Active Vibration Isolation of the 6-RSS Parallel ... - Science Direct

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ScienceDirect Procedia Engineering 174 (2017) 941 – 946

13th Global Congress on Manufacturing and Management, GCMM 2016

Active Vibration Isolation of the 6-RSS Parallel mechanism Based on Grey-Fuzzy Control Kunquan Li, Rui Wen* Henan Institute of Engineering, No. 1, Xianghe Road, Xinzheng, Zhengzhou 451191, China

Abstract In this paper, the 6-RSS parallel mechanism is studied to deal with multiple DOF vibration isolation problems of precision control, the active isolation vibration control is investigated. Based on the dynamics model of active isolation vibration system, a Grey Fuzzy Control (GFC) approach is proposed to improve the performance and stability of the 6-RSS parallel mechanism. It proves that GFC is a better design method in the presence of constraints. And the control algorithm utilizes the grey predictive theory to predict the system next-step output response for pre-correction. Simulation results demonstrate the feasibility of the GFC algorithm for real-time implementation. 2016The TheAuthors. Authors. Published by Elsevier Ltd. is an open access article under the CC BY-NC-ND license ©2017 © Published by Elsevier Ltd. This (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of the 13th Global Congress on Manufacturing and Management. Peer-review under responsibility of the organizing committee of the 13th Global Congress on Manufacturing and Management Keywords: grey fuzzy control; 6-RSS parallel mechanism;active isolation vibration; pre-correction;

1. Introduction Active vibration control is a significant topic and a very useful strategy to increase the performance of certain parallel mechanisms, sometimes damping of the vibrations by passive treatments is not sufficient or even impossible, i.e. large space structures, high-precision machines, etc. In this paper, an example of a 6-RSS parallel mechanism, which consists of six chains and an upper and lower plate, see Figure 1, is used to demonstrate the procedure of modeling and active vibration controllers design. The upper plate is fixed to the surroundings and external disturbances, which causes the vibration of the lower plate, enter the mechanism here. The lower plate can be used for example as a support of vibration isolation equipment. Therefore, the main objective of the controllers is to

* Corresponding author. Tel.: +86-0371-62508765; fax: +86-0371-62508765. E-mail address: [email protected]

1877-7058 © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of the 13th Global Congress on Manufacturing and Management

doi:10.1016/j.proeng.2017.01.245

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reduce the motion of the lower plate relative to the upper plate so that the precision of the vibration isolation equipment can be improved. A 6-RSS parallel mechanism can be modeled as a dynamic plant comprised of mass and damping elements [1-3]. It produces dynamic outputs in response to random excitation inputs. A active vibration isolation control system consists of the mechanical and electrical components [4-6]. This system influences the dynamics of the 6-RSS parallel mechanism and is the main factor in determining the quality of vibration isolation. The key characteristic of the active vibration isolation system is that an external power source is used to achieve the desired vibration isolation goal [7]. The actuator is placed as a part of the active vibration isolation system for the 6-RSS parallel mechanism. The controller drives the actuator based on the designed control law. The active vibration isolation system provides the freedom to adjust the entire vibration isolation system, and the control force can be introduced locally or globally based on the system state [8, 9]. Hence, the model-based integrated system controller is hard to design, making it unsuitable for effectively improving control performance of the 6-RSS parallel vibration isolation system. Fuzzy logic control strategies have been applied extensively in recent years. Such strategies possess the special feature of the controller development being model-free. Therefore, fuzzy logic control strategies can also be applied to control complicated nonlinear parallel mechanism dynamic model. Although fuzzy logic control has many advantages to deal with complicated system, the application of fuzzy logic control still needs considerable efforts to identify the appropriate membership functions and fuzzy rules, particularly when the system is complicated or rapidly changing. These features substantially increase the difficulty in designing a Traditional Fuzzy Controller (TFC) for the parallel mechanism to obtain good performance. To enhance the performance of the controller, a grey prediction algorithm has been employed to estimate the next-step output of the integrated parallel mechanism for TFC [10-12]. These proposed control strategies reduce the difficulty of the fuzzy controller implementation and improve the control performances of the TFC, by using a predictor for pre-compensating the system output error, rather than using post-compensation. This paper is organized as follows: in Section 2, the model of 6-RSS parallel mechanism is built. The grey prediction algorithm and TFC are stated respectively, and then integrated into Grey-Fuzzy Controller (GFC) in Section 3. In Section 4, the simulation results which illustrate the usefulness and advantage of the proposed methodology are given in Section 3. And the conclusion is drawn in the final section. 2. Dynamics model establishment of the 6-RSS parallel mechanism For the design of active vibration controllers for a parallel mechanism, the first aspect that needs to be considered is the analysis and modeling of the real mechanism. A simplified mechanical and mathematical model is usually sought in order to produce useful and realistic information and to avoid an unnecessarily complicated analysis. This paper deals with the 6-RSS parallel mechanism as shown in Fig. 1.

Fig. 1. Schematic diagram of a 6-RSS parallel mechanism.

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The form of the equations of dynamics model is

q  2WEVq  E 2 q Cq q  Cwq With

0T

)

Bqu

(1)

y

\ 7 0\

.T

\ 7 .\

E2

M q1K q

H

M q1Dq1E 1 2

Where q is the vector of generalized displacements, u is the vector of input forces, y is the output vector, M is the mass matrix, D is the damping matrix, K is the stiffness matrix, Cq is the output displacement matrix, and Cw is the output velocity matrix, E is the matrix of eigenfrequencies and ψ is the transformation matrix. For the controller design the model is transformed to state-space form by defining the state variables as

x

ªq º «q » ¬ ¼

(2)

Resulting in the equations of motion in state-space form

x y

Ax  Bu Cx

(3)

Where

ª 0 A « 2 ¬ E

I º »,B  2 HE¼

ª0 º «B » , C ¬ q¼

>C

q

Cw

@

(4)

The matrix A represents the system matrix, B the input matrix, and C the output matrix. 3. Grey-fuzzy controller 3.1. Grey prediction algorithm The grey model was firstly proposed by Deng in 1982. A grey system is a partially known and partially unknown system. This system employs the data generation method to obtain a more regular generating sequence from the initial random data. The grey prediction thus is used to establish a grey model that extends from past to future information, based upon both past and present known or undetermined information. The unique feature of establishing a grey model is the use of discrete time sequence data to establish an ordinary differential equation. The accumulated generating operation (ATO) and inverse accumulated generating operation (IATO) provide the basic tools for searching the grey model. The orders of ATO and IATO operations not only depend on the order of the differential equation of the grey model but the number of grey variables. The general form of a grey model is GM (n, m), where n denotes the order of the ordinary differential equation of the grey model and m represents the number of grey variables. The generation time increases exponentially with increasing equation order n and variable number m. Moreover, using large n and m values does not assure improved prediction accuracy. Hence, the GM (1, 1) model is widely employed in various grey systems for prediction application and the basic prediction algorithm is summarized as follows: Step 1: Given the initial data,

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Kunquan Li and Rui Wen / Procedia Engineering 174 (2017) 941 – 946

X (0) [ x(1), x(2),", x(n  1), x(n)]

(5)

Where [N corresponds to the system output at time k. We try to predict the next [Q  L L t  . Step 2: From the initial ;  a new sequence ;  is generated by the accumulated generating operation 1 1 1 (AGO), where X (1) [ x (1), x (2),", x (n), ] , and is derived as follows: n

x1 (k )

¦ x(m)

(6)

m 1

;  we can form the following first order differential equation,

Step 3: From

dx1  cx1 dt

u

(7)

Where c is development coefficient of GM (1,1) and u is grey input and both of them are constants requiring determination in the model. Step 4: From step 3 we have,

xˆ1 (k  1)

( x(1)  u c)eck  u c .

(8)

The parameters of c and u are obtained by Least Square approximation method as:

aˆ

ªc º «u » ¬ ¼

( BT B) 1 BT Yn

(9)

ª z1 (2) « 1 « z (3) « # « 1 «¬ z (n)

1º » 1» 1 1 1 Where Yn , z (k ) D k x (k )  (1  D k ) x (k  1) . » » 1»¼  The generating coefficient D N is usually given as >@. Which means that D N is affected by [ N and  [ N   equivalently. ª x(2) º « x(3) » », B « « # » » « ¬ x ( n) ¼

Step 5: Restore the predictive value from the inverse AGO, we have,

[ЦN   Here

[ЦN    [ЦN

  HF >[  X F@H FN

(10)

[ЦN   is the predicted value of [N   at time k+1.

3.2. Fuzzy logical controller The fuzzy set theory was firstly proposed by Zadeh, and numerous control applications subsequently have been developed by using this theory. The main design objective of most of these applications is to construct a fuzzy system to approximate the desired control action. The structure of a TFC design consists of the definition of input/output fuzzy variables, decision-making related to fuzzy control rules, fuzzy inference logic and

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defuzzification. Fig. 2 presents a traditional fuzzy control for the integrated parallel manipulator system. The control system variables are defined as,

e(k ) r (k )  y(k )

(11)

ec(k ) e(k )  e(k  1)

(12)

Where HN and HFN are the system output error and the error change respectively; represent the reference input of the system and the output response of the system respectively.

UN and \N

Fig. 2. Traditional fuzzy controller.

3.3. Grey-fuzzy controller Fuzzy controllers require significant effort to find appropriate membership functions and robust fuzzy rules for improving control performance. And it exists time delay as well, especially when the fuzzy inference system is complex. Consequently, this study introduces the grey predictive theory into the TFC to predict the next output error of the parallel manipulator system and the error change, rather than the current output error of the system and the current error change, as the input variables of the TFC. This control strategy is expected to improve the performance of TFC and reduce the difficulty of its implementation. Fig. 3 shows the grey-fuzzy control strategy for the robot system. Here \ЦQN   is the predicted next-step output response of the robot system, estimated using the grey X , q are used to indicate the grey-fuzzy controller prediction algorithm described previously. Their subscripts Q of the active isolation vibration of the 6-RSS parallel mechanism.

Fig. 3. Grey-fuzzy controller.

4. Simulation result In order to verify the performance of the proposed control strategy and algorithm, the simulation is produced towards the parallel mechanism. In the 6-RSS parallel mechanism, the components quality and the rotation inertia may be achieved by Pro/E computation. During the course of this study, a cubic configured 6-RSS parallel platform using the actuators is built incorporating a unique vibration control, and a Grey-fuzzy Control (GFC) algorithm is implemented in a real-time simulation using the 6-RSS parallel mechanism. As shown in Fig. 4.

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Fig. 4. Grey-fuzzy control simulation of the lower plate.

5. Conclusions Active vibration isolation with Grey-fuzzy controller provides significant improvement in system performance over a purely active suspension system. The 6-RSS parallel mechanism is studied to deal with multiple DOF vibration isolation problems of precision control. GFC proves to be a better design in the presence of constraints since the constraints are kept satisfied exactly in the GFC design process. The simulation results for the GFC are extremely encouraging in terms of large improvements in the precision control and mobility. Simulation results demonstrate the feasibility of the GFC algorithm for real-time implementation. Acknowledgment This work is jointly funded by the National Natural Science Foundation of China (No.50375004). References [1] D. Steward, A platform with six degree of freedom, Proc.Inst. ech. Eng, London, 1965. [2] Y. Takeda, H Funabashi, M Kimura, Development of a spatial six-degree-of-freedom in-parallel actuated worktable with rolling spherical bearings, Proc. of the Ninth Int, Conf. on Adv. Robotics, (1999) 551-556. [3] C. E. Garcia, D.M. Prett, M. Morari, Model Predictive Control: Theory and Practice-a Survey, Automatica. 25 (1989) 335-348. [4] J. Richalet, Industrial Application of Model Based Predictive Control, Automatica. 5 (1993) 1251-1274. [5] A. AST, P. EBERHARD, Control Concepts for a Machine Tool with an Adaptronic Actuator, Proc. of the Multibody Dynamics 2007, ECCOMAS Thematic Conference, by C. Botasso (Ed.), Milano, Italy, 2007. [6] V.P. Phan, N.S. Goo, H.C. Park, Vibration suppression of a flexible robot manipulator with a lightweight piezo-composite actuator, International Journal of Control, Automation and Systems. (2009) 243-251. [7] N. Hara, Y. Fukazu, Y Kanamiya, D Sato, Singularity-Consistent Torque Control of a Redundant Flexible-Base Manipulator, Motion and Vibration Control. (2009) 103-112. [8] Y. Yun, Y. Li, Design and analysis of a novel 6-DOF redundant actuated parallel robot with compliant hinges for high precision positioning, Nonlinear Dynamics. (2010) 829-845. [9] W. Chen, Study on multi-variable adaptive integrated control of automobile electric power steering and active suspension systems, Journal of Vibration Engineering. 3 (2005) 360-365. [10] R. Lian, B Lin, A grey prediction fuzzy controller for constant cutting force in turning, International Journal of Machine Tools & Manufacture. 45 (2005) 1047-1056. [11] Y. Huang, C Huang, The integration and application of fuzzy and grey modelling methods. Fuzzy sets and Systems. 78 (1996) 107-119. [12] H.Z. Zhang, H.X. Wu, The relationship study between adjusting factors and grey prediction, in Proc. 5th Conference on the Grey System Theory and Application. (2000) 289-298.