Kinematic design of large displacement precision XY positioning stage by using cross strip flexure joints and over-constrained mechanism (2).doc

article in pressavailable online at sciencedirectmechanism and machine theory xxx (2007) xxxxxxmechanismandmachine t/locate/mechmtkinematic design of large displacement precision xypositioning stage by using cross strip flexure joints andover-constrained mechanismyeong-jun choi *, s.v. sreenivasan, byung jin choidepartment of mechanical engineering, university of texas at austin, austin, tx 78712, usa received 14 february 2005; accepted 22 may 2007abstractflexures are widely used in precision machines since they offer frictionless, particle-free, and low maintenance operation, and they provide extremely high resolution. a large displacement precision xy positioning stage is designed by using cross strip flexure joints. an over-constrained mechanism is used to incorporate symmetry to cancel out the effects of center shifting in large-motion flexures. advanced kinematic techniques such as screw system theory are used to achieve a good kinematic design. existing flexure-based translation stages usually have motion range to size ratios of less than 0.01 as compared to 0.25 or higher in this research. it is believed that large-motion flexure-based xy stages can be a cost-effective solution for semiconductor applications, particularly the ones that operate in vacuum. 2007 elsevier ltd. all rights reserved.keywords: flexure; motion stage; screw system theory; over-constrained mechanism1. introductionas an effort to eliminate undesirable characteristics such as friction and backlash in traditional joints, a flexure joint was proposed in the 1960s 6. flexure joints do not have stickslip friction or backlash. additional advantages of flexure joints are that they are wear-free and can be made as a monolithic element. if the forcedisplacement curves are known, then the displacements that are continuous at all ranges can be calculated from the external force. however, these flexures have disadvantages such as the limited range of motion.some researchers have developed high precision positioning systems using flexure joints. rong and zhu 8 designed and analyzed a flexure-hinge mechanism that has a motion range of 100 lm and a positioning accuracy of 0.1 lm. single-axis flexures and piezoelectric actuators were used in the motion stage. yang et al. 13corresponding author. present address: molecular imprints, inc., 1807 west braker ln., c-100, austin, tx 78758, usa. tel.: +1 512 339 7760; fax: +1 512 339 3799.e-mail address: (y.-j. choi).0094-114x/s - see front matter 2007 elsevier ltd. all rights reserved. doi:10.1016/j.mechmachtheory.2007.05.009please cite this article in press as: y.-j. choi et al., kinematic design of large displacement precision xy ., mech. mach. theory (2007), doi:10.1016/j.mechmachtheory.2007.05.009article in press2y.-j. choi et al. i mechanism and machine theory xxx (2007) xxx-xxxdeveloped a micro-positioning stage that was actuated by piezo actuator and guided by structure based on flexure joints. the motion range of that stage was only 200 (im. the comparison of static and dynamic characteristics between analytical model and fem model was completed in that research. ryu et al. 9 developed a flexure hinge based xy9 stage which has the total range of 41.5 x 47.8 (im. they presented an optimal design method. tajbakhsh et al. 11 used flexures to make a three d.o.f. optic mount with the motion limit of 100 (im. commercially physik instrumente 7 are selling p-731 series xy piezo flexure nano positioners which can travel in ranges of 100 x 100 (im. it uses low voltage pzts (0-100 v) and flexures are used as the drive and guiding system. integrated capacitive position feedback sensors provide sub-nanometer resolution. the flexures provide zero sticktion/friction, ultra-high resolution and exceptional guiding precision. all positioning stages developed so far can give a few hundreds of micrometer motion range, because notch type flexure joints are adequate for small motion range.in order to use flexure joints in large-motion range, some researchers made dual servomechanism that has a fine motion stage mounted on a coarse motion stage. they incorporated flexure joints in fine motion stages. lee and kim 3 presented an ultra precision three d.o.f. stage for alignment of wafers in micro lithography. for high precision, they adopted a dual servo system and used flexures and piezoelectric actuators in the fine motion stage. the working range was 200 x 200 mm. lee et al. 4 developed an ultra precision positioning system using a dual servomechanism that consists of the global stage and the micro stage. the global stage can travel 40 cm and include a ball screw that has the position accuracy of 5 (im. piezoelectric actuator actuates the micro stage connected by flexures. dual servo stages make the whole stage complex, and errors associated with coarse motion stage degrade the performance of the whole stage.fig. 1 depicts a crossed strip type flexure joint that provides large rotation. the flexural pivot made by lucas aerospace 5 is a commercially available large deformation revolute flexure joint. this flexure does not have friction or backlash, and provides a large rotation of 60. the kinematic and dynamic characteristics of crossed strip type flexure joint are described in 1,12 well. these flexure joints do not provide exact rotary motion over the entire 60 motion range. they can however, used in conjunction with symmetric kinematic designs, yield exact linear motion stages.since linear motors are completely non-contact devices, there is no friction, no cogging, and no parts to wear. as a linear-motor-based system can provide high speeds and accelerations, linear motors are becoming the best actuators for ultra-high precision applications.fig. 1. crossed strip type flexure joint.please cite this article in press as: y.-j. choi et al., kinematic design of large displacement precision xy ., mech. mach. theory (2007), doi:10.1016/j.mechmachtheory.2007.05.009a large displacement flexure-based precision xy stage for vacuum-based semiconductor equipment is developed in this research. the weight support mechanism of this motion stage is made of links and flexure joints, and a linear motor is used as the actuator. until now, no researches have been done on positioning system that can move large displacement only with flexure joints without using dual servo stages. this research is, to our knowledge, the first work for developing a macro motion stage that can support the weight of the stage and guide the motion by a mechanism based purely on flexure joints.article in pressy.-j. choi et al. i mechanism and machine theory xxx (2007) xxx-xxx32. design conceptfig. 2 shows a double compound notch type small motion rectilinear spring that is used as the basic configuration for the new xy stage design. since semi-circular notch type flexures are ideal only for small motion range, large-motion flexural pivots such as the one in fig. 1 are used here. the flexural pivot has many advantages such as no rolling or coulomb friction, no backlash, no lubrication, and applicability in vacuum. however, center shift may introduce inaccuracies in the positioning of a mechanical system using flexural joints. for the case of complex loading, since the center shift is a function of the deflection angle and the proposed motion stage has a symmetrical structure, it is assumed that the center shift does not make any significant error in the direction perpendicular to the moving direction of the motion stage. an over-constrained mechanism can make no benefit when there is sticktion or friction. since purely compliant system is employed in the current design, there is no sticktion or friction. temperature variation can cause thermal expansions, but temperature controlled environment such as within 0.01 c is available in semiconductor industry. however, there is a lack of analysis methods to understand and predict such over-constrained mechanisms performances.the moving body in fig. 2 has one degree-of-freedom in its nominal configuration and has been used for small motion applications 10. the nominal configuration as shown in fig. 2 is defined as the configuration with minimum strain energy. however, a mobility analysis based on screw system theory 2 shows that the moving body has two degrees-of-freedom in its off-nominal configurations (see fig. 3). this mobility analysisfig. 2. double compound notch type small motion rectilinear spring.i3tfixed bodyyxmoving bodyfig. 3. two degrees-of-freedom motion.please cite this article in press as: y.-j. choi et al., kinematic design of large displacement precision xy ., mech. mach. theory (2007), doi:10.1016/j.mechmachtheory.2007.05.009article in press4y.-j. choi et al. i mechanism and machine theory xxx (2007) xxx-xxxis described in section 3. therefore, the undesirable degree-of-freedom must be eliminated for a large-motion application.3. mobility analysisthe mobility of a double compound notch type rectilinear spring shown in fig. 2 is expressed asm = 3( - 1) - 2y= 3(12 - 1)-2(16) = 1(1)where mis the mobility, n is the number of links, and/is the number of joints. this is only applicable when the stage is in its nominal configuration.the actual mobility of the mechanism in its off-nominal position can be found by using screw system theory. more detailed descriptions related to screw system theory can be found in 2. a brief introduction of screw theory is included in this paragraph. the advantage one can achieve by using screw theory in the field of robotics has been repeatedly emphasized. it has been known to provide geometric insight into the kinematics and static force analyses and syntheses of spatial mechanisms. an instantaneous screw axis $ can be represented by a vector pair, or motor: $=(o)t;t)t, where co represents three angular velocity components and fi represents three linear velocity components of the point on the rigid body instantaneously coincident with the origin of the reference frame.fig. 4 shows the schematic for the mobility analysis. points a, b, c, and d are located in the center of link 2 or the edge of the moving body.the motor of point a of link 2, va, can be obtained simultaneously starting at joint 1 or joint 3. the resultant motors from the two starting points should be same.the screw of point a by joint 1, $1, can be represented as$1p1 x w1(2)wherew1p1 x w1 =ij khx + h hy 000 1hy=(l1x + 12)0(3)where l1 is the length of link 1 and also equal to half of the length of link 2 (l2), l1x = l1cosh, l1y = l1sinh, q1 is the position vector of point a relative to the joint 1, and w1 is the screw axis direction by joint 1.link 1 joint 1fixed bodylink 2 a joi nt 4joint 2a i joint 4jtj bt 1?m? joint 3moving bodyjoint 5joint 6cjoint 8fig. 4. schematic for mobility analysis.please cite this article in press as: y.-j. choi et al., kinematic design of large displacement precision xy ., mech. mach. theory (2007), doi:10.1016/j.mechmachtheory.2007.05.009article in pressy.-j. choi et al. / mechanism and machine theory xxx (2007) xxxxxx5the screw of point a by joint 2, $2, can be expressed as$2w2 p2 x w2(4)wherew2p2 x w2 =ijk/20000100(5)where q2 is the position vector of point a relative to joint 2, and w2 is the screw axis direction by joint 2. the resultant screw axes $1 and $2 can be found as 0 01my-i 1x + h) 000(6)$1$20 1 0-h 0 )the motor of point a of link 2, va, starting at joint 1 can be written as(7)va = tt1$1 + tt2$2where x1 and x2 are the scalar angular velocities of the screw axes $1 and $2, respectively.the motor of point a of link 2, va, starting at joint 3 can be calculated by using a similar procedure. the screw of point a by joint 3, $3, can be represented as$3w3p3 x w3(8)wherew3f3 x w3 =i j k 1yhx - h hy 0 2 1x0 0 1 0(9)where q3 is the position vector of point a relative to joint 3, and w3 is the screw axis direction by joint 3. the screw of point a by joint 4, $4, can be expressed as$4w4 p4 x w4(10)wherew4p4 x w4 =ij k 0h00= h00 10(11)where q4 is the position vector of point a relative to joint 4, and w4 is the screw axis direction by joint 4. the resultant screw axes $3 and $4 can be found as$3/ 0 01hy12 hxv 0 /$4 0 0 1 0h 0(12)please cite this article in press as: y.-j. choi et al., kinematic design of large displacement precision xy ., mech. mach. theory (2007), doi:10.1016/j.mechmachtheory.2007.05.009article in press6y.-j. choi et al. i mechanism and machine theory xxx (2007) xxx-xxxthe motor of point a of link 2, va, starting at joint 3 can be represented asva = tt3$3 + tt4$4where oo3 and oo4 are the scalar angular velocities of the screw axes $3 and $4, respectively. since eqs. (7) and (13) should be equal, the following relation can be obtained.(13)0 0oo3 + oo4c03hy003(12 hx) + w4/2 000(14)o01 + oo2o01hy-o01(l1x + i2) oo2i2 0c a0eq. (14) can be rewritten as(15)001+ 002 = oo3 + oo4oo1l1y = oo3l1y a1(/1x + h) 002i2 = 003(12 hx) + 004i2 therefore,002= 004(16)(17) (18)(19)001= oo3002= o01the motor of link 2, va, can be represented asva = (0 0 0 oo1l1y (q1l1x 0) similarly, the motor of point b of the moving body, vb, relative to the link 2 can be expressed as0)tv(0 0 0 -oo2l1y oo2l1xwhere oo2 is the scalar angular velocity of the screw axis of point b by joint 2. the motor of point b of the moving body, vb, is the sum of va and vb.ky -oo1hx - o02l1x 0fvb = (0 0 0 oo1l1y oo2l1 similarly, the motor of point d of the moving body, vd, starting at joints 5 and 7 can be expressed asvd = (0 0 0 oo5l1y ooj1y oo5l1x oo6l1x 0)t(20)where oo5 and oo6 are the scalar angular velocities of the corresponding screw axes. since eqs. (19) and (20) should be equal, the following relations can be derived as0)5o01oo1 oo2 = 005 + oo6(21)since the number of equations is two and the number of arbitrary variables is four, the degree-of-freedom of this system is two. as shown in fig. 3, the stage can move in the x and y directions, which interpret two de-grees-of-freedom motion physically.4. calculation of design parametersthe side links shown in fig. 5 eliminate the undesirable degree-of-freedom, which makes this stage one degree-of-freedom. however, when two stages are stacked orthogonal to each other to result in the xy stage, the actuation of one stage causes undesirable orthogonal excitations to the moving body. it is necessary to useplease cite this article in press as: y.-j. choi et al., kinematic design of large displacement precision xy ., mech. mach. theory (2007), doi:10.1016/j.mechmachtheory.2007.05.009article in pressy.-j. choi et al. / mechanism and machine theory xxx (2007) xxxxxx7additional linkages to eliminate the undesirable motion and constrain the moving body along the linear motion direction. therefore, side links are installed at both sides of the moving body.fig. 5. modified one degree-of-freedom stage.23fig. 6. linkage schematic of stage.fig. 7. proposed xy stage including side links.please cite this article in press as: y.-j. choi et al., kinematic design of large displacement precision xy ., mech. mach. theory (2007), doi:10.1016/j.mechmachtheory.2007.05.009article in press8y.-j. choi et al. / mechanism and machine theory xxx (2007) xxxxxxassuming that constraining linkages exist, the mechanism in fig. 5 can be optimized to lead to the smallest footprint for a 300 mm motion range. fig. 6 shows a schematic of the basic linkage in fig. 5. links 1 and 3 are of the same length. when the moving body moves along x dir