexible arm of the experimental system. When determining the acceleration of the acceleration and deceleration sections, the trapezoidal wave velocity setting should also take into account the load-bearing capacity constraints of the flexible arm of the experimental system. If the speed is too high, the acceleration time will be longer, which will damage the flexible manipulator experimental system. Therefore, if the speed is much higher, the thickness of the flexible arm should be increased appropriately. To suppress the residual vibrations of the two-link flexible manipulator, an appropriate acceleration and deceleration time of the motor angular velocity profile are selected. The main aims are to realize the motion and to excite the appropriate vibrations of the flexible links. Without losing generality and after several tests, the desired angular velocity of the two motors are chosen as x1 ¼ 138 r/min and x2 ¼ 69 r/min; the constant angular velocity time is tcon ¼ 6:64 s, t acc ¼ t dec ¼ 80 ms. The acceleration and deceleration time spans of the two axes are the same. According to the resolution

0

t Fig. 7. The angular velocity profile of motors.

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10

5

5

2.5

Voltage (V)

Voltage (V)

of the encoder and speed ratios of the two reducers, the rotation angular degree of the shoulder link and the elbow link are chosen as h1 ¼ 87:98 and h2 ¼ 43:99 , respectively. The motors are controlled according to the above planned velocity profile. Then, the vibrations of two flexible links are excited due to the inertia of motion during the start and stop period. The time-domain vibration responses of two links without active control are shown in Fig. 8. Fig. 8(a) and (b) are the excited vibration of link-1 and link-2 due to motion, respectively. Fig. 9 shows zoom in on the time axis of the time-domain vibration responses of Fig. 8. Fig. 8 shows that the bending vibration of link-1without active control lasts longer than 20 s. Similarly, the vibration of link-2 also lasts a long time. From Fig. 9, namely the vibration of each link from eleventh to fourteenth seconds, one sees that the coupling effects between link-1 and link-2. Therefore, the vibration control of the flexible links is more difficult than that of the single link, especially for the elbow link. Fig. 10 shows the velocity response curve of each motor without vibration control. Here, two motors are only controlled to track the joint planned trajectories. Fig. 11 shows the control voltage applied to each motor servo amplifiers without vibration control. Power spectral density (PSD) curves of the vibration signals shown in Fig. 8 are obtained and shown in Fig. 12. Fig. 12(a) and (b) correspond the PSD curve of link-1 and link-2, respectively. From Fig. 12, it can be seen that the vibration of the each link is dominated by the first two vibration modes. The natural frequencies of the first two vibration

0

0

-2.5

-5

-5

-10 0

5

10

15

20

25

0

30

5

10

15

Time (s)

Time (s)

(a) Link-1

(b) Link-2

20

25

30

13

13.5

14

10

5

5

2.5

Voltage (V)

Voltage (V)

Fig. 8. The time-domain vibration response without active control.

0

-5

-10 11

0

-2.5

11.5

12

12.5

13

13.5

14

-5 11

11.5

12

12.5

Time (s)

Time (s)

(a) Link-1

(b) Link-2

Fig. 9. Zoom in on the time axis of the time-domain vibration responses of two links without active control.

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150

100

80

Speed (r/min)

Speed (r/min)

100

50

60

40

20

0

0 0

5

10

15

20

25

30

0

5

10

15

Time (s)

Time (s)

(a) Motor-1

(b) Motor-2

20

25

30

20

25

30

1.2

0.6

0.7

0.3

Voltage (V)

Voltage (V)

Fig. 10. The velocity curves of two motors without vibration control.

0.2

0

-0.3

-0.3

-0.6

-0.8 0

5

10

15

20

25

30

0

5

10

15

Time (s)

Time (s)

(a) Link-1

(b) Link-2

Fig. 11. Control voltage applied to each motor without vibration control.

modes are 1.33 Hz and 2.69 Hz. They are the coupled vibration modes of link-1 and link-2. Neither of the two natural frequencies is the natural frequency of either link independently. Just as shown in Fig. 9, the vibration curve trend for the measured PZT sensors signal of two links are similar. 4.2. Vibration PD control algorithm In Section 4.1, two motors are only controlled to track the trajectory. Vibration suppression is not considered. The subsequent experiments will implement active vibration suppression using the adopted control strategies during motion. After some trials of tests, the PD control parameters are achieved for a better motor position and vibration suppression control. The units of angle position and angle velocity are rad and rad/s. Then, the control gains for the motor rotation are selected as k0p1 ¼ k0p2 ¼ 0:637, k0d1 ¼ k0d2 ¼ 0:127. These gains in motor position control aims to keep the motor following the planned trajectory. For vibration suppression, the control gains are chosen as kp1 ¼ 0:03, kp2 ¼ 0:012, kd1 ¼ 0:002, kd2 ¼ 0:0012. The experimental results of vibration responses under PD vibration control are shown in Fig. 13. It is obvious that the large amplitude vibration can be suppressed effectively. However, the control effect to the small amplitude vibration is not significant. This is the reason why the control parameters in the PD controllers are fixed and not able to tuning along with the vibration amplitude. Furthermore, the control voltage generated by the liner PD controller is small for the small

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40

40

20

20 0

Magnitude (dB)

Magnitude (dB)

0 -20 -40 -60

-40 -60 -80

-80

-100

-100 10

-20

-120 0

1

2

10

10

10

0

1

2

10

Frequency (Hz)

10

Frequency (Hz)

(a) Link-1

(b) Link-2

10

5

5

2.5

Voltage (V)

Voltage (V)

Fig. 12. Power spectral density of vibration response of each link without control.

0

-5

0

-2.5

-10

-5 0

5

10

15

20

25

30

0

5

10

15

Time (s)

Time (s)

(a) Link-1

(b) Link-2

20

25

30

Fig. 13. Vibration response curve of each link under PD control.

amplitude vibration. This level of control voltage cannot overcome the problem of the dead zone of the driving actuators with speed reducers, such as the backlash. Thus, the small amplitude residual vibration will last for a long time. Fig. 14 shows the speed response curve of each motor under PD vibration control. Fig. 15 illustrates the control voltage applied to each motor under PD vibration control. Compared Fig. 14 with Fig. 10, and compared Fig. 15 with Fig. 11, it can be obviously seen that the velocity and control voltage curves have effects for vibration suppression. Fluctuation phenomena on the velocity and control voltage curves are just the feedback control effects for vibration suppression. 4.3. T-S FNN control algorithm In order to deal with these problems caused by the fixed parameters controller, adaptive control approaches are considered. Since the control parameters of a constant gain PD controller are fixed, it cannot achieve great suppression performance in both the large-amplitude and the small-amplitude vibration at the same time. Therefore, a neural networks controller based on T-S fuzzy model is utilized, combining the advantages of fuzzy control and neural networks control. A BP algorithm is applied to adjust these parameters on-line, including the connection weights after the piece parts, the mean and variance of the Gaussian membership function in the antecedent part.

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170

80

Speed (r/min)

Speed (r/min)

120

70

50

20

20

-30

-10 0

5

10

15

20

25

30

0

5

10

15

Time (s)

Time (s)

(a) Motor-1

(b) Motor-2

20

25

30

20

25

30

Fig. 14. The speed response curves of motor under PD vibration control.

1.2

0.6

0.8

Voltage (V)

Voltage (V)

0.3 0.4

0

0

-0.3 -0.4

-0.8

-0.6 0

5

10

15

20

25

30

0

5

10

15

Time (s)

Time (s)

(a) Motor-1

(b) Motor-2

Fig. 15. Control voltages applied to motors under PD vibration control.

According to the T-S FNN controller designed in Section 3, the input of the network is the measured signal of the flexible manipulator and its derivative with respect to time for each link used the PZT sensors. The output of the T-S FNN controller is _

the vibration control voltage. The network inputs are normalized by x1 ¼ maxeðfkeÞðkÞg, x2 ¼ maxeðfke_ÞðkÞg. Here, maxfeðkÞg and maxfe_ ðkÞg are the largest value and obtained experimentally. For the shoulder link T-S FNN vibration controller,
 0:9  0:6  0:3 0 0:3 0:6 0:9 c1 ¼ is chosen in antecedent work networks. For each basis function, the parameter 0:9  0:6  0:3 0 0:3 0:6 0:9 b1 is chosen to have same initialized value of 0.6; the parameters in the consequence network is initialed to 2 3 zerosð1; 49Þ p1 ¼ 4 0:1  onesð1; 49Þ 5, and the learning rates are chosen as a1 ¼ 0:009, b1 ¼ 0, g1 ¼ 0:05. For the elbow link T-S FNN 0:9  onesð1; 49Þ vibration controller, the fuzzy partitions of the input is set to 9, and the parameters in antecedent work networks are chosen
 0:9  0:6  0:3  0:1 0 0:1 0:3 0:6 0:9 . For each basis function, the parameter b2 is chosen to have same as c 2 ¼ 0:9  0:6  0:3  0:1 0 0:1 0:3 0:6 0:9

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3 zerosð1; 81Þ initialized value of 0.4, and the parameters in the consequence network is initialed to p2 ¼ 4 0:4  onesð1; 81Þ 5. The learn0:4  onesð1; 81Þ ing rates are specified as a2 ¼ 0:009, b2 ¼ 0, g2 ¼ 0:06. Fig. 16 shows the experimental vibration suppression results under the designed T-S FNN control algorithm. Fig. 17 shows the speed response curve of each motor under T-S FNN vibration control. The control voltage of each motor under T-S FNN vibration control is shown in Fig. 18. Compared Fig. 16 with Fig. 13, it is not hard to find that the designed T-S FNN controller works well in damping the TFLM vibration by comparing with the experiment result with the PD controller. Both the largeamplitude vibration and the small-amplitude are suppressed more quickly under the T-S FNN controller, especially for the small amplitude residual vibration. This is the reason why the T-S FNN controller has the feature of nonlinear adaptive algorithm. The parameters is adjusted by using the error back propagation learning algorithm, which increasing the system robustness to the disturbance. Furthermore, it can solve the problem that PD controller cannot attenuate the smallamplitude vibration quickly. The T-S FNN controller increases the motion stability of the end-point during the TFLM working, owing to the learning capacity and nonlinear characteristic.

10

5

5

2.5 Voltage (V)

Voltage (V)

2

0

-5

0

-2.5

-10

-5 0

5

10

15

20

25

30

0

5

10

15

Time (s)

Time (s)

(a) Link-1

(b) Link-2

20

25

30

20

25

30

170

80

100

50 Speed (r/min)

Speed (r/min)

Fig. 16. Vibration response of each link under T-S FNN vibration control.

30

-40

20

-10 0

5

10

15

20

25

30

0

5

10

15

Time (s)

Time (s)

(a) Motor-1

(b) Motor-2

Fig. 17. The speed response curve of each motor under T-S FNN vibration control.

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0.6

1.2

0.4 0.8

Voltage (V)

Voltage (V)

0.2 0.4

0

0

-0.2 -0.4

-0.4

-0.8

-0.6 0

5

10

15

20

25

30

0

5

10

15

20

Time (s)

Time (s)

(a) Motor-1

(b) Motor-2

25

30

Fig. 18. Control voltage of each motor under T-S FNN vibration control.

2

0.5

0

Center value c

Center value c

1

0.5

-0.5 c c

c

1_1

c

1_3

0

10

c

c

2_1

c

2_3

2_2 2_4

c

2_5

-0.5

1_2 1_4

c -1

c

0

1_5

-1

20

30

0

Time (s)

10

20

30

Time (s)

(a) Central value corresponding to error

(b) Central value corresponding to error derivative

Fig. 19. Part of central values adjustment process of link-1.

It is noted that the FNN controller does not work immediately at the beginning of the each motor’s rotation. This is the reason why the neural networks are adjusting the parameters as introduced in the previous section. This process is transient. After several sampling intervals later, the T-S FNN controller begins to work, and the parameters are tuned at the same time. The learning rate of the variance of the membership function is very small, which increases the robustness of the controller to the system while the network adjusting the parameters. The adjustment process for part central values and connecting weight values for link-1 are shown in Fig. 19 and Fig. 20, respectively. Part of central values and connecting weight values adjustment process for link-2 are shown in Fig. 21 and Fig. 22, respectively. Finally, the center values converge to




c1 ¼



0:90  0:67  0:57  0:47 0:32 0:59 0:96


c2 ¼

0:42  0:42  0:29 0:02 0:26 0:59 0:91

1:12  0:87  0:45  0:23  0:09 0:12 0:26 0:74 1:08 0:93  0:57  0:58  0:71  0:74  0:13 0:43 1:70 4:9



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0.2

0.5

p p

p

1_1

p

1_9

p

1_3

p

1_19

1_22

p

2_10

p

2_4

p

p

2_13

2_18

-0.1

Parameters p

1n

2n

0

Parameters p

2_1

-0.5

-0.4

-0.7

-1

-1.5

-1 0

10

20

30

0

10

Time (s)

20

30

Time (s)

(a) Connecting weights corresponding to error

(b) Connecting weights corresponding to error derivative

Fig. 20. Parts of the connecting weights adjustment process of link-1.

1.2

1.2 c

Center value c

1

c

1_1

c

1_5

c

1_3

c

1_9

2_5

c c

c

2_3

2_7

2_6

0.6

1_7

2

0.6

c

Center value c

c

2_1

0

-0.6

0

-0.6

-1.2

-1.2 0

10

20

30

0

Time (s)

10

20

30

Time (s)

(a) Central value corresponding to error

(b) Central value corresponding to error derivative

Fig. 21. Part of central value adjustment process for link-2.

4.4. GMVSTC control algorithm As described in Section 3, the GMVSTC algorithm is a kind of adaptive algorithm. The control parameters are adjusted automatically, and the RELS is applied to adjust these parameters. Considering that the vibration of the each link is dominated by the first two vibration modes, the parameters of the GMVSTC vibration controller in CRAMA model for link-1 are selected as na1 ¼ 3 and nb1 ¼ 2, and those for link-2 are na2 ¼ 2 and nb2 ¼ 1.  The initial values of weighting polynomials are set as Q z1 ¼ q0 for simplifying the calculation without reducing the control accuracy or stability. It is noted that q0 should be adjusted according to the requirements. A large value of q0 will lose the constraint performance, which makes it difficult to guarantee control stability. On the contrary, it is difficult to achieve the optimal control performance when it is selected too small. After comparing the results of several vibration control exper  iments, q0 is selected as 2.1 for link-1 and 2.8 for the link 2. P z - 1 ¼ 1 and R z1 ¼ 2 are the same in each vibration controller of the two links without losing the stabilization and precision. Then, the vibration control output obtained by the specific process introduced in Section 3 can be used in real-time control. The experimental results show that these parameters are reasonable and effective for the vibration control of flexible links. Fig. 23 shows the experimental results on vibration responses using the GMVSTC algorithm. Fig. 24 shows the speed

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0.8

-0.01 p

p

1_8

1_13

p

p

1_14

p

1_19

p

p

p

p

2_1

1_25

2_7

p

2_8

2_13

-0.05

Parameters p

Parameters p

2n

1n

0.4

2_4

0

-0.4

-0.09

-0.13

-0.8

-0.17 0

10

20

30

0

10

Time (s)

20

30

Time (s)

(a) Connecting weights corresponding to error

(b) Connecting weights corresponding to error derivative

10

5

5

2.5

Voltage (V)

Voltage (V)

Fig. 22. Parts of the connecting weights adjustment process of link-2.

0

-5

0

-2.5

-10

-5 0

5

10

15

20

25

30

0

5

10

15

Time (s)

Time (s)

(a) Link-1

(b) Link-2

20

25

30

Fig. 23. Vibration response of each link under GMVSTC.

response curve of each motor under the GMVSTC algorithm. The control voltage of each motor under the GMVSTC is shown in Fig. 25. It is apparently to see that vibration suppression using the GMVSTC control algorithm is faster than that of the utilized PD control algorithm in suppressing the large amplitude vibration to small, and the vibration of shoulder link can be suppressed very quickly. The large amplitude vibration of elbow link can be damped out fast correspondingly, which helps the flexible arms keep stability in the motion process. It is easy to find that the small amplitude vibration of the flexible links is suppressed to zero in a short time owing to the adaptive adjusting of parameters in the controller. The controller parameters identifying process by the RELS is shown in Fig. 26 and Fig. 27. Aiming at this experimental system, different motor speeds were tried during the experiment. When the speed is specified high, the acceleration time in the trapezoidal wave speed is longer; so the vibration amplitude is relatively large in the start-up and stop phases of the motion process. Judging from the experimental results after being tested at different operating speeds, the vibration suppression control trend is almost identical. Therefore, the experimental results of this paper are to carry out active control experimental research considering one speed and different control algorithms. The experimental results of vibration suppression are recorded and compared during motion and residual vibration control after motion stop, using different control algorithms under one speed.

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170

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Fig. 24. The speed response curve of each motor under the GMVSTC.

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Fig. 25. Control voltage of each motor under the GMVSTC.

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1 f

10 g

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Fig. 26. Identification parameters of GMVSTC for link-1.

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Fig. 27. Identification parameters of GMVSTC for link-2.

5. Conclusions Experiments on active vibration control of a two-link flexible manipulator are conducted, using the constructed experimental apparatus actuated by AC servo motors with speed reducers. Three control algorithms are employed for the flexible links vibration control experiments, namely, PD control, T-S FNN control and GMVSTC control algorithm. The experimental results demonstrated that the designed control strategies can attenuate the vibration of the two-link flexible manipulator effectively, and the adaptive controllers have better performance than that of the PD controller, especially for the small amplitude residual vibration. The FNN controller combines the T-S model fuzzy control with the learning capacity and nonlinear characteristic of neural networks to adjust the controller parameters; therefore, it performs better in suppressing the small amplitude residual vibration than the PD controller. The parameters in the GMVSTC controller can be adjusted online; thus, both the large and the small amplitude residual vibration can be damped out more quickly under the self-tuning vibration control than PD vibration control. The robustness to the disturbance is also improved. Acknowledgements This work was partially supported by the National Natural Science Foundation of China, China (Grant Nos. 51775190, U1501247), partially supported by ‘‘the Fundamental Research Funds for the Central Universities, SCUT, China” (2018PY14), partially Supported by State Key Laboratory of Robotics Foundation, China (2017-O012). The authors gratefully acknowledge these support agencies. References [1] H. Supriyono, M.O. Tokhi, Parametric modelling approach using bacterial foraging algorithms for modelling of flexible manipulator systems, Eng. Appl. Artif. Intell. 25 (5) (2012) 898–916. [2] S.K. Dwivedy, P. Eberhard, Dynamic analysis of flexible manipulators, a literature review, Mech. Mach. Theory 41 (7) (2006) 749–777. [3] M. Khairudin, Z. Mohamed, A.R. Husain, M.A. Ahmad, Dynamic modelling and characterisation of a two-link flexible robot manipulator, J. Low Freq. Noise Vibr. Active Control 29 (3) (2010) 207–219. [4] K. Lochan, B.K. Roy, B. Subudhi, A review on two-link flexible manipulators, Ann. Rev. Control 42 (2016) 346–367. [5] W.J. Book, O. Maizza-Neto, D.E. Whitney, Feedback control of two beam, two joint systems with distributed flexibility, J. Dyn. Syst. Meas. Contr. 97 (4) (1975) 424–431. [6] M. Khairudin, Z. Mohamed, A.R. Husain, R. Mamat, Dynamic characterization of a two-link flexible manipulator: theory and experiments, Adv. Rob. Res. 1 (1) (2014) 61–79. [7] M. Abdollahpouri, G. Takács, B. Rohal’-Ilkiv, Real-time moving horizon estimation for a vibrating active cantilever, Mech. Syst. Signal Process. 86 (2017) 1–15. [8] T.T. Jiang, J.K. Liu, W. He, Boundary control for a flexible manipulator based on infinite dimensional disturbance observer, J. Sound Vib. 348 (2015) 1– 14. [9] Z. Mohamed, J.M. Martins, M.O. Tokhi, J.S.D. Costa, M.A. Botto, Vibration control of a very flexible manipulator system, Control Eng. Pract. 13 (3) (2005) 267–277. [10] H.C. Shin, S.B. Choi, Position control of a two-link flexible manipulator featuring piezoelectric actuators and sensors, Mechatronics 11 (6) (2001) 707– 729. [11] E. Mirzaee, M. Eghtesad, S.A. Fazelzadeh, Maneuver control and active vibration suppression of a two-link flexible arm using a hybrid variable structure/Lyapunov control design, Acta Astronaut. 67 (9–10) (2010) 1218–1232.

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