Strojniški vestnik - Journal of Mechanical Engineering 62(2016)9, 494-502 © 2016 Journal of Mechanical Engineering. All rights reserved. D0l:10.5545/sv-jme.2015.3016 Original Scientific Paper Received for review: 2015-09-13 Received revised form: 2016-04-08 Accepted for publication: 2016-05-09 The Bond Graph Method for Analysis of the Micro-Motion Characteristics of a Micro Gripper Chao Lin1* - Yi-hang Ren1 - Jiu-xiang Ji1 - Li-zhong Cai1 - Ji-ming Shao2 1Chongqing University, The State Key Laboratory of Mechanical Transmission, China 2Shanghai Key Laboratory of Spacecraft Mechanism, China A full-flexure micro gripper with three-stage amplification has been designed. The pseudo-rigid-body (PRB) model of a bridge-type amplification mechanism (BTAM) and micro gripper are established; the input/output stiffness of the BTAM are deduced on the use of the compliant mechanism, Castigliano's theorem and the PRB method. The characteristic equations and state-space equations of the micro gripper are derived on the basis of bond graph theory. The displacement simulation curve and flexible hinge angular simulation curve of the micro gripper are acquired through Matlab R2013b programming. Ansys13.0 finite element simulation software is utilized for simulation analysis of the micro gripper micro-motion. The micro gripper is 3D printed using laser rapid prototyping technology, and a test bench has been set up. The experimental value, finite element analysis value, and Matlab simulation value are comparatively analysed, and the change rules are essentially the same. As a result, the validity of the bond graph model of the micro gripper is verified, and providing a new effective method for the flexible mechanism analysis. Keywords: micro gripper, pseudo-rigid-body (PRB), flexure hinge, bond graph, Matlab R2013b, 3D print Highlights • A full-flexure micro gripper with three-stage amplification has been designed. • The bond graph model of the micro gripper has been set up, the characteristic equations and state equations of the micro gripper are derived. 0 INTRODUCTION In recent years, with the rapid development of science and technology, micro/nanotechnology has been becoming a key direction in research around the world. The demand for micro-motion devices has increased in many industrial applications, particularly in the fields of manipulating biological cells, microsurgical operation and assembling micro-machines. With the operation objects trending towards miniaturization, developing flexible, high precision and easily control microrobots are becoming an urgent requirement, and micro-manipulating robots combined with micro-positioning technology and robot technology are receiving increased attention [1] and [2]. According to the driving principle, micro grippers can be roughly divided into five categories: electrothermal actuators [3], electrostatic actuators [4], piezoelectric actuators (PZTs) [5], shape memory alloy (SMA) actuators [6], and electromagnetic actuators [7]. PZTs are widely used because they have many advantages such as stable output displacement, great output force, high resolution, high response speed and are easy to control [8] and [9]. Nah and Zhong [9] designed a monolithic compliant-flexure-based micro gripper, of which the displacement amplification and maximum stroke are 3.0 ^m and 170 ^m, respectively. Zubir et al. [10] developed a compliant-based micro gripper for high precision manipulation; a high displacement amplification and a maximum stroke of 100 ^m can be achieved. Ramadan et al. [11] proposed a chopstick-like two-fingered micromanipulator based on a hybrid mechanism for cell manipulation. Xiao et al. [12] presented a micro gripper which has absolutely parallel movement of the gripping arms; the displacement amplification ratio is about 10, ^m, and the micro gripper arm maximum stroke is 300 ^m. Yu et al. [13] studied a piezoelectric-based micro gripper of which the displacement amplification of one side can be 10.6, and the maximum output displacement is 261 ^m. Furthermore, other micro grippers have been designed, such as a thin-walled copper tube by Li et al. [14], a compliant piezoelectric actuator based micro gripper by Jain et al. [15], a single active finger ionic polymer metal composite (IPMC) micro gripper by Ford et al. [16], a rotary micro gripper with locking function by Hao et al. [17], a piezoelectric actuator for micro gripper by Jain et al. [18], and a piezoelectric-actuated micro gripper with a three-stage flexure-based amplification by Wang et. al [19]. The transmission of flexible micro gripper motion depends on the deformation of the material, so the kinematic modelling accuracy is difficult to guarantee. Furthermore, it is difficult to obtain the specific state change of a single internal component of the micro gripper. The bond graph is put forward by Paynter, describing the transmission, storage, conversion and consumption of power [20]. The nonlinearity of 494 *Corr. Author's Address: State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing, China, linchao@cqu.edu.cn Strajniski vestnik - Journal of Mechanical Engineering 62(2016)9, 494-502 systems can be intuitively considered in the bond graph method, and state variables in the model are all physical variables; thus, the state changes of the system can be further described. Moreover, the superiority of consistency between the bond graph and systematic state equations is significant. The corresponding mathematical model can be acquired by the bond graph of the system [21]. At present, bond graph modelling has been applied in the analysis of the Stewart platform mechanism [22]; however, the use of bond graph theory to analyse the flexible mechanism remains to be further developed. Given the advantages of the bond graph in complex system physical model analysis, the bond graph model of the micro gripper is built after the analysis of working principles of the flexible micro gripper, stiffness of the bridge type amplification mechanism (BTAM) and parallel-guiding mechanism (PGM). The characteristic equations and statespace equations are derived. Finally, the output displacement simulation curve and flexible hinge angular displacement simulation curve of the micro gripper are obtained by using Matlab simulation and analysis. 1 STRUCTURE DESIGN AND WORKING PRINCIPLE ANALYSIS OF THE MICRO GRIPPER According to the basic theory of compliant mechanism and mechanical principles, a micro gripper mechanism with large output displacement is designed (as shown in Fig. 1). The dimension is 106 mm x 76 mm x 10 mm. The micro gripper is driven by PZTs; the input displacement provided by PZTs is amplified by BTAM, and further enlarged by a two-stage flexible lever. As a result, the micro gripper arm has a large stroke. Meanwhile, the parallelguiding mechanism ensures gripper arm maintains parallel movement and makes the grip jaw clip object more stably and firmly. 2 STATICS ANALYSIS OF THE MICRO GRIPPER 2.1 Input Stiffness of the Micro Gripper According to the PRB method, a BTAM can be divided into 16 components, and each component is connected by a flexure hinge without clearance. The flexure hinge is equivalent to torsional spring, and the rest of the components are equivalent to a rigid rod. A quarter of the BTAM is taken for analysis because it is entirely symmetrical in structure. The PRB model and force analysis of BTAM are shown in Fig. 2. Fig. 2. Graph of PRB model and 1/4 model force analysis of BTAM Using Castigliano's theorem, input stiffness and output stiffness of BTAM were deduced by Lobontiu. The derivation process of input stiffness is the same as The Bond Graph Method for Analysis of the Micro-Motion Characteristics of a Micro Gripper 495 Strojniski vestnik - Journal of Mechanical Engineering 62(2016)9, 494-502 in reference [23], in this paper, however, in order to simplify the calculation, the shear force of flexure hinge is ignored. The output stiffness can be obtained in the same way while assuming the force F6y is applied to point 6' when solving output stiffness. Flexible hinges 2 and 4 have the same dimension; hence, the compliance along the x-axis and around the z-axis of flexible hinge 2 are equal to that of flexible hinge 2 correspondingly. That is, CX-P =C^-P, C =C ^g-M ^ g-M ■ Based on the analysis and theoretical derivation, the input stiffness and output stiffness of BTAM can be expressed by: _ F0 x _ 2 U0x 4Cx-F +LlC6-M F k = (1) (L L )2C2 u4l ^e-M where C2e_M=12L2/(Ebt3), C2X_F = L2/(Etb), t, L, and L2 to L5 are structural parameters of BTAM as shown in Fig. 2, b is thickness of BTAM. According to the movement of the micro gripper, when the micro gripper is driven by PZTs, not only the BTAM produces elongation deformation, flexible hinge 16, 19, 26 and 29 also achieve bending deformation in the driving force direction. As a consequence, the input stiffness of the micro gripper includes the input stiffness of BTAM and the bending stiffness of flexure hinge, as, Kn =K +£ cm F y-F (2) where m = 16, 19, 26, 29, and 1/C^ is bending stiffness of flexure hinge m when it is under the force F6V which is parallel to the cross section, and 1/C™_F = EbftI(612), l is the length of flexure hinge. 3 THE BOND GRAPH MODEL OF MICRO GRIPPER The energy conversion and transmission of interactional components are the basic foundations to set up the system bond graph in the engineering system. In the bond graph theory, the system with the co-existence of different forms of energy can be processed in a unified way; any system can be summarized into four state variables: flow variables ft), effort variables e(t), generalized momentum p(t), and generalized displacement q(t) . Based on the theory and method of the bond graph, combined with the characteristics of the micro gripper driver and motion transmission, firstly the bond graph models, respectively, of BTAM, leverage, and PGM are set up; then the bond graph model of each part to set up the bond graph model of the microclamp is assembled. In the process of setting up the bond graph model of PGM, the rotary angles of the flexible hinges of PGM are assumed to be the same. For BTAM, it works as one end fixed while the other stretches, but in the bond graph model, both endsare assumed to stretch. According to the transmission characteristics of the system and the causal relationships among different parts, considering generalized displacement q(t) of capacitive component and generalized momentum p(t) of inertial component as the state variables, e(t) and f(t) as the effort and flow of each bond., numbering each key from 1 to 82, the bond graph model of the micro gripper was established. The model is displayed in Fig. 4. In Fig. 4, '0' connection and '1' connection stand for serial and parallel energy connections; the arrows indicate energy flow directions of the system. I, C and R represent the inertial component, capacitive component, and resistive component, respectively. Transformer (TF) helps to convert one type of motion to another. Se is driving force, Kin is input stiffness of the micro gripper, ki, Mi, Ji and ui represent stiffness, mass, the moment of inertia and damping coefficient, and the subscript label is the series number of parts in Fig. 1b. k'=km = 1/ClM (m = 2, 4, 6, 8, 9, 11, 13, 15) is bending stiffness of flexure hinge of BTAM, k" =kn=1/ Cg6M (n = 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 35) is bending stiffness of flexure hinge of PGM. J' = Jj (j = 3, 7, 10, 14) is the moment of inertia of the rigid rod of BTAM, J"=Jk (k = 18, 20) is moment of inertia of the rigid rod of lever, J'"=Jp ( p = 24, 28, 30, 34) is the moment of inertia of the rigid rod of PGM. By mechanics of materials, J = ml2 / 12. 4 THE ESTABLISHMENT OF THE STATE-SPACE EQUATIONS The motion state of the system can be described by a set of state variables. State-space equations are the time-varying physical quantities of the system's internal state. As seen in Fig. 4, there are 30 energy-storage components in the system, including 16 capacitive components and 14 inertial components. In these components, I63 and I82 have differential causality, and the rest have integral causality. The BTAM has equivalent damping in the direction of a driving force, as do both the left and right parts of micro gripper; as a consequence, the system also has resistive components. From Fig. 4, characteristic equations can be obtained as follows. k 2 U6 y 1 496 Lin, C. - Y.H. Ren - Ji, J.X. - Cai, L.-Z. - Shao, J.-M. Strajniski vestnik - Journal of Mechanical Engineering 62(2016)9, 494-502 micro gripper left part C:ll(ki6+kn) /1/11 SeF(t) t\ 5o\ The BT AM paît vu ClVkis ^ {h+hyie IsKU-h) L^tt^^^M,, 33 ;il ml ^ V, "I 1/h Uli C.Vka I'Jn C:Vki! 4:1 -a isl -4.1, .-TO C : 1/h I JwC Vku Sij" ]nr ' 1 73^ 1 1 Tt^ Q |79 „ Czl/Qai-tha ■ ^ (h-w<, is/çt-h) i' Il____________I vu 7°l F-Mr. Riu, micro gripper right part Fig. 3. The bond graph model of micro gripper The characteristic equations of the resistive component are: e4 /4 R4 , e58 f58 R58 e =f R e =f R 24 J 24Jl24> 70 J101 70 e5l=f5lR5l, e77=f77 R77 (3) The characteristic equations of the energy-storage components with integral causality are: fi = -1 Pi 12 e3 c ei6 c q!6 C16 /i8=~r Pi _ 1 e60 = c ^60 C60 f =_L J 64 j p64 f76 j p76 1 e79 = c ^79 C79 (4) The characteristic equations of the energy storage components with differential causality are: I e63=!63 f63 '82 I82f82 (5) Assuming the system state variable is * = ((