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Optimal design of a 1-rotational DOF flexure joint for a 3-DOF H-type stage Abstract A 3-DOF H-type stage using a flexure joint to accomplish rotational motion about the Z axis of a gantry stage is presented. To employ the rotational motion about the Z axis in an H-type stage, a 1-rotational DOF flexure joint is proposed. The proposed flexure joint in the H-type stage has high off-axis stiffness and adequate durability against high thrust force. A 6-DOF stiffness equation of the proposed flexure joint is obtained by analysis of leaf spring stiffness. To satisfy the required stage dynamic performance, the optimal design is executed on the geometric parameters of the proposed flexure joint using Sequential Quadratic Programing. The results of the optimal design are verified by experiment on the actual flexure joint. Keywords · H-type stage; · Flexure joint; · DOF analytical stiffness equation; · 6-Optimal design; · System mode analysis 1. Introduction Information technology has developed dramatically in the world since the invention of the PC. The development of information technology requires large and high resolution displays. The long range precision positioning stage is the essential system for implementing large and high resolution displays like LCDs, OLEDs and PDPs. The long range stage requires high thrust force to satisfy the high throughout. Consequently, to support the development of the display industry or semiconductor industry, the required performances are the high thrust, precise accuracy, and long range motion. The H-type stage has generally been used as the precision positioning system. The H-type stage has been developed to achieve larger range, more precision accuracy, and higher thrust force. In particular, accomplishing long range in the H-type stage conflicts with high accuracy due to manufacturing errors and assembly errors. A rotation error about the Z axis in the gantry stage, which is the guide for the X axis drive, is the dominant position error in the H-type stage. Recently, the monolithic flexure hinge has been used to guide a high precision motion system. There have been many efforts to compensate the Z rotational error. In et al. [1] developed a planar-type redundantly actuated parallel mechanism that rotated the moving platform about the Z axis. It had poor angular accuracy because it used the bearing at the revolute joint. Shinno et al. [2] proposed the VCM (Voice Moil Motor) stage with aerostatic levitation. It was used for positioning the nano-machining process. The Z rotational motion was arose from 8?VCM modules. It had 0.1?μrad accuracy. In this paper, we propose an H-type stage using flexure joints to compensate for the Z rotation error. The flexure guide mechanism has many advantages: negligible backlash and stick–slip friction; smooth and continuous displacement; adequate amplification for the output displacement of actuation; and inherently infinite resolution. Therefore, the flexure joint has been used for various applications, such as a micromanipulation system [3], an atomic force microscope (AFM) [4], and a dual actuation system for the flat panel display process [5]. The flexure joint, which is placed between the gantry stage and the tandem linear motors, makes the gantry stage rotate in the θz axis. The flexure hinge mechanism needs to have high stiffness except for the Z rotational DOF to guarantee the motion range required for the H-type stage. High off-axis stiffness in the flexure joint is important, because unwanted motions in the H-type stage appear by low stiffness components in the stage. There are many flexure joints for a 1-rotational DOF. The most basic flexure joint is the notch-type flexure joint [6]. A notch-type flexure joint has a low rotation angle due to the stress concentration around the central pivot point. A cross strip flexure joint, axial strip flexure [6], and cartwheel flexure [7] have been proposed to increase the motion of the flexure joint. These flexure joints are not enough for the off-axis stiffness, and are against a high thrust force. However, the proposed flexure joint, which is an over-constrained structure, can give high structural stiffness. There have been many efforts to derive stiffness modeling of the flexure joint. Paros and Weisbord’s model [8] was developed to calculate the spring rates of a single-axis flexure hinge mechanism. Ryu et al. [9] developed a stiffness modeling process for a whole flexure system using a compliance equation of the flexure joint in 6-DOF. This method can be used in a complex flexure system. However, this method is complex and it is difficult to find modeling error when calculation errors arise in the coordinate change and modeling errors emerge in the stiffness modeling of the flexure mechanism. Recently, a computer-based FEM method has been used to analyze the elasticity, natural frequency, and dynamic characteristics of a whole flexure mechanism automatically [10]. In this paper, we analyzed the stiffness of the flexure joint in 6-DOF using a simple stiffness calculation method. We present the optimal design of the proposed flexure joint to satisfy the desired specifications of the H-type stage. Using MATLAB’s Sequential Quadratic Programing (SQP), the size of the flexure joint is optimally designed for the required rotation motion about the Z axis and high off-axis stiffness. The stiffness equation of the proposed flexure joint is verified by the FEM results and experiment. 2. System configuration of 3-DOF H-type stage Fig. 1 shows the configuration of the proposed stage, which consists of the gantry stage, tandem Y axis motors, and slider. The proposed flexure joints are placed between the gantry stage and tandem Y axis motors to allow rotational motion of the gantry stage about the Z axis. The slider and tandem Y axis motors use a linear motor as an actuator with an air bearing guide, which is generally used in precision positioning systems. The guide mechanism uses the magnet preload mechanism to strengthen the stiffness of air bearing used in the slider and tandem Y axis motors. The sensors for position feedback in the X, Y1 and Y2 axes are an optical linear encoder with 1-nm resolution by 12-bit interpolation. Fig. 1.?3-D modeling of the H-type stage. The tandem Y axis motors are separated into master axis and slave axis as the reference for accuracy. The master axis (Y1) motor supports the reference of the Y axis motion to determine the accuracy and motion error in the Y axis. When the gantry stage rotates about the Z axis, the translational motion in the X axis is required to release a large rotation of the gantry stage. Fig. 2 shows the required freedoms in the H-type stage for Z rotational DOF. In the master axis (Y1), rotational joint implements θZ DOF between the gantry stage and the master axis motor. In the slave axis (Y2), the translational guide is equipped with a rotational flexure joint to allow X translation motion. There is no translational motion between the master axis (Y1) and the gantry stage. Fig. 2.?Schematic of the H-type stage for analysis of DOF. Fig. 2.?Schematic of the H-type stage for analysis of DOF. View thumbnail images The translational mechanism is implemented using a linear motion (LM) guide. The LM guide makes the gantry stage decouple from the slave axis motor (Y2). The LM guide prevents contact with the air bearing stage when the gantry stage is rotating. 3. 1R-DOF flexure joint for a 3-DOF H-type stage 3.1. Introduction to the proposed flexure joint The proposed flexure joint has 1R-DOF to implement rotational motion of the gantry stage about the Z axis. Fig. 3 shows the flexure joint which has a wheel shape. Even though the flexure joint is an over-constraint structure, the rotational motion of the gantry stage about the Z axis occurs with the elastic deformation of a leaf spring. Fig. 3.?3-D modeling of the flexure joint. The stiffness of the flexure joint influences the structure stiffness of the stage, which determines the whole dynamics of the H-type stage. It is important that the off-axis stiffness of the flexure joint is maximized to obtain high precision motion. Thus, the proposed flexure joint is adequate for implementation in the H-type stage due to the high off-axis stiffness. To satisfy the desired specifications of the H-type stage, 6-DOF stiffness modeling of the proposed flexure joint is required to perform an optimal design. 3.2. 6-DOF stiffness modeling of the proposed flexure joint The proposed flexure joint is symmetric in the X and Y axes, so the compliance matrix of the flexure joint has a diagonal matrix form as in Eq. (1), and the 6-DOF stiffness equations of the flexure joint reduce to four equations. (1) The proposed flexure joint is composed of the same eight leaf springs. The stiffness modeling of the flexure joint can be derived from the stiffness of each leaf spring. To derive the stiffness modeling, it is necessary to analyze the 6-DOF stiffness of one leaf spring. There have been many studies to determine precise stiffness modeling of a leaf spring, but it is hard to derive the stiffness modeling of the flexure joint whole motion range due to the complexity of finite element behavior. Smith [4] analyzed motion stiffness of a leaf spring in desired motion. Kang [7] derived a 6-DOF stiffness equation for a clamped leaf spring.Fig. 4 is a clamped leaf spring. The compliance matrix for a leaf spring is as [7]. (2) where b, l, and t are the height, length, and thickness of the leaf spring respectively in Fig. 3, E is Young’s modulus, G is the shear modulus, k2 is the modeling coefficient determined by b/t. 6-DOF stiffness equations of the flexure joint were derived by analyzing the deformation of all the leaf springs. In the next section, we present methods to determine 6-DOF stiffness equations for the flexure joint. Fig. 4.?Parameters and axis definition of a leaf spring [7]. View thumbnail images 3.3.1. Translational stiffness equation in the X axis For convenience, the flexure joint can be separated into?+?shaped leaf springs and an × shaped leaf spring as shown in Fig. 5. Eq. (3) represents the translational stiffness of a?+?shaped leaf spring and Eq. (4) represents the translational stiffness of an × shaped leaf spring. (3) (4) where E is Young’s modulus, G is the shear modulus, and φ is the angle between the leaf spring and the X axis. Using Eqs. (3)?and?(4), the translational stiffness in the X direction (dFx/dx) of the proposed flexure joint is shown in the following equation: (5) Fig. 5.?(a) a?+?shaped leaf springs and (b) an × shaped leaf springs. View thumbnail images 3.3.2. Rotational stiffness about the Z axis When the flexure joint rotates about the Z axis, the leaf springs encounter the axial force, normal force and moment about the Z axis as shown in Fig. 6. The sum of the total axial force in the flexure joint is zero due to the cancelation of all the force components, and the normal force is also zero. Therefore, the rotational stiffness about the Z axis is derived from the sum of the moment about the Z axis of all the leaf springs. Eq. (6) is the rotational stiffness of the flexure joint about the Z axis. (6) where ri is the radius of the inner body. Fig. 6.?Deformation of the flexure joint by Mz. View thumbnail images 3.3.3. Translational stiffness in the Z axis The Z translational stiffness of eight leaf springs makes the Z translational motion of the flexure joint. Thus, it can be modeled easily in the following equation: (7) 3.3.4. Rotational stiffness about the X axis The X rotational motion, which is out-of-plane motion has a complex deformation in leaf springs. Fig. 7 shows the deformation of each leaf spring. Fig. 7.?Deformation of flexure joint under Mx(a) Flexure joint under the moment Mx, (b) Free body diagram of 3, 7 leaf spring, (c) Free body diagram of 1, 5. leaf spring, (d) Free body diagram of 2, 4, 6, and 8 leaf spring. View thumbnail images The X rotational stiffness of the flexure joint is derived from three types of deformation of a leaf spring. The first deformation of the leaf spring is like 3, 7 leaf springs which undergo torsional moment about the X axis as shown in Fig. 7b. The second deformation is like 1, 5 leaf springs which are deformed by moment Myl1 and Myl2. The third deformation is like 2, 4, 6, 8 leaf springs, which are influenced by the previous two deformations equally. Eq. (8) indicates the rotational stiffness of the flexure joint about the X axis. where le is the effective length of the leaf spring. 3.3.5. Maximum stress The maximum stress occurs at the end position of the leaf spring when the flexure joint undergoes Z rotational deformation. Eq. (9) shows the maximum stress of the flexure joint. (9) where σmax is the maximum stress of the flexure joint and Kt is the stress concentration factor given by Peterson and co-workers [11]. 3.4. Verification of the flexure joint modeling To check the effectiveness of the stiffness equations of the flexure joint, it is verified by a FEM program named Pro Engineering/MECHANICA. Table 1 gives the results of the verification. The parameters of the flexure joint are ri?=?60?mm, l?=?40?mm, b?=?30?mm and t?=?2.5?mm. Table 1. Verification results of the stiffness modeling of the flexure joint. Unit Analytic model FEM simulation Error (%) kx N/μm 441.630 416.899 5.6 ky N/μm 441.630 416.899 5.6 kz N/μm 168.013 148.691 11.5 kθx Nm/μrad 0.69294 0.60492 12.7 kθy Nm/μrad 0.69294 0.60492 12.7 kθz Nm/μrad 0.02497 0.02282 8.6 σmax MPa 202.05 221.851 9.8 Full-size table The model of the flexure joint shows a reasonable prediction of the stiffness modeling with less than 13% errors. 4. Parametric analysis To create the optimal design, a parametric analysis on the flexure design is required to investigate how the design parameters of the flexure joint affect the 6-DOF stiffness and the maximum stress in the flexure joint. The results of the parametric analysis will ensure that the result of the optimal design is reasonable The design parameters of the flexure joint are as follows: Height of the leaf spring: b. Length of the leaf spring : l. Thickness of the leaf spring: t. Radius of the inner body: ri. We can derive the sensitivity analysis on the flexure joint with respect to variations of the design parameters. The parametric analysis results are shown in Fig. 8. Fig. 8.?Parametric analysis results of the flexure joint. View thumbnail images The length of the flexure joint(l) is the most sensitive design parameter in designing the flexure joint as shown in Fig. 8. The most sensitive property of the flexure joint is the rotational stiffness about the Z axis and the maximum stress as shown in Fig. 8. The design parameter b does not affect the maximum stress from Fig. 8b. The design parameter ri affects only the rotational stiffness about the Z axis and the maximum stress. 5. Optimal design of flexure joint The flexure joint is used to accomplish the yaw motion in the H-type stage. The rotational angle is important in the H-type stage to compensate for yaw errors. Thus, the flexure joint must have a low enough rotational stiffness about the Z axis. When driving the H-type stage, the settling time is determined by the stiffness of the system structure. The flexure joint in the H-type stage plays the role of a bottleneck in designing the H-type stage. To obtain a desired stiffness of the flexure joint that does not disturb the required specifications of the H-type stage, the optimal design is required to obtain enough rotational stiffness about the Z axis and the high off-axis stiffness. The design variables for the flexure joint are l, b, t, and ri. The influence of each design variable was discussed in the previous section. The cost function of the optimal design is determined to minimize the rotational stiffness about the Z axis and maximize the off-axis stiffness. Eq. (10) shows the cost function of the optimal design for the flexure joint. (10) where c1, c2, c3, and c4 are coefficients to make the design variables unity. The optimal design minimizes the cost function. The problem includes several constraints. For example, the maximum stress in theflexure joint should be less than the yield stress. There are also constraints for the system size and the off-axis stiffness. (11) where Sf1 is the safety factor of the maximum stress constraint, σyield is the yield strength of the flexure joint, and θzd is the desired Z rotation angle. Second, it is necessary that the rotation angle(θz) about the Z axis be larger than the desired specification, when the normal force of the linear motor is applied to the stage. θz should be as (12) where Sf2 is the safety factor for the rotational angle constraint and Mz is the drive moment about the Z axis by the thrust of the Y axis linear motor. Third, the flexure joint dominantly affects the dynamics of the H-type stage, because the flexure joint has the smallest stiffness in the H-type stage system. The settling time influenced by the system dynamics is an important specification in determining how fast the stage drives. Thus, to achieve the required settling time, the flexure joint must have enough stiffness in all degrees of freedom. When applying the X motion to the slider, the flexure joint deformed to the X translation and rotational motion about the Y axis. When applying the Y motion, deformation about the X rotational motion occurs in the flexure joint. The constraints for the settling time of the stage are as follows by the dynamic modeling of the H-type stage [12]. (14) where kx and kθx are the desired stiffness values for the X axis and the rotational X axis, respectively. There are the constraints for geometry of the flexure joint. The geometric constraints are listed as follows: (15) g5=ri+2l-C5?0 In summary, the described optimization problem can be rewritten as and The flexure joint is optimally designed using above optimal design work. Table 2 shows the constants for the optimal design of the flexure joint. Table 2. Constants for the optimal design of the flexure joint. Constant Unit Value σyield Mpa 503 E Gpa 68.95 θz Degree 0.3 kxd N/m 200 kθxd Nm/μrad 0.45 Kt – 15.7 Sf1 – 2.0 Sf2 – 2.0 Sf3 – 2.0 Sf4 – 2.0 Full-size table 6. Design results To accomplish the optimal design, we adopted SQP which uses a positive define quasi-Newton approximation of the Hessian of the Lagrange function implemented on the MATLAB program. This method generally guarantees the local minimum [13]. Fig. 9 shows the convergence profile of the cost function. The cost function value gradually converged to certain value. Fig. 10 shows that the final cost function values with various initial points converge to the same value. This proves that the optimal design result reaches sub minimum. Fig. 9.?Convergence profile of cost function. View thumbnail images Thus, the optimal design results with diffident initial points were checked. Table 3 shows the design variable sets for the optimal design. However, because of manufacturing cost effect of Electrical Discharge Machining (EDM), the design variables were chosen as in Table 4. Table 5 shows the calculated characteristics from modeling of the flexure joint. Table 3. Design variable sets. Design variables Start points Optimum results Unit S1 S2 S3 S4 Sopt l mm 25 30 40 50 42.10 b mm 20 25 30 40 40.00 t mm 1.5 1.8 2.0 2.5 1.92 ri mm 55 60 70 75 61.56 Full-size table Table 4. Final dimension of the design variables. Final dimension l (mm) b (mm) t (mm) ri (mm) Value 42.0 40.00 1.90 61.60 Full-size table Table 5. Simulated characteristics of the flexure joint. Start poi