IROS2019国际学术会议论文集 0545

Design and Analysis of a New 3 DOF Active Type Constant Force Compliant Parallel Stage Xiaozhi Zhang Qingsong Xu Senior Member IEEE and Yuzhang Wei Abstract This paper presents the design analysis and testing of a novel three degree of freedom 3 DOF compli ant parallel kinematic active constant force stage The active constant force property enables a large travel and constant driving property which is enabled by introducing symmetrical bistable fl exure hinges The folded fl exure mechanism is adopted to guide the driving input and to balance the stiffness of the stage to zero In addition leaf fl exure hinges are employed to decouple the cross axis motion of the 3 DOF parallel stage Analytical modeling is conducted to evaluate the stage perfor mance To verify the performance of the constant force property and motion decoupling fi nite element analysis simulation study is carried out By minimizing the fl uctuation of the constant force value design optimization of the stage parameters is implemented with multi objective genetic algorithm Moreover a prototype is fabricated for demonstration of the proposed concept design Index Terms Compliant mechanism fl exure mechanism parallel mechanism constant force mechanism mechanical de sign I INTRODUCTION Flexure based compliant stages have been widely used in precision positioning applications In view of kinematic scheme the stages can be divided into the serial kinematic and parallel kinematic stages 1 The serial stage can offer larger travel range and allows relatively straightforward con trol But the high inertia existence of parasitic motion large cumulative error and relatively lower natural frequency affect the high accuracy of the stage and narrow its application 2 On the contrary the parallel stage turns to be an expected alternative to tackle the shortcomings of serial stage The parallel stage have been applied in diverse domains Due to the advantage of high load carrying capacity it has been adopted in the equipment such as fl ight simulators and earthquake mobile sensors 3 The parallel mechanism also owns good dynamic performance So it is widely adopted in robotic fast transportation operation such as pick and place operation 4 Replacing conventional mechanical joints with fl exure hinges the resulted compliant parallel mechanisms have been extensively applied in ultrahigh precision engi neering applications such as precise positioning operation 5 This work was supported in part by the National Nature Science Founda tion of China under Grant 51575545 and the Macao Science and Technology Development Fund under Grant 179 2017 A3 and the Research Committee of the University of Macau under Grant MYRG2018 00034 FST The authors are with Department of Electromechanical Engineering Faculty of Science and Technology University of Macau Avenida da Universidade Taipa Macau China Corresponding author Q Xu phone 853 8822 4278 fax 853 8822 2426 e mail qsxu umac mo To achieve a high precision the commonly adopted ac tuator for compliant parallel stage is piezoelectric actua tor PZT 6 7 8 However the travel range of the parallel stage is usually limited to micrometer range To broaden the travel range of PZT displacement amplifi ers are employed In reference 9 the bridge type amplifi er is proposed with the analysis of amplifi cation ratio The lever type displacement amplifi er is analyzed in 10 However neither of the aforementioned amplifi er can overcome the limitation of travel in less than millimeter level So the voice coil motor VCM is commonly adopted to provide larger actuation travel In reference 11 the proposed XY parallel stage obtains the travel of centimeter with compound parallel fl exures CPF For a 3 DOF parallel stage the travel is achieved as 1 1 mm in reference 12 The travel is constricted by the actuation force of the VCM To eliminate the dependance on large actuation force the constant force mechanism is proposed in reference 13 As compared with conventional mechanism the constant force mechanism can reduce the required actuation force by around 63 As for the constant force mechanism it can be divided into active type and passive type The passive constant force mechanism can protect the operated object from overloading 14 which can be adopted in safe grasping operation 15 While the active constant force mechanism can broaden the travel and eliminate the use of large force actuator The difference between the active and passive types of constant force structures is described in 16 The constant force mechanism is usually devised by the combination of positive stiffness mechanism and negative stiffness offered by bistable mechanism mechanism The active type needs special boundary condition to constrain the bistable fl exure mechanism In reference 17 a structure fabricated with aluminium alloy is adopted to fi x the negative stiffness mech anism In reference 18 two symmetrically located sticks are added between group 1 and 2 to constrain unexpected travel of the bistable beams However no multi DOF active constant force compliant parallel stage has been reported in the literature To this end a novel 3 DOF active constant force stage is proposed in this work In particular two fi xing structures are adopted to offer tight fi xing on two ends of the negative stiffness mechanism Moreover to obtain a fi ne constant force property the fl uctuation of the constant force value needs to be reduced In reference 19 the pseudo rigid body model is adopted to generate the force displacement relationship fi rstly Then the simplifi ed crank slider model is employed to reduce the nonlinear optimization routine Finally the optimized 2019 IEEE RSJ International Conference on Intelligent Robots and Systems IROS Macau China November 4 8 2019 978 1 7281 4003 2 19 31 00 2019 IEEE2708 Fig 1 CAD model of the designed 3 DOF active constant force parallel stage 1 VCM 2 folded guiding mechanism 3 negative stiffness fl exure mechanism 4 decoupling fl exure mechanism 5 output platform 6 constant force fi xing structure and 7 baseboard constant force mechanism is generated by determining the di mensional parameters To reduce the fl uctuation of constant force value a parametric study is proposed in reference 20 In this paper the genetic algorithm GA is adopted to obtain the constant force property via a design optimization process with FEA simulation The following parts of the paper are organized as follows The mechanical design is presented in Section II Then the analytical modeling is derived in Section III Finite element analysis FEA simulation study is conducted to verify its performance in Section IV Section V gives the conclusion II MECHANICALDESIGN In this section the modular design process is presented To reduce the cross axis motion the decoupling design is proposed A Modular Design The modular design has been widely adopted in complex design process such as life cycle engineering 21 complex web applications 22 axial fl ux generators 23 In this paper the modular design solution is an effi cient way to realize the complex structure of the 3 DOF active constant force parallel stage As shown in Fig 1 the stage is composed of seven modules The VCM is adopted to offer the actuation force To bear the coil part and guide the movement direction of VCM the folded guiding fl exure mechanism is employed To obtain the constant force property the negative stiffness fl exure mechanism is applied The decoupling fl exure mechanism is adopted to reduce the cross axis motion for the parallel stage The platform is located upward to offer a target for collecting the data for laser sensor in experimental testing For the active constant force mechanism it needs to be specially fi xed So the constant force fi xing structure is proposed In addition the baseboard is fi xed on the optical table by four bolts Fig 2 Kinematic model for the fl exure mechanisms B Constant force Mechanism Design The constant force property is produced by resorting to zero stiffness in an acceptable travel So the zero stiffness of the mechanism is the key point For this purpose the negative stiffness mechanism and positive stiffness mech anisms are combined together The zero stiffness can be obtained by changing the parameters of the folded guiding mechanism C Motion Decoupling Design The parallel leaf fl exures are employed to build the decou pling fl exure mechanism due to the relatively simple structure and good decoupling ability in three dimension In this work fi ve decoupling fl exure mechanisms are orthogonally located to reduce the cross axis errors III ANALYTICALMODELING To evaluate the constant force property analytical stiffness modeling is conducted Besides the decoupling model is proposed to test the performance of the decoupling fl exure mechanism A Stiffness Model The stage is symmetrical along X and Y axes So the stiffness models for X and Y are the same Along X axis as the folded guiding mechanisms decoupling fl exure mechanisms and negative stiffness mechanism are connected in parallel the force relationship can be get as Ft 2Fn Ff 3Fd 1 where Ftdonates the actuation force Fn Ffand Fdare the reaction force of the negative stiffness mechanism folded guiding mechanisms and decoupling fl exure mechanism re spectively The detail is illustrated in Fig 2 which illustrates the fl exure elements 2 and 3 as denoted in Fig 1 2709 With actuation along X axis as shown in Fig 3 a the stiffness equation can be obtained kt 2kn kf 3kd 2 Concerning the motion decoupling fl exure mechanism the translational stiffness Kdcan be derived as Kd 4Ebt3 l3 1 3 where b is the width and l1is the length of the decoupling fl exure hinges Regarding the negative stiffness mechanism the move ment is nonlinear So the elliptic integral approach is em ployed to describe the stiffness equation With reference to 18 the equation of the displacement X is derived X l2sec 1 2 cos cos 1 cos 2 sin 2E 2 2E 1 F 2 F 1 4 where the fi rst kind incomplete elliptic integral is F the second kind one is E and the amplitude of elliptic integral is represented by F 1 2 Z 1 2 0 dx p 1 2sin2x 5 E 1 2 Z 1 2 0 p 1 2sin2x dx 6 In addition is the non dimensional force and FnL2 cos EI The modulus of equations 5 and 6 varies from 0 to 1 As the moments M1and M3 M 1 and M 3 are balanced and two sides are constrained by a connected beam for the constant force fi xing mechanism the movement of the output end in middle part along Y axis is turned to zero Thus a boundary condition can be added cos 1 2 sin cos 1 cos 2 cos 2E 2 2E 1 F 2 F 1 7 As the negative stiffness beams are orthogonal to the fi xed structure one has sin 1 2 cos 2 8 Thus 1and 2 are the solutions to the fi rst and second modes which have the relationship as follows 1 2 9 2 1 2 10 The stiffness of the negative stiffness mechanism is calcu lated below kn 8 EIx L2 1dx cos 1sin 1 sin 1cos 1 11 a b Fig 3 Stiffness model of the 3 DOF stage with a actuation along X axis and b actuation along Z axis For the folded fl exure mechanism the reaction force Fp of the positive stiffness fl exure hinge can be expressed based on pseudo rigid body model Ff F aF b F bF c F aF c F a F b F c 12 where F irepresents the reaction force of positive stiffness fl exure hinges with folded beams along x axis which can be expressed by F i 2EI i l2 i 13 with I tot3 i 12 where to is the out of plane thickness of the layer and ti is the in plane width of the fl exure beams Based on the geometry relationship ican be derived as follows la 1 cos a lb 1 cos b lc 1 cos c 0 lasin a lbsin b lcsin c x 2710 TABLE I MAINPARAMETERS OF THEDESIGNEDSTAGE Parametersl1lalblc Dimension53 mm41 mm43 mm51 mm Parameterstb l2 Dimension1 0 mm4 0 mm3 1 55 5 mm a c Thus the stiffness equation of the folded fl exure mecha nism is described as Ff 2EI a bl2 c a cl2b b cl2a al2 bl2c bl2 al2c cl2al2b 14 The total stiffness equation can be obtained as kt 12Ebt3 l3dx 16 EIx L2 1dx cos 1sin 1 sin 1cos 1 2EI a bl2 c a cl2b b cl2a al2 bl2c bl2 al2c cl2al2b 15 With actuation along Z axis as shown in Fig 3 b the total stiffness k t can be expressed as k t 2kn 2kf 4kd 16 Similarity concerning the negative stiffness fl exure hinge the total negative stiffness in Z axis can be calculated by k t 16Ebt3 l3dx 16 EIx L2 1dx cos 1sin 1 sin 1cos 1 4EI a bl2 c a cl2b b cl2a al2 bl2c bl2 al2c cl2al2b 17 IV SIMULATIONSTUDY WITHFINITEELEMENT ANALYSIS In this section simulation study is conducted with fi nite element analysis to investigate the constant force property To verify the constant force property the MOGA is adopted Then the decoupling property of the parallel end effector is analyzed The main parameters of the 3 DOF CFM mecha nism are given in Table I A Investigation of Constant Force Property As the stage is symmetrical along X and Y axes the constant force properties in the two motion directions are the same Thus FEA simulation study is conducted for the CFM along X axis below Analytical model shows that the stiffness is very sensitive to the parameters and la So a parametric study is fi rstly conducted on to verify this phenomenon To perform the parametric study on a time function of displacement from 0 to 7 0 mm is adopted Besides a probe is located at the same time to detect the reaction force to get the force displacement relationship Then the stiffness curve for the stage about the parameter is obtained as is shown in Fig 4 For 2 5 3 5 and 4 5 the fl uctuation of the force is very large in these parameters and the constant force 00 511 522 533 544 555 566 57 Displacement mm 0 2 4 6 8 10 12 14 16 18 Force reaction N 2 5 3 5 4 5 3 1 MOGA result Fig 4 Force displacement results of parametric study on parameter 00 511 522 533 544 5 Displacement mm 0 0 5 1 1 5 2 2 5 3 3 5 Force reaction N Analytical result FEA result 1 49 N 1 53 N 1 57 N Constant force travel Constant force travel 1 53 N Fig 5 Analytical model and FEA simulation results of the constant force property property is not evident Thus to decrease the fl uctuation and to get a good constant force property optimization design is conducted based on simulation study by ANSYS Workbench software B MOGA Method The multi objective genetic algorithm MOGA method is chosen as it is one of the most effi cient way to optimize the multi objective problem The objective function for mini mization is assigned as the fl uctuation of constant forcevalue Two key parameters are determined based on static structural analyses One parameter is the maximum force F1in the travel of 0 to 2 mm Another one is the maximum force F2 in the travel from 0 to 3 mm The fl uctuation can be expressed as F1 F2 18 The objective is to minimize the fl uctuation The boundary conditions for the parameters and l1are shown as follows l2 30 0 5 i i 0 1 60 2 5 0 1 j j 0 1 20 19 2711 00 511 522 533 544 5 Displacement mm 2 1 0 1 2 3 Percentage of cross axis travel Y axis Z axis Fig 6 Simulation results of cross axis motion testing The constant force property of MOGA with 3 1 and l2 55 5 mm is shown in Fig 4 It can offer a constant force of about 1 51 N with the fl uctuation of 0 02 N i e 1 3 The constant force ranges from 1 5 to 3 0 mm For the conventional stage which does not have constant force property with an input force of 1 51 N the stage can only obtains the travel around 1 5 mm With the proposed active constant force stage it can obtain a travel range up to 3 0 mm This reveals that the active constant force property can eliminate the use of large force actuator to obtain a large travel As shown in Fig 5 the analytical model result is calcu lated using the optimized parameters given by MOGA As compared with the analytical result 1 55 N the FEA result of the constant force value is about 0 04 N smaller The deviation of the constant force value 2 6 which reveals the effectiveness of the developed model for the prediction of the constant force property C Motion Decoupling Testing To verify the motion decoupling property of the designed mechanism FEA simulation is conducted Specifi cally the motion decoupling testing is conducted by defi ning three probes Each probe is placed on the platform to detect the travel along X Y and Z axes individually The simulation results are shown in Fig 6 As the output platform translates along X axis the cross axis motion along Z axis ranges from 0 9 to 2 7 In the constant force travel from 1 5 to 3 0 mm the decoupling ratio ranges from 0 2 to 1 4 The maximum decoupling ratio is 1 6 between X and Y axes within the test travel of 4 5 mm The decoupling ratio varies from 1 2 to 0 6 in the constant force travel from 1 5 to 3 0 mm The FEA results reveal a good decoupling property be tween X and Y axes The decoupling ratio between Z and X axes is relatively larger while still gets good decoupling performance in the constant force travel The fi ne decoupling between X and Y axes is attributed to the symmetrical design of the stage Fig 7 Simulation results of distribution of equivalent stress Fig 8 Fabricated prototype of the stage D Stress Analysis Results Additionally stress analysis is conducted to test the stage performance using ABS plus material The stress distribution is shown in Fig 7 Within the travel of 4 5 mm the maximum stress experienced by the stage is 10 1 MPa While the allowable tensile stress of the adopted material P430 ABS plus for prototype fabrication is 37 MPa which is much larger than exerted stress Hence the stage obtains a safety factor of 3 7 which is high enough for normal operation As shown in Fig 8 the prototype of stage is fabricated by 3D printer for demonstration where one VCM is mounted to drive the stage to translate in X axis ABS plus is chosen as the material for fabricating the prototype with high elastic property The experimental study will be conducted to test the performance of the stage in the next step V CONCLUSION A novel 3 DOF active type parallel constant force stage is proposed in this paper The analytical model and simulation results reveal a