Micro-mechanical stages with enhanced range

Transcription

1 Micro-mechanical stages with enhanced range International Journal of Advances in Engineering Sciences and Applied Mathematics ISSN Volume 2 Combined 1-2 Int J Adv Eng Sci Appl Math (2011) 2:35-43 DOI / s

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3 Int J Adv Eng Sci Appl Math (Jan-June 2010) 2(1 2):35 43 DOI /s IIT, Madras ORIGINAL RESEARCH Micro-mechanical stages with enhanced range M. Dinesh G. K. Ananthasuresh Published online: 14 January 2011 Ó Indian Institute of Technology Madras 2011 Abstract We present concepts and an optimization-based methodology for the design of micro-mechanical stages that have not only high precision but also an enhanced range. Joint-free distributed compliant designs provide high precision and easy manufacturability at macro and micro scales. The range of motion is enhanced by using displacementamplifying compliant mechanisms (DaCMs). The main issue addressed in this paper is how to retain the decoupling between the X and Y motions in the stage when it is equipped with DaCMs. The natural frequency of the stage is also not compromised in enhancing the range. The optimized design has 2.5 times more range than the designs reported in the literature. Furthermore, the sensitivity improved by a factor of two when the stage is optimized for an accelerometer. Keywords Micro-actuators Compliant mechanisms Optimization methods Inertial sensors 1 Introduction Mechanical stages ought to have decoupled movements in the three orthogonal directions: X, Y, and Z. The decoupled translating motion is easily achieved with sliding joints as M. Dinesh Indian Institute of Science, Bangalore, India Present Address: M. Dinesh General Electric, Bangalore, India G. K. Ananthasuresh (&) Mechanical Engineering Department, Indian Institute of Science, Bangalore , India suresh@mecheng.iisc.ernet.in can be seen in milling and drilling machines and microscopes. However, sliding joints suffer from backlash, friction, and wear. They also need lubrication and are not always suitable for operation in vacuum conditions. Therefore, it is common to use flexural joints in highprecision stages (see Fig. 1) and micromachined stages. Many commercial stages use flexure-based designs. The range of motion in flexures is limited by the stress that can be tolerated by the material they are made of. Stress in flexures is limited to a short and narrow necked region as can be seen in Fig. 1. This is called lumped compliance because the elastic flexibility is restricted to a small portion of the body. Distributed compliance, where elastic flexibility is spread over a large portion of the body, is an improvement over flexures [1]. See Fig. 2a, b for a comparison of lumped and distributed compliant designs. In this work, we consider distributed compliant designs for XY stages which have decoupled motion in the two directions in a plane. The focus of this work is to further enhance the range of motion by using Displacement-amplifying Compliant Mechanisms (DaCMs). Compliant mechanisms [2] use elastic deformation to transmit and transform motion, force, and energy. Amplification of the displacement in actuators (e.g. [3]) and sensors [4] is one of the applications of compliant mechanisms. DaCMs, which give substantially large displacement at the output point when force/displacement is applied at the input point, make this possible without introducing any complications in manufacturing at any scale and without using excessive space around the intended device. Some problems do arise when DaCMs are used in compliant XY stages. Before discussing those problems and how we resolve them, we begin with a brief review of some of the existing designs and provide the motivation for using DaCMs.

4 36 Int J Adv Eng Sci Appl Math (Jan-June 2010) 2(1 2):35 43 Fig. 1 An XYZ positioning stage based on Roberts linkage in which kinematic joints are replaced with flexural joints. A photograph taken from the website of PI (Physik Instrumente GmbH) physikinstrumente.com/en/products/nanopositioning/nanopositioning_ basics.php#anker2. Some of the flexural hinges are encircled Fig. 3 A compliant XY stage with decoupled motion. Redrawn from [5] Fig. 2 a A lumped compliant design, and b a distributed compliant design of an inverting compliant mechanism. Although both designs give similar motion, the stress is more uniformly distributed in the distributed compliant design. It is thus less prone to failure and gives longer range of motion than the lumped design Many compliant XY stages are reported in the literature and some are available commercially. Figures 3, 4, and 5 show some representative designs. The stages can also be used in micromachined inertial sensors such as two-axis accelerometers and gyroscopes. This variant is shown in Fig. 5. A detailed literature review of the stages can be found in [6, 8]. The basic principle of these designs is shown schematically in Fig. 6. Figure 6 shows that four sliding joints provide the necessary decoupling. When X-actuation is given at A, the stage moves only in the X direction as the slider C also moves. Sliders B and D do not undergo any relative movement. The same is true for Y actuation: sliders A and C have no relative motion while B and D move resulting in Fig. 4 A compliant XY stage with decoupled motion. Redrawn from [6] pure Y motion of the stage. The compliant designs of Figs. 3, 4, 5 follow this general principle by replacing the sliding joints with their compliant equivalents. The perpendicularity of the axes of the sliding joints is crucial for the stage to work. Similarly, the translational stiffnesses in the X and Y directions as well as the rotary stiffness about the Z axis should be appropriately designed in these stages. Symmetry is used in the designs of Figs. 3, 4, 5 to achieve the perpendicularity and parallelism among the axes of the compliant equivalents of the sliding joints. One drawback of these stages is that their range is just as much as or less than that of the actuators used to move them. Unfortunately, actuators have limited motion at the micro scale. The current work aims to remove this limitation by adding DaCMs to these stages.

5 Int J Adv Eng Sci Appl Math (Jan-June 2010) 2(1 2): Fig. 5 A compliant XY stage with decoupled motion used as an accelerometer. Redrawn from [7]. This is called a serial arrangement as opposed to the parallel arrangement of Figs. 3 and 4. Here, it is shown for a two-axis accelerometer application instead of an XY stage. The inertial force experience by the proof mass in either direction results in motion in that direction, which is sensed capacitively using the differential electrostatic comb fingers shown in Fig. 7. When a physical embodiment in the form of a DaCM is used for the displacement-amplifying sliding joint, a problem becomes apparent. In order to see this, we observe the design and its deformed profile in Fig. 8. It can be seen that the DaCM building block does not have the necessary high stiffness in the cross-axis (i.e., when we consider the X motion, the cross-axis is Y and vice versa) and the rotary direction about the out-of-plane Z axis. This problem does not get resolved even if we use symmetry and add two more arms to the stage on the right and bottom sides as is done in Fig. 4. If we design a DaCM that has zero stiffness and considerable amplification in the intended axis, and infinite stiffness and no amplification in the other axes, the arrangement shown in Fig. 7 would work. But it is not easy Complications arise when DaCMs are added to the compliant XY stages. These are as discussed in Sect. 2 along with our new designs that overcome them. A topology and shape optimization-based method for designing XY stages with enhanced range is presented in Sect. 3. Section 4 contains the details of an optimized XY stage and its prototype. Concluding remarks are in Sect. 5. Fig. 7 Schematic illustration of an XY stage with amplified motion from the actuator to the stage. The trapezium indicates a DaCM 2 Compliant XY stages with DaCMs 2.1 Problems associated with the addition of DaCMs Let us consider a conceptual arrangement in which the four sliding joints of Fig. 6 are replaced with joints that not only provide sliding but also amplification. This is symbolically Fig. 6 Schematic illustration of the decoupled motion of the XY stages Fig. 8 Imperfect cross-axis stiffness and rotational stiffness when a physical embodiment of a DaCM is substituted for the building blocks in the arrangement shown in Fig. 7. The DaCM corresponding to block C in Fig. 7 is rotating and translating down instead of simply moving to the right when the actuation is given in the X direction

6 38 Int J Adv Eng Sci Appl Math (Jan-June 2010) 2(1 2):35 43 even when zero stiffness and infinite stiffness are interpreted as very low stiffness and very high stiffness. In lieu of that we propose a novel arrangement that circumvents this problem even with a DaCM that has finite stiffness along all the axes. Such a DaCM is feasible in practice. The other problems that arise with the addition of DaCMs to an XY stage are that the natural frequency of the system changes and the overall in-plane size increases. In fact, the natural frequency may decrease because of increased mass and reduced stiffness (which is a consequence of amplified displacement). This will affect the bandwidth of the system and its dynamic behavior at high speeds. These are addressed in Sect. 3 by imposing constraints in the optimization problem. 2.2 Novel arrangement of DaCMs Decoupling of the two axes in compliant XY stages is contingent upon two criteria: stage isolation and actuator isolation [6]. Stage isolation requires that the stage does not under go any motion in the Y direction due to actuation in the X direction and vice versa. Similarly, actuator isolation requires that the point of actuation for the Y direction does not undergo any type of undesirable motion due to actuation in the X direction and vice versa. The arrangements of eight building blocks (i.e., the compliant equivalents of the sliding joints) shown in Figs. 3, 4, 5 have good stage and actuator isolations. As illustrated in Fig. 8 and mentioned in the earlier section, this is not possible with only eight DaCM building blocks. This is because the amplified motion of one building block should be transferred to the others that move in the same direction (see Fig. 7 where sliders A C and B D move in unison). Hence, we propose an arrangement shown in Fig. 9 with 12 building blocks each which is a DaCM [9]. This arrangement might occupy a large area around the stage, which is sometimes undesirable. Hence, we propose an alternative two-layer arrangement with six DaCMs in each layer [9]. The details of this arrangement become clear in Sects. 3 and 4. 3 Optimization-based design method 3.1 The spring-mass-lever (SML) modeling Because of the arrangement with identical building blocks, the design of the stage reduces to that of the design of the building block alone. However, the entire arrangement must be considered in the analysis for the determination of stage isolation, actuator isolation, the natural frequency, and the range. Finite element analysis of the entire stage becomes computationally prohibitive especially because it Fig. 9 A novel arrangement of 12 building blocks so that desired stage and actuator isolations are achieved while amplification of the actuator s displacement is achieved at the stage in both the directions is needed in each iteration of the optimization method used to design the stage. Hence, we use a lumped model for the building block to facilitate simplified, but accurate, analysis and optimization-based design. This model is referred to as a spring-mass-lever (SML) model [4]. An SML model of a typical DaCM is shown in Fig. 10. It has five parameters: input and output side stiffnesses (k ci and k co ) and inertias (m ci and m co ); and an inherent amplification ratio (n), which is the ratio of output and input displacements when there are no external attachments to the DaCM at the input and output points. These five parameters are the terminal (i.e., input and output) characteristics of the DaCM. They accurately capture the static and dynamic behavior of the DaCM in the linear elastic regime. When there are any attachments or loads at the input and output points of DaCM, they can be added as shown in Fig. 10. One of the attachments on the input side is an actuator, which will have its own stiffness and inertia. These are shown as k a and m a in Fig. 10. A stage usually has no stiffness of its own but will have significant inertia. But when the stage is used as an inertial sensor, it will have stiffness and inertia associated with the sensing side, which will be the output side. For the sake of generality, both stiffness and inertia are shown as k s and m s in Fig. 10. Using the SML model of each building block, the entire assembly of Fig. 9 can now be analyzed. The arrangement of Fig. 9 is such that only six DaCMs of either direction need to be considered when analyzing the motion in that direction. Hence, we show the SML model-based lumped representation of the assembly of Fig. 9 in Fig. 11 using

7 Int J Adv Eng Sci Appl Math (Jan-June 2010) 2(1 2): Fig. 11 A lumped model representation of the arrangement shown in Fig. 9 for DaCM building blocks that control the motion in one direction, either X or Y Fig. 10 a A physical embodiment of a DaCM, b the SML model. For the sake of generality, the stiffness, inertia, and the force are shown at the input (actuator side) and output (stage side) of the DaCM only six building blocks. This representation is amenable for analyzing the behavior of the stage in terms of the five SML model parameters. The static displacement at all points in the elastic system of Fig. 11 can be obtained by using the principle of minimum potential energy. Likewise, by considering the kinetic energy and using the Lagrangian method, the dynamic equation can also be written. By equating the derivatives of the potential energy PE with respect to the displacements involved; and then solving them, we get: PE =k ci x 2 in þ k 2þkco 2 co x stage nx in x stage nx 3 2þkci ð1þ +k co x stage nx 4 ðx 3 þ x 4 Þ 2 F in x in x stage ¼ 1 6 nf in k ci x in ¼ 1 F in ðn 2 k co þ 3k ci Þ 6 k ci ðn 2 k co þ k ci Þ ð2þ ð3þ NA ¼ x stage ¼ n ð n2 k co þ k ci Þ x in n 2 ð4þ ð k co þ 3k ci Þ The net amplification (NA) is of interest in the design of the stage. Note that not only the inherent amplification (n) but also the input and output side stiffnesses (k ci and k co ) influence NA. Without going into the details of the Lagrangian, which by the way is simple and routine for this elastic system, we give the matrix whose eigenvalues give good estimates of the natural frequencies of the stage. See [8] for details. B ¼ 2m 1 ci 0 2 ðk ci þ n 2 K co Þ 2 nk co 0 6m co þ M stage 2 nk co 6 K co 3.2 Topology optimization using beam ground structure ð5þ Now that we have shown how the entire assembly can be analyzed using the SML model of the DaCM, the design problem can be stated as follows: Maximize NA w Subject to X N i¼1 tl i w i V 0 Ustage U stage 0 x stage x stage 0 U cross Ucross 0 and Equilibrium Equations in the two load cases ð6þ We now explain all the terms in the above problem. First, the objective is to maximize NA. Its expression is given in Eq. 4. The maximization is done with respect to the design variables, w. These are the in-plane widths of the beams that form what is called a ground structure in the topology optimization literature [10]. A ground structure is an exhaustive set of non-overlapping beams arranged as a rectangular grid of given resolution. If the optimization algorithm takes the width of a beam to its lower limit a value that is numerically equal to zero it means that that beam is absent. There will also be an upper limit on the beam as dictated by the grid size, the aspect ratio limitation

8 40 Int J Adv Eng Sci Appl Math (Jan-June 2010) 2(1 2):35 43 in the beam theory, and the manufacturing technique used. The algorithm finds the optimum values of the in-plane widths as per the given upper limit on the volume, V *.This is incorporated in the first constraint in the problem where t is the uniform thickness of the structure and l i is the length of the ith beam. Some beams that reach the lower limit disappear from the ground structure while the others remain at different values below the upper limit. This determines the topology of the DaCM. The size of the design domain is specified such that the overall assembly of the stage is not excessive. The second and third constraints impose a lower limit on the range (U stage ) and the natural frequency (x) of the stage. This is to ensure that the algorithm gives at least that much range and frequency. In lieu of these constraints either or both of these characteristics may be too low to be useful. This may happen because we use gradient-based optimization (nonlinear programming) methods that are efficient in finding only local minima. These two constraints ensure satisfactory performance even when only a local minimum is reached. The fourth constraint addresses the cross-axis stiffness requirement. Here, U cross is the displacement of the stage in the intended direction when the actuation is provided in the other direction. Naturally, the upper limit U cross is a small value as we wish to have good cross-axis stiffness. Finally, we also need to add the static equilibrium conditions under the two actuations (intended and crossaxis). The solution of these equations gives the SML parameters which are needed to evaluate all other quantities. Since the objective function and all the constraints are differentiable in terms of the design variables, it is straightforward to compute the gradients. We use the method of moving asymptotes [11] to solve the optimization problem stated in Eq. 6. The result of the topology optimization for the DaCM building block is shown in Fig. 12a. As can be seen here, many beams in the ground structure have been removed by Fig. 12 a Optimized topology, b shape and size optimized design. The topology of a was interpreted as four segments in each symmetric half and then it was optimized for shape and size the algorithm. Although the result in Fig. 12a shows what type of structural form gives the intended behavior of the stage, it is cumbersome to manufacture as is. Hence, we interpret this optimal topology to simplify it and then perform shape and size optimization. 3.3 Size and shape optimization We observe four segments in the right and left symmetric halves of the topology given in Fig. 12a. The widths of these beams as well as the coordinates of their end points are made design variables in the shape and size optimization. This is called shape and size optimization because the orientations and locations of the beams change (i.e., the shape of the overall structure) in addition to the widths of the beams (i.e., sizes) [12]. The optimization problem is the same as that in Eq. 6 except the design variables. Here, unlike in the topology optimization problem, the lower limit is not zero but a value that is dictated by the manufacturing technique. Here, we had imposed 200 lm because the wire-cut electro discharge machining (EDM) had this limit in our machine. The resulting solution is shown in Fig. 12b. Another important point not mentioned thus far is that we assumed a compliant slider arrangement at the input side of the DaCM. This is to ensure sufficient cross-axis stiffness in the overall assembly. This, in conjunction with the fourth constraint in the optimization problem of Eq. 6, gives sufficient decoupling for the stage. The compliant slider gets appended as the actuator stiffness and inertia in the lumped representation involving the SML models. The optimized DaCM along with the compliant slider is shown in Fig. 13a for one layer of the assembly. This layer gives the motion in one direction while another layer rotated by 908 takes care of the other direction. The two layers are attached to each other at the stage. This was the alternative arrangement noted at the end of Sect It should also be noted that all the 12 DaCMs and their respective compliant sliders can be arranged in one layer if it is desired. Half of such an arrangement is shown in Fig. 14. But the alternate arrangement is compact and hence is more practical than the single-layer design. The optimized stage has an amplification factor of 4.9. This means that the range of the stage is nearly five times more than that of the actuator. The output side stiffness of the stage in the intended direction is 608 N/m whereas that in the cross-axis direction is 5,882 N/m. The natural frequency was 50 Hz. The frequency and cross-axis stiffness, because of the imposed constraints, were not compromised. The precision is also as high as the other designs if not more because of the distributed compliance of the new design. The new design with its two layers superposed on each other is shown in Fig. 13b. The force required to

9 Int J Adv Eng Sci Appl Math (Jan-June 2010) 2(1 2): Fig. 14 Left symmetric half of the optimized 12-building block design. Note the amplification of the displacement of the stage as compared with the actuator s displacement; it is nearly five times while its output is attached to the capacitive sensing side. Here, we used a deep reactive ion etcher (DRIE) as the basis for specifying the overall size, thickness, and the minimum feature size. The resulting design (without the compliant sliders) and its deformed profile are shown in Fig. 15a b. This design gives a sensitivity that is twice as good as the one without it. 4 prototyping Fig. 13 a One layer of the optimized stage with six building blocks, b two layers of the optimized stage actuate, actuator movements and the stage movements for a stage movement of 11.6 mm (5.8 mm away in both directions from the un-actuated position) are simulated and summarized in Table A design for the two-axis accelerometer By reversing the DaCM to amplify the movement of the stage at the sensing point, the same methodology described above can be used to design a two-axis accelerometer. Here, the input of the DaCM is attached on the stage side The stage design presented in Sect. 3 was intended for manufacturing using aluminium while the two-axis accelerometer is meant for silicon with DRIE. Here, since the focus is on micro-mechanical stages, we present only a prototype of the stage. A 5 mm thick sheet of aluminium (3003-H18) was cut to shape on a wire-cut EDM (Maxicut e). Two layers were made separately and then joined as shown in Fig. 16. It occupies a space of mm and has a minimum feature size of 200 lm. The range of this stage was limited by the compliant slider s beams: the range is as much as the gap between them so that they do not touch each other. This was nearly 10.5 mm. This, after amplification, corresponds to a specific range (equal to the range divided by the square root of the footprint area of the entire stage) of 4.2%. The highest reported value of specific range in the literature is 1.67% [6]. Thus, the new design has nearly 2.5 times more range than the best among the

10 42 Int J Adv Eng Sci Appl Math (Jan-June 2010) 2(1 2):35 43 Table 1 Performance summary of the final X Y stage conceived for actuation along the X-axis S. no. Actuation force (N) Actuator displacement (lm) Stage movement (lm) Amplification factor Fig. 16 Aluminium prototype made using wire-cut EDM similar stages. The prototype s performance was tested by moving the state manually with a graph sheet pasted underneath it. The measured amplification was close to 5, which is in agreement with the simulated value of Closure Fig. 15 a Optimized design for the two-axis accelerometer, b simulated displacement in the vertical direction where amplification by a factor of two can be observed: the proof-mass movement is half of the output (i.e., sensing) side. So, by measuring the amplified displacement instead of the movement of the proof-mass, the sensitivity of the accelerometer is doubled. The cross-axis sensitivity and the natural frequency are not compromised The design of commercial micro-mechanical stages is generally perceived as an art because their details are not known in the literature. Recent work [6] reveals that there is scope for systematic design of XY stages. In this work, we use topology, shape, and size optimization methods to design a new stage that has 2.5 times more range than the best design reported in the literature. More importantly, since the focus on our work is presenting a systematic design rather than a specific new design, it should be noted that better performance than the prototype design reported here is indeed possible. The main challenge addressed here is how to systematically achieve decoupled XY motion with displacement amplification from actuator to the stage

11 Int J Adv Eng Sci Appl Math (Jan-June 2010) 2(1 2): without compromising on the natural frequency, stage isolation, and actuator isolation. In the process of this work, we have shown the utility of spring-mass-lever model that circumvents the finite element analysis of the entire arrangement, which is computationally prohibitive. The same arrangement is suitable for two-axis inertial sensors such as accelerometers and gyroscopes. The example of the two-axis accelerometer indicates that the sensitivity is doubled in the new design. References 1. Yin, L., Ananthasuresh, G.K.: Design of distributed compliant mechanisms. Mech. Based Des. Struct. Mach. 31(2), (2003) 2. Howell, L.: Compliant Mechanisms. Wiley, New York (2001) 3. Canfield, S., Frecker, M.: Topology optimization of compliant mechanical amplifiers for piezo-electric actuators. Struct. Multidiscip. Optim. 20, (2000) 4. Krishnan, G., Ananthasuresh, G.K.: A systematic method for the objective evaluation and selection of compliant displacement amplifying mechanisms for sensor applications. J. Mech. Des. 130(10), : :9 (2008) 5. Alper, S.E., Azgin, K., Akin, T.: High-performance SOI-MEMS gyroscope with decoupled oscillation modes. In: MEMS 2006, Jan 2006, Istanbul, Turkey 6. Awtar, S.: Synthesis and analysis of parallel kinematic XY flexure mechanisms. Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge (2006) 7. Yazdi, N., Ayazi, F., Najafi, K.: Micromachined inertial accelerometers. Proc. IEEE 86, (1998) 8. Dinesh, M: Design of two-axis displacement-amplifying compliant mechanisms using topology optimization. Masters thesis, Mechanical Engineering, Indian Institute of Science, Bangalore (2008) 9. Ananthasuresh, G.K., Dinesh, M.: Compliant platforms to generate amplified displacements, compliant platform for sensing applied motion and method of designing DaCMs. Indian Patent Application No: 01136/CHE/ Bendsøe, M.P., Sigmund, O.: Topology Optimization Theory, Methods, and Applications. Springer-Verlag, New York (2003) 11. Svanberg, K.: The method of moving asymptotes a new method for structural optimization. Int. J. Numer. Method Eng. 24, (1987) 12. Hetrick, J., Kota, S.: An energy formulation for parametric size and shape optimization of compliant mechanisms. Trans. ASME J. Mech. Des. 121, (1999)