Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 52, 2, pp. 459-468, Warsaw 2014 ANALYTICAL MODELING OF DUAL ACTUATED COMPLIANT BEAM MICROGRIPPER SYSTEM Nayyer Abbas Zaidi Faculty of Electronic Engineering, Ghulam Ishaq Khan Institute of Engineering Sciences and Technology, Pakistan e-mail: nayyer@giki.edu.pk Shafaat A. Bazaz Department of Computer Science, Centre for Advance Studies in Engineering (CASE), Islamabad, Pakistan e-mail: bazaz@case.edu.pk This paper presents the analytical modeling for the design of a microgripper system that comprises dual jaw actuationmechanismwith real-time contact sensing. The interdigitated lateral comb-drive based electrostatic actuator is used tomove the gripper arms. Simultane- ous contact sensing is achieved through a transverse combbased capacitive sensor, to detect the contact between the jaws andmicroobject. The detailed analytical modeling of the mi- crogripper reveals that the stresses induced in the structure is well below the maximum yield stress of 7000MPa for single crystal silicon. The fabricated microgripper produced a displacement of 16µm at gripper jaws for the applied actuation voltage of 45V, which is approximately the same as predicted by the analytical model. Keywords: compliant structure, contact sensing, linear amplification, microgripper, micro- manipulation 1. Introduction Expansion inmicro and nanotechnologies has raised the need for the development of themicro- tools to handle small scale objects. MEMS based microgrippers are the major focus for many applications like microassembly, micromanuplation and microsurgery. Microgrippers based on the electrostatic actuation are ideal because of low power consumption, satisfactory amount of force, no hysteresis and ease in fabrication. Microgrippers based on the electrostatic actuation were presented in Volland et al. (2002), Beyeler et al. (2007). Both of these designs operate on high DC voltages. Moreover, in these designs only one gripper jaw is actuated while other is connected to the capacitive force sensor. The first jaw keeps dragging and pushing the object during its motion till it touches the second jaw and transfers the force to it. This single jaw movement operation increases the probability of damaging the micro-object. In this paper, an electrostatically actuated microgripper design, fabricated using standard micromachining process SOI-MUMPs is developed (Miller et al., 2004). This microgripper ope- rates at 0 to 45V actuation voltages. The lateral comb-drive actuation mechanism is used to produce movement in both jaws to enhance the gripping force. The microgripper is integrated with the transverse comb-drive which acts as a contact sensor. The purpose of this contact sensor is to avoid producing an excessive force at the jaws. This makes it possible to grasp mi- croobjects like biological cells automatically using a closed loop control which makes the whole micromanipulation process easy and time efficient. The rest of the paper is organized as follows: Section 2 explains the complete operation of the proposedmicrogripper design. The analytical modeling of microgripper parts which include the cantilever beam design with concentrated end loading, quad-clamped beam design with central distributed mass and gripper arm design are described in Section 3. The actuator and sensor 460 N.A. Zaidi, S.A. Bazaz design is presented in Section 4. Experimental characterization of the microgripper is carried out in Section 5. Finally Section 6, provides the conclusion of our work. 2. Theory of operation Theproposedmicrogripper, as shown inFig. 1, consists of three parts: a) actuator, b) sensor and c) gripper jaws. The design uses dual electrostatic actuation system, i.e. there are two separate electrostatic actuators for the simultaneous movement of two jaws. Each actuator consists of a set of stator combs interdigitated with a set of rotor combs. DC voltages applied to both the actuators simultaneously, which moves the central beam in upward direction. This movement of central beam is amplified at the gripper jaws due to the integrated action of the gripper arm and the vertical beam. A transverse comb-drive based capacitive contact sensor is included along each of the two central beams in between the gripper arm and actuator as shown inFig. 1. The change in the gap between the overlap length of the sensor combs results in the change in capacitance. This change in capacitance ismeasured through universal capacitance readout chip MS3110 that produces voltage proportional to the change in capacitance (Universal Capactive readout, 2004).When theobject is grippedbetween the gripper jaws, thennocapacitance change is detected by the universal capacitance readout chip. This indicates that the object has been grasped and thus, to avoid any damage to the object, no further actuation voltage is applied. Fig. 1. Complete microgripper design with the integrated capacitive contact sensor 3. Microgripper design Themicrogripper beam system is primarily composed of: a) cantilever beamswith concentrated end loading, b) quad-clamped beams with central distributedmass, c) gripper arm. Analytical modeling of dual actuated compliant beam microgripper system 461 3.1. Cantilever beam design The single layer cantilever beamwith concentrated end loading is shown inFig. 2a.Assuming that F is the loading force caused by the mass M attached to the free end of the beam, the resulting equation for the bendingmoment of the cantilever beam is (Bao, 2005) M(x) =−F(L−x) (3.1) The differential equation for the bending moment as a function of moment of inertia I and displacement function w(x) is expressed as (Bao, 2005) M(x) =−EIw ′′ (x) (3.2) After solving the above equation and applying boundary conditions w(0) = 0, w ′ (0) = 0 and w′′ (L) =0, we get displacement function as w(x) = 2F(3L−x)x2 Ebh3 (3.3) Fig. 2. (a) Single layer cantilever beamwith concentrated end loading, (b) stress in the guided cantilever beam at the top surface, (c) stress in the guided cantilever beam at the bottom surface Themaximum displacement will occur at the free end of the cantilever beam i.e., at x = L and is given as wmax = 4FL3 Ebh3 (3.4) Using Hook’s law, the restoring force produced by the beam is given as F = kwmax (3.5) The spring constant from the above equation is k = Ebh3 4L3 (3.6) 462 N.A. Zaidi, S.A. Bazaz wherewidth, thicknessand lengthof thebeamare b,hand L, respectively.Thestressesproduced in the cantilever beamat the top andbottom surfacewith (I = bh3/12) are given as (Bao, 2005) T(x) ∣ ∣ ∣ Top = 6F bh2 (L−x) T(x) ∣ ∣ ∣ Bottom = −6F bh2 (L−x) (3.7) Figures 2b and 2c show stress variations at the top and bottom surface with respect to the distance x from the fixed end of the beam as the force F is applied at the free end. The stress has itsmaximumvalue at the insertion point of the top of the cantilever beamwith the support and it decreases linearly to zero at the end where the mass is attached. The maximum values of stress calculated for the top surface of the horizontal and vertical beams are 424.8MPa and 708MPa, respectively. Since the bottom surface of the beam is com- pressedduringbending, so themaximumstress at thebottomsurface for both thehorizontal and vertical beam is zero and theminimum stress is −424MPa for horizontal beam and −708MPa for the vertical beam. These stress values, calculated for both the top and bottom surfaces of the beam, are much less than the yield strength of the single crystal silicon which is 7000MPa (Petersen, 1982). 3.2. Quad-clamped beam design The microgripper system is suspended on the quad-clamped beam with a central mass. A schematic diagram of the structure is shown inFig. 3. Due to the symmetry of the beam system, Fig. 3. Single layer quad-clamped beamwith central load (a) top view, (b) side view only one fourth of the structure is to be considered. The bending moment for the bottom left beam is expressed as (Bao, 2005) M(x) = Fox−mo− x ∫ 0 Mbg L (x−s) ds (3.8) where Fo is the supporting force at the four ends of the quad-clamped beam, mo is the reaction moment at the clamped end of the beam to balance the bendingmoment caused by the loading force F. The differential equation for bendingmoment as a function ofmoment of inertia I and displacement function w(x) is expressed as (Bao, 2005) M(x) =−EIw ′′ (x) (3.9) The displacement of the beam can be calculated by applying the boundary conditions w(0)= 0, w′(0) = 0, w′(lf) = 0, w ′′(lf/2) = 0 and assuming that the central mass is much wider and thicker than the beams (the bending of the mass can be negligible) the expression for this displacement is w(x) = F(3x2lf −2x 3) 48EI (3.10) Analytical modeling of dual actuated compliant beam microgripper system 463 It will be maximum at x = lf, therefore wmax = Fl3f 48EI (3.11) Using Hook’s law, the beam stiffness is derived as F = kwmax (3.12) The spring constant is given as kf = 4Ebh3 l3 f (3.13) The stresses due to expansion of the top surface and compression of the bottom surface of the beam and with I = bh3/12 are derived as T(x) ∣ ∣ ∣ Top = 3F 4bh2 (lf −2x) T(x) ∣ ∣ ∣ Bottom = −3F 4bh2 (lf −2x) (3.14) The distribution of the stress along the length of the beam is shown in Fig. 4. The stresses on the beam surface vary linearly fromapositivemaximumat one end to the negativemaximumat the other end. Themaximum andminimum stresses calculated for the top and bottom surfaces of the quad-clamped beam are 39.85MPa and −39.85MPa, respectively. These stress values are less than the yield strength of the single crystal silicon which is 7000MPa (Petersen, 1982). This ensures that the beamwill not undergo permanent deformation for the maximum applied force and will remain in the elastic range. The stress is zero at the midpoint x = lf/2 of the beam both at the top and at the bottom surfaces. Fig. 4. Stress plot of two beams at one side of the single layer quad-clamped beam on (a) top surface, (b) bottom surface 3.3. Gripper arm design Anewgripper armand jawwith the compliant structureas shown inFig. 5hasbeendesigned for graspingmicro sized objects. The design includes a horizontal and vertical beam to produce the elastic restoring force in the horizontal and vertical direction simultaneously. The vertical beam additionally provides support against the out of planemovement of the gripper arms. The 464 N.A. Zaidi, S.A. Bazaz structure is designed in such a way that it will maintain an angle of 90◦ between the gripper arm and the horizontal beam. The inner face of the jaw makes the angle ψ with the y-axis so that both jaws join parallel to each other upon the application of the force as shown in Fig. 6. Additionally, the jaw moves a small distance along the y-axis during grasping due to the direction of the applied force. This action ensures that the object completely comes between the jaws while grasping. Two stoppers have been placed near the point of application of the force in order to stop any extra movement of the central beam after the object has been fully grasped. In this design, the vertical displacement produced in the central beam is amplified by a constant factor. When a force is applied by central beam at point Q of the gripper arm, a moment is producedwhichmakes the rigidmass of the gripper arm to rotate about the point O. The horizontal and vertical beams act as the moment arms. Fig. 5. Gripper arm and beam system design (a) with no force (b) under applied force Fig. 6. (a) Right section of the gripper, (b) gripper armmovingmechanism In order to join both jaws completely and firmly on the point of application of the force at F, the jaw angle ψ is designed to be equal to the angle θ subtended at O when the force is applied at the point Q. Hence ψ = φ = θ (3.15) Analytical modeling of dual actuated compliant beam microgripper system 465 4. Gripper actuator and sensor design The electrostatic force required for the actuation is given as F = Nn 2 ε tV 2 d (4.1) where nN is the total number of comb-drives connected in parallel, t is the thickness, d is the gap between the comb-drives and V is the applied voltage. Thehorizontal and the quad-clampedbeamsare connected in parallel, and it is assumed that the force and displacement produced at the free end of the horizontal beam is approximately equal to the force and the displacement produced at the free end of the vertical beam. Thus the spring constant of the overall system is given as K = Et3 (4bf l3 f + bh 4l3 h + bv 4l3v ) (4.2) where bf is width of flexure connected to vertical beam, lf is length of that flexure, bh is width of the horizontal flexure and bv is width of the vertical flexure. Using Hook’s law, the resultant force on the vertical beam is given as F = Et3 (4bf l3 f + bh 4l3 h + bv 4l3v ) y (4.3) where y is the displacement of the central beam along the y-axis, and from Eq. (4.3) displace- ment expression is derived as y = Nn 2 εtV 2 d Et3 ( 4bf l3 f + bh 4l3 h + bv 4l3v ) (4.4) Considering the fact that tangents of ψ, φ and θ in Fig. 6b are equal, the displacement of the gripper jaws in the x-axis is derived as X = L lQ Nn 2 εtV 2 d Et3 ( 4bf l3 f + bh 4l3 h + bv 4l3v ) (4.5) where L/lQ is the amplification factor of the displacement at the tip of the gripper jaws, where L =2.1mmis the lengthof themicrogripper jawsand lQ =150µmis the lengthof thehorizontal flexure as shown in Fig. 6a. In the transverse comb-drive based differential sensor, the capacitance on one side of the comb-drive increases and decreases by same proportion on the opposite side of the comb-drive. These capacitances are given as Cs1 = Nn ε(tl) d0+ L LQ X +Cfringe Cs2 = Nn ε(tl) d0− L LQ X +Cfringe (4.6) where d0 is the initial gap between the transverse combs, Cs1 is the decreased capacitance and Cs2 is the increased capacitance in the transverse comb sensor corresponding to the gap change y =(L/LQ)X and Cfringe is the capacitance produced due to the fringe fields.When the object 466 N.A. Zaidi, S.A. Bazaz is gripped between the gripper jaws, no change in capacitance is detected by the electronic control circuitry. This indicates that the object has been grasped with avoiding any damage to the object due to excessive force applied through the actuators. The total capacitance change between differential transverse comb-drive fingers will be ∆C = Cs2−Cs1 (4.7) The force at the tip of the jaw can be calculated in terms of the change in capacitance of the sen- sor by equating the total force on the electrostatic actuator fingers to the change in capacitance on the sensors fingers. During the calculation it is assumed that the central beam movement is much less than the initial gap spacing between the sensor combs fingers, i.e. y2/d20 ≪ 1, the relationship is given as F = Ed20t 3 ( 4bf l3 f + bh 4l3 h + bv 4l3v ) ∆C 2Nnε(tl) (4.8) where E is themodulus of elasticity of thematerial, d0 is the initial gap between sensor fingers, t is thickness of the device, ∆C is change in the capacitance, N is the number of comb-drives and n is the number of capacitors. UsingEq. (4.5) and (4.8), the analytical relationship between the voltage, single jaw displacement and the force at the jaw is shown in Fig. 7. It reveals that a total of 7.42µm displacement and force of 215.14µN is obtained at the operating voltage of 45V. Fig. 7. Voltage vs. jaw displacement vs. force at the tip of the jaw 5. Design implementation and experimental results The proposed design of the microgripper with a structural layer of 25µm thick single crystal silicon is fabricated using the standardMEMSprocess, SOIMUMPs, offered by theMEMSCAP Inc. (Miller et al., 2004). Figure 8 shows the Scanning Electron Microscope (SEM) image of the fabricated microgripper. The microgripper actuator was tested at different voltages in the range of 0 to 45V and the displacement of gripper jaws was obtained. At the actuation voltage of 45V, a total displacement of 16µm was achieved at both the jaws as shown in Fig. 9. It reveals that each jaw moved by a displacement of 8µm that is almost the same as predicted by Eq. (4.5). Thus, the microgripper could accurately hold an object of size between 54µm to 70µm with the actuation voltage in the range of 0 to 45V at the comb-drive based actuator. Analytical modeling of dual actuated compliant beam microgripper system 467 Fig. 8. SEM image of the fabricatedmicrogripper Fig. 9. Displacement at the microgripper jaws at actuation voltages of (a) 0V, (b) 35V, (c) 45V 6. Conclusion Design, analytical modeling, fabrication and testing of an electrostatically actuated microgrip- per is presented in the paper. The microgripper is fabricated using the standard SOI-MUMPs technology with a device layer of single crystal silicon having thickness of 25µm. Lateral comb- drive based dual electrostatic actuation is used to produce displacement in both gripper arms simultaneously. An integrated cantilever and quad-clamped beam based compliant structure produces four times linear amplification of the displacement at the gripper jaws. The experi- ments demonstrate that the gripper jaws displace by 16µm under the applied voltage of 45V. The jaw design ensures full and firm grasping of micro-objects of size between 54µm to 70µm with the actuation voltage in the range of 0 to 45V. Acknowledgment This researchworkwas supportedbyHigherEducationCommission(HEC)ofPakistanunder research grant No. 1012. References 1. Bao M., 2005,Analysis and Design Principles of MEMS Devices, 1st edition Elsevier 2. Beyeler F., Neild A., Oberti S., Bell D.J., Sun Y., Dual J., Nelson B.J., 2007,Mono- lithically fabricated microgripper with integrated force sensor for manipulating microobjects and biological cells aligned in an ultrasonic field, Journal of Microelectromechanical Systems, 16, 1, 7-15 3. Miller K., Cowen A., Hames G., Hardy B., 2004, SOIMUMPs Design Handbook Version 4, Memscapinc, available at: http://www.memscap.com/products/mumps/soimumps/reference- material 468 N.A. Zaidi, S.A. Bazaz 4. Petersen K.E., 1982, Silicon as a mechanical material,Proceedings of the IEEE, 70, 5, 420-457 5. Universal Capactive Readout, 2004, Universal Capactive ReadoutTM IC (MS3110)Manual Irvine Sensor Coropration, available at: http://www.irvine sensors.com 6. Volland B.E., Heerlein H., Rangelow I.W., 2002, Electrostatically driven microgripper, Journal of Microelectronic Engineering, 61, 1015-1023 Manuscript received February 26, 2013; accepted for print November 6, 2013