Microgripper Based on Simple Compliance Conﬁgurations, Improved by Using Parameterization

: The design of a novel electrothermal microgripper device is shown, which is based on an improved chevron type actuator developed considering their elements parameterization, whose resistive model is also provided. The performance of the microgripper’s parameters, such as displacement, force, and temperature distribution, with convection for the voltage range from 0 up to 5 V, is evaluated through numerical and analytical simulation. Microgripper design was also improved with aid of parameterization. The e ﬀ ect on the microgripper performance due to its thickness is also analyzed, ﬁnding a considerable increment in force, when thickness increases. Its main advantage is given by the simplicity of the compliance arrangement of the microgrippers jaws. Considering convection, when 5 V are applied, 37.72 ◦ C was generated at the jaw’s tips of the Improved Microgripper 2 (IMG2), implemented with silicon, this relatively low temperature increases its capabilities of application. When the IMG2 is implemented with polysilicon, its response is competitive comparing with a more complex microgripper, increase of displacement (50%) is shown, but a decrement of force (30%). The diameters allowed for the subjection objects are found between 84.64 µ m and 108 µ m, with weights lower than 612.2 µ g. Some tests of subjection were performed using microcylinders of Au, glass ceramic, polycarbonate and carbon ﬁber, showing a permissible stress on them, considering its Young’s modulus, as well as the total reaction force induced. All simulations were done on Ansys software. The results demonstrate the feasibility of the future microgripper fabrication.


Introduction
Microelectromechanical systems (MEMS) involve both electronic and non-electronic elements, and perform functions that can include signal acquisition, signal processing, actuation, display and where the intermediate stage, which is called Detailed Design and Optimization [17], is directly related to parametric analysis. 26-28 mA [21] Microgripper based on chevron actuator 71.5 N/A~3500 × 3120 × 45 Copper-SU-8-Copper 195 mV [22] In this paper, determinant elements of chevron actuator have been parameterized, such as its inclination angle, and the width of its shuttle, considering their effects on the displacement and the reaction force on the shuttle, for each case. In addition, the parameterization of the complete microgripper will be carried out considering the effect of the change in the values of the angles of the compliance arrangements that are decisive in the operation, as well as the length of the arms effect.
Our aim is the design of a novel and simple microgripper based on a chevron actuator, and a compliance arrangement. The content of this work is organized as follows. In Section 2, design of the elements of the microgripper are provided, as well as the conventional mathematical model of chevron actuator and the basis of the microgripper. Simulation results of the main parameters of the microgripper, a comparison with another microgripper, and some objects holding tests, are provided in Section 3. Finally, in Section 4, some concluding remarks are given.

Materials and Methods
Silicon is used as structural material for the microgripper design, its parameters are shown in Table 2. Polysilicon was also chosen for generalizing the obtained results and compare the performance of microgrippers, its parameters are also given in Table 2. Table 2. Physical and mechanical properties of silicon [23][24][25] and polysilicon [26]. The schematic diagram of the proposed electrothermic chevron, which actuates the microgripper is depicted in Figure 1. Its performance will be analyzed, with the aid of an improvement process based on parametric analysis, described in the flux diagram shown in Figure 2, to take decisions about any change on the geometry. In this established process, at first, the performance parameters to improve, must be determined, as they will guide to the user's decisions. The knowledge of the possible device applications will be very useful in this determination. As second step, it is necessary to identify the determinant geometric elements on the initial device performance, because, individually, all of them will be parameterized. The graphical deployment of their results (step 3) will be useful to user, for the determination of the most suitable dimensions of the determinant elements of the geometry (step 4). Some design rules of the fabrication process, previous knowledge and expertise should be also considered in the decision making. The enhanced geometry will be determined with this selection of elements dimensions (step 5). Later, it is necessary to simulate the improved device in order to analyze its performance parameters (step 6). If the appropriated parameter values have been obtained (step 7), the process will be finished, or it will be necessary to go back to step 4 and continue again since this point, until the chosen performance parameters would have suitable values.

Parameters
Actuators 2020, 9, × FOR PEER REVIEW 4 of 22 of the fabrication process, previous knowledge and expertise should be also considered in the decision making. The enhanced geometry will be determined with this selection of elements dimensions (step 5). Later, it is necessary to simulate the improved device in order to analyze its performance parameters (step 6). If the appropriated parameter values have been obtained (step 7), the process will be finished, or it will be necessary to go back to step 4 and continue again since this point, until the chosen performance parameters would have suitable values. Table 2. Physical and mechanical properties of silicon [23][24][25] and polysilicon [26].    Actuators 2020, 9, × FOR PEER REVIEW 4 of 22 of the fabrication process, previous knowledge and expertise should be also considered in the decision making. The enhanced geometry will be determined with this selection of elements dimensions (step 5). Later, it is necessary to simulate the improved device in order to analyze its performance parameters (step 6). If the appropriated parameter values have been obtained (step 7), the process will be finished, or it will be necessary to go back to step 4 and continue again since this point, until the chosen performance parameters would have suitable values. Table 2. Physical and mechanical properties of silicon [23][24][25] and polysilicon [26].    As it can be observed, automated sweeps, quickly performed, generating useful information to analyze performance trends and, therefore, to select the most appropriate values of the variables under analysis, in accordance with the established requirements and conditions. Parameterization was performed without consider convection. Dimensions of the proposed chevron are given in Table 3. The properly microgripper section of the complete device was initially designed considering two structures based on simple compliance arrangements. Jaws tips were designed to favor grip, using a toothed shape. The schematic diagram of the jaws' section is shown in Figure 3, and its dimensions are given in Table 4. Thickness of the microgripper is 70 µm. Later, this microgripper would be improved, considering the determinant geometry elements on its displacement and reaction force.

Parameters
Actuators 2020, 9, × FOR PEER REVIEW 5 of 22 As it can be observed, automated sweeps, quickly performed, generating useful information to analyze performance trends and, therefore, to select the most appropriate values of the variables under analysis, in accordance with the established requirements and conditions. Parameterization was performed without consider convection. Dimensions of the proposed chevron are given in Table 3. The properly microgripper section of the complete device was initially designed considering two structures based on simple compliance arrangements. Jaws tips were designed to favor grip, using a toothed shape. The schematic diagram of the jaws´ section is shown in Figure 3, and its dimensions are given in Table 4. Thickness of the microgripper is 70 µm. Later, this microgripper would be improved, considering the determinant geometry elements on its displacement and reaction force.  In accordance with [11], the here proposed microgripper can be considered as part of the class 1, as it is based on lumped compliance mechanism. In this case, on a compliance rhombus frame. In [11], inside class 1, there are also shown several cases of microgrippers actuated by chevron. In [27], other example of microgripper actuated by chevron is provided, with thickness  In accordance with [11], the here proposed microgripper can be considered as part of the class 1, as it is based on lumped compliance mechanism. In this case, on a compliance rhombus frame. In [11], inside class 1, there are also shown several cases of microgrippers actuated by chevron. In [27], other example of microgripper actuated by chevron is provided, with thickness of 50 µm. Its application as part of a micropositioner is also provided. The thickness of the device here chosen (70 µm) is near to that value and is feasible to fabricate using SOI process.
Previous simulations of microgripper parameters, displacement, and force, lets us envisage the elements to be parametrized, due to immediate influence on the microgripper performance:

•
The internal angles, represented by α i and β i , of the two symmetrical simple compliance arrangements, which are the bases of the jaws.

•
The length of the arms or jaws, L 2 , which initially have a length of 480 µm.

•
The length of the beam, L 1 , which connects the simple compliance arrangement to the anchor. Its length change has a significant impact on the displacement and force.

•
The angles represented by χ i and δ i , also have a determinant influence on the microgripper performance.

Chevron Model
In this section, the modeling of the displacement and force for the chevron actuator is given. These variables are necessary as they support the jaw's performance.
Displacement is calculated by [28]: where N is the number of beams, E is the Young's modulus, L is the beam length, F y is the actuation force, A is the area of the cross-section, I is the inertia moment, S and C are sinθ and cosθ, respectively, and θ is the aperture angle of each beam, as it is shown in Figure 1. Inertia moment is calculated as follows [29]: where w is the beam's width, and t is the device thickness. The force is obtained by [30]: where all variables where previously defined, except ε, the thermal deformation, which provides the increment in the body's dimensions, due to the increment in temperature, caused by the voltage supply: where α is the thermal expansion coefficient of the corresponding material, and ∆T denotes the temperature change. Room temperature in this paper was considered as 22 • C. Newton's second law was used to calculate the maximum mass that the micro-gripper can support. To obtain the displacement amplification factor (DAF), it is used [31,32]: where d 1 is the microactuator displacement and d 2 corresponds to displacement between tips. In the literature, there are reported amplification factors in the range from 2.85 up to 50, for piezoelectric microgrippers, with applied voltages from 0 to 700 V, giving output forces from 1 µN up to 1.87 N [31,33]. For electrothermal microgripper, with nickel, amplification factor of 7.89 was reported, when two stacks of chevron actuators are used to drive two gripper arms [34].
For the electrical properties used in the microgripper design, the intensity of the electrical current I can be calculated by using Ohm's equation. The electrical resistance (R) can be determined from the well-known equation: where ρ is resistivity, l is the length and A is the cross-section area, respectively. The total resistance of each device is calculated in accordance with its configuration.

Resistive Modeling of the Chevron Microactuator of 8 Beams
Schematic model of chevron is given in Figure 4. Its volumetric resistive and electric models are shown in Figure 5a,b, respectively. Actuators 2020, 9, × FOR PEER REVIEW 7 of 22 where ρ is resistivity, l is the length and A is the cross-section area, respectively. The total resistance of each device is calculated in accordance with its configuration.

Resistive Modeling of the Chevron Microactuator of 8 Beams
Schematic model of chevron is given in Figure 4. Its volumetric resistive and electric models are shown in Figure 5a, b, respectively. Resistances of each beam have the same value (RB1 = RB2 =…= RB8 = RB), simplified as: where  is the resistivity, Lb, and Wb are length and width of beams, and t the thickness. The general expression that describes the total resistance (RT) of chevron microactuator is given by: Actuators 2020, 9, × FOR PEER REVIEW 7 of 22 where ρ is resistivity, l is the length and A is the cross-section area, respectively. The total resistance of each device is calculated in accordance with its configuration.

Resistive Modeling of the Chevron Microactuator of 8 Beams
Schematic model of chevron is given in Figure 4. Its volumetric resistive and electric models are shown in Figure 5a, b, respectively. Resistances of each beam have the same value (RB1 = RB2 =…= RB8 = RB), simplified as: where  is the resistivity, Lb, and Wb are length and width of beams, and t the thickness. The general expression that describes the total resistance (RT) of chevron microactuator is given by: Resistances of each beam have the same value (R B1 = R B2 = . . . = R B8 = R B ), simplified as: where ρ is the resistivity, L b , and W b are length and width of beams, and t the thickness. The general expression that describes the total resistance (R T ) of chevron microactuator is given by: where N is the numbers of beams, W a and L a , are the width and length of anchor, and n is the number of the pairs of beams. It is important to mention that in the simulations, the effect on the anchor's resistance is also considered. Other equation to calculate total resistance of a chevron with a pair of beams is given in [35]: Actuators 2020, 9, 140 8 of 21 where L = L bx /(cosθ), L bx is the horizontal beam component (adjusted notation), ρ e is the electrical resistance, A c is the cross-sectional area (W × t), L, W, and t are length, width, and thickness of the beam, respectively. θ is the pre-bend angle. Equations (8) and (9) are equivalents, if in Equation (8), N = 2, n = 1, and L a = W a = 1. That means, without considering the anchor. Calculation of stiffness constant can be obtained from [36]:

Microgripper
The microgripper shown in Figure 3 is normally open, and with the chevron actuation tends to be closed. Figure 6 shows the geometry of one of its symmetrical arms. From its left side, it is observed that a small cantilever is directly joined from its guided end to a compliant rhombus frame, which is also direct joined to the extreme of the cantilever's shuttle, which provides the displacement and force to the compliant rhombus frame. This rhombus frame provides the force and displacement to jaws.
Other equation to calculate total resistance of a chevron with a pair of beams is given in [35]: where L = Lbx/(cosθ), Lbx is the horizontal beam component (adjusted notation), e is the electrical resistance, Ac is the cross-sectional area (Wxt), L, W, and t are length, width, and thickness of the beam, respectively. θ is the pre-bend angle. Equations (8) and (9) are equivalents, if in Equation (8), N = 2, n = 1, and La = Wa = 1. That means, without considering the anchor.
Calculation of stiffness constant can be obtained from [36]:

Microgripper
The microgripper shown in Figure 3 is normally open, and with the chevron actuation tends to be closed. Figure 6 shows the geometry of one of its symmetrical arms. From its left side, it is observed that a small cantilever is directly joined from its guided end to a compliant rhombus frame, which is also direct joined to the extreme of the cantilever´s shuttle, which provides the displacement and force to the compliant rhombus frame. This rhombus frame provides the force and displacement to jaws. The longitudinal force applied to the rhombus frame, from the shuttle, is given by: where φ is equal to 59.8°.
As it is known, rigidity depends on configuration, and the case of a rhombus frame is generically flexible [37,38]. This advantage was used in this design. However, it is necessary to mention that extreme cases of large deflection will not be generated, by this rhombus frame, because the magnitude of the applied force, Fin, is lower than 3 mN, which is not enough to tapered drastically the rhombus frame sides and change the apex angle (α) significatively, but the deflection of prismatic beams is enough to generate the jaws displacement. The longitudinal force applied to the rhombus frame, from the shuttle, is given by: where ϕ is equal to 59.8 • .
As it is known, rigidity depends on configuration, and the case of a rhombus frame is generically flexible [37,38]. This advantage was used in this design. However, it is necessary to mention that extreme cases of large deflection will not be generated, by this rhombus frame, because the magnitude of the applied force, F in , is lower than 3 mN, which is not enough to tapered drastically the rhombus frame sides and change the apex angle (α) significatively, but the deflection of prismatic beams is enough to generate the jaws displacement.

Chevron Actuator Improvement
To improve the Silicon initial microgripper, a parameterization process was first applied to chevron actuator and later, to gripper geometry. Chevron has been widely analyzed then, we only focus on its parameterization, which will be performed considering a single pair of bent beams.
It is well-known, when a potential is applied across the anchors, it causes a temperature difference due to joule heating. Variables will be called as follows: T M is the maximum temperature, T 0 , the temperature of one anchor, V, the voltage applied across anchors, Idc, the current flow through it. The change in L, due to thermal expansion, is denoted by ∆L, and ∆y is the movement of the shuttle [39]. In Figure 7, graphs obtained by parameterization of the determinant elements using Ansys, as well as the analytical calculations for chevron actuator are given. These graphs provide the trends of the actuator performance.  From Figure 7a, it is observed that with beams length of 600 µm, the shuttle displacement is approximately 6 µm, which is found acceptable, considering the total length of the device, in addition, this length is feasible for its fabrication using SOI technology. The pre-bending angle is chosen as 1 • , privileging displacement over force. If we consider an angle near the intersection point of graph shown in Figure 7b, the total displacement of the microgripper is seriously decreased. From Figure 7c, it is seen that the displacement is inversely proportional to beam width, while force is directly proportional. Selection of a width beams of 5 µm was performed, again focused on displacement and maintaining an acceptable width of the structure.
Trends of displacement and force with respect to structure thickness is shown in Figure 7d, as it can be observed a linear increment of force against thickness, 70 µm is found convenient, and it compensates the selection of pre-bending angle and beam width. In addition, adequate SOI prime wafers are commercialized (Ultrasil Corp. Hayward, CA, USA), which can be used for the possible fabrication. From Figure 7e, simulated results of displacement and force against shuttle width are given, 30 µm are found suitable, as changes in force are not significant, and displacement changes are also not determinant due to the chosen scales. After this parameterization analysis, it was decided to maintain the initial values, given in Table 3, without changes.

Chevron Resistance
In Table 5, a comparison of the total resistance of chevron, is provided, with an error of 0.88%, which is an indicator of the analysis precision.

Initial Microgripper
The initial proposal of microgripper, shown in Figure 3, was determined by means of try and failure, without convection. ANSYS™ tool was used to improve the determinant elements of the microgripper design by using parameterization through Finite Element Analysis (FEA), without convection.
Temperature distribution generated by applying a maximum voltage of 2 V at one anchors of the chevron actuator, obtaining a maximum temperature of 112 • C, on the shuttle. One anchor is considered at room temperature (22 • C). At this maximum temperature, The displacement obtained applying a maximum voltage of 2 V is 9.02 µm in each jaw. The thickness of the structure corresponds to 70 µm, and the reaction force obtained is 3795 µN. When the simulation is performed considering a thickness of 15 µm, the reaction force is 795 µN. As can be seen, for a thickness of 70 µm, the increase in force is considerable (477.36%).

Performance of Initial Microgripper
In the performance analysis of the initial and improved microgrippers, three conditions were considered: 1.
Anchors are fixed 2.
Room temperature was applied to one of the anchors (300 • K) 3.
Convective heat flux was also considered (20,000 W/m 2 • K) Under thermal convection, the heat is transferred by a moving fluid, and is usually the dominant form of heat transfer in liquids and gases. Condition number 3, which was considered as a rule of thumb, establishes that for dimensions less than 1 mm, there will likely not be any free convective currents [40]. If the surrounding fluid is a liquid, the range of the forced heat transfer coefficients are much wider. For free convection in a liquid h ≈ 50 to 1000 W/m 2 • K is the typical range. Constant h is the film coefficient of heat transfer coefficient. For forced convection, the range is even wider h ≈ 50 to 20,000 W/m 2 • K. In this case, air was considered as the fluid.
In Ansys, the procedure to perform convection analysis, is described as follows: in steady-state thermal-electric conduction, convection is inserted, frontal device's face is chosen, later h, the value of fluid considered in forced convection, is added, and finally the ambient temperature is stablished.
Temperature distribution of initial microgripper against applied voltage is given in Figure 8, under convection. From Figure 8, is possible to observe the variation of the maximum temperature, with the applied voltage. A voltage sweep from 0 to 10 V was performed, which generates a maximum temperature of approximately 497 • C, on the shuttle. The applied voltage parameterization was carried out, in this case, to analyze its effect on the temperature on shuttle and jaws (Figure 8a,b), as well as on the displacement generated in the jaws of the clamp and their reaction force (Figure 9). Due to this response, it was selected to apply only 5 V. Temperature distribution in the microgripper, obtained with this voltage, is shown in Figure 8c.
Under thermal convection, the heat is transferred by a moving fluid, and is usually the dominant form of heat transfer in liquids and gases. Condition number 3, which was considered as a rule of thumb, establishes that for dimensions less than 1 mm, there will likely not be any free convective currents [40]. If the surrounding fluid is a liquid, the range of the forced heat transfer coefficients are much wider. For free convection in a liquid h ≈ 50 to 1000 W/m 2 °K is the typical range. Constant h is the film coefficient of heat transfer coefficient. For forced convection, the range is even wider h ≈ 50 to 20,000 W/m 2 °K. In this case, air was considered as the fluid.
In Ansys, the procedure to perform convection analysis, is described as follows: in steadystate thermal-electric conduction, convection is inserted, frontal device´s face is chosen, later h, the value of fluid considered in forced convection, is added, and finally the ambient temperature is stablished.
Temperature distribution of initial microgripper against applied voltage is given in Figure 8, under convection. From Figure 8, is possible to observe the variation of the maximum temperature, with the applied voltage. A voltage sweep from 0 to 10 V was performed, which generates a maximum temperature of approximately 497 °C, on the shuttle. The applied voltage parameterization was carried out, in this case, to analyze its effect on the temperature on shuttle and jaws (Figure 8a,b), as well as on the displacement generated in the jaws of the clamp and their reaction force (Figure 9). Due to this response, it was selected to apply only 5 V. Temperature distribution in the microgripper, obtained with this voltage, is shown in Figure 8c.  In Figure 9, a voltage sweep from 0 to 10 V is also applied. A non-linear increase in displacement is observed. The maximum displacement at 5 V, is 12.2 µm, considering both jaws, so the maximum diameter of the clamping objects can be of 24.4 µm. Some examples of microtubes in this range can be found at [41]. In the same Figure, the graph of voltage vs. force is displayed, with similar tendency. About the relationship between thickness versus reaction force, from Figure 10, a quasilinear dependence is observed. General data are given in Table 6. In Figure 9, a voltage sweep from 0 to 10 V is also applied. A non-linear increase in displacement is observed. The maximum displacement at 5 V, is 12.2 µm, considering both jaws, so the maximum diameter of the clamping objects can be of 24.4 µm. Some examples of microtubes in this range can be found at [41]. In the same Figure, the graph of voltage vs. force is displayed, with similar tendency. About the relationship between thickness versus reaction force, from Figure 10, a quasilinear dependence is observed. General data are given in Table 6.    In Figure 9, a voltage sweep from 0 to 10 V is also applied. A non-linear increase in displacement is observed. The maximum displacement at 5 V, is 12.2 µm, considering both jaws, so the maximum diameter of the clamping objects can be of 24.4 µm. Some examples of microtubes in this range can be found at [41]. In the same Figure, the graph of voltage vs. force is displayed, with similar tendency. About the relationship between thickness versus reaction force, from Figure 10, a quasilinear dependence is observed. General data are given in Table 6.

Parameterization of the Initial Microgripper Elements without Convection
Graphs of parametrization results of angles α, β, χ, and δ, and lengths L 1 and L 2 , versus displacement and force, are given in Figure 11. They are useful to determine improved geometries of the initial microgripper.
the case of force, which determines the new value of this angle (150°). About the L1 length ( Figure  11e), it is observed that it is at the right point to obtain the greatest displacement, although the force is slightly less than that the achieved at 100 µm. Then, its initial value was not changed.
For L2 (Figure 11f), the initial value is close to the maximum displacement, without greatly sacrificing force (Improved Microgripper 1, IMG1). If the length value that allows maximum displacement is selected, the initial opening of the gripper would be very closed, which depends on the purpose for which it is intended.
If it is desired to maintain an opening like the initial one, the length of L1 value corresponding to Improved Microgripper 2 (IMG2) can be chosen, where, in addition, provides an improvement in strength, which may be desirable for another application. Dimensions of improved microgrippers 1 and 2, as well as their parameters are shown in Table 6.

Performance of Improved Microgripper 2, (IMG2)
Dimensions for IMG2 are given in Table 7. Its displacement response and reaction force when a voltage sweep from 0 to 10 V is applied are given in Figure 12. Its temperature distribution at 5 V is given in Figure 13. General data of the IMG2 performance are also given in Table 7. Table 7. Dimensions of the improved microgripper 2 (IMG2).

Element Value Element
Value g 108 µm f 270.5 µm About the angle's selection, from Figure 11a,b that the initial value of the angles α and β, are very close to the maximum values indicated in the corresponding graphs, so it was decided to conserve them. In the case of χ, shown in Figure 11c, although the change in displacement is not so great, it is slightly significant in the case of force, so it was considered to take a value of 110 • .
For the δ angle, the behavior of displacement is like the previous case, but it is significant for the case of force, which determines the new value of this angle (150 • ). About the L 1 length (Figure 11e), it is observed that it is at the right point to obtain the greatest displacement, although the force is slightly less than that the achieved at 100 µm. Then, its initial value was not changed.
For L 2 (Figure 11f), the initial value is close to the maximum displacement, without greatly sacrificing force (Improved Microgripper 1, IMG1). If the length value that allows maximum displacement is selected, the initial opening of the gripper would be very closed, which depends on the purpose for which it is intended.
If it is desired to maintain an opening like the initial one, the length of L 1 value corresponding to Improved Microgripper 2 (IMG2) can be chosen, where, in addition, provides an improvement in strength, which may be desirable for another application. Dimensions of improved microgrippers 1 and 2, as well as their parameters are shown in Table 6.

Performance of Improved Microgripper 2, (IMG2)
Dimensions for IMG2 are given in Table 7. Its displacement response and reaction force when a voltage sweep from 0 to 10 V is applied are given in Figure 12. Its temperature distribution at 5 V is given in Figure 13. General data of the IMG2 performance are also given in Table 7.  From Table 6, it is observed that both improved grippers exceed the initial gripper force and displacement parameters. The total displacement between the jaws is greater for IMG1, while the force in IMG1 and IMG2 is greater than initial microgripper. The initial and final opening between  From Table 6, it is observed that both improved grippers exceed the initial gripper force and displacement parameters. The total displacement between the jaws is greater for IMG1, while the force in IMG1 and IMG2 is greater than initial microgripper. The initial and final opening between the jaws is greater in the case of IMG2.
In all cases, from directional displacement on Z axis, it is observed that the out-of-plane deflection is minimal, providing a minimal cross axis sensitivity in this direction [42], which could represent a great advantage, because this characteristic would avoid the realignment during the  From Table 6, it is observed that both improved grippers exceed the initial gripper force and displacement parameters. The total displacement between the jaws is greater for IMG1, while the force in IMG1 and IMG2 is greater than initial microgripper. The initial and final opening between the jaws is greater in the case of IMG2.
In all cases, from directional displacement on Z axis, it is observed that the out-of-plane deflection is minimal, providing a minimal cross axis sensitivity in this direction [42], which could represent a great advantage, because this characteristic would avoid the realignment during the operation. For the case of IMG2, the cross-axis sensitivity in this axis is 0.04%.

Comparison of Microgrippers
When IMG1 and IMG2 are implemented in polysilicon, their DAF, are 2.6 and 3.26, respectively, applying 2 V. A comparison of IMG2 with other microgrippers were performed (top part of Table 8). With Polysilicon, IMG2, comparing with the one shown in [20], allows to observe that at 50% of applied voltage, IMG2 shown larger value of displacement (75%), but lower force (30%). Temperature on chevron is similar, but in jaws, temperature is lower (15%). It is important to mention that [5] is actuated by two chevrons, and implemented in metal, for this reason, it shows, in general, a better performance at lower applied voltage. An additional advantage of the proposed gripper is its simplicity.   The bottom part of Table 8 provides a comparison of IMG1 and IMG2, implemented in silicon and polysilicon. With the last material, the grippers provide bigger values on displacement and force, than in the case of silicon, at lower voltage. With polysilicon, improved MG2 shows larger parameters than improved MG1, simulated with the same material.
On the other hand, technical details about FEA analysis are provided in Table 9.

Frequency Response of IMG2, Implemented in Silicon
Modal analysis is performed to know the 3D microgripper mode shapes and its natural frequencies (Table 10, Figure 14). This knowledge allows us to avoid the vibration of the microgripper at any of its natural frequencies.

Frequency Response of IMG2, Implemented in Silicon
Modal analysis is performed to know the 3D microgripper mode shapes and its natural frequencies (Table 10, Figure 14). This knowledge allows us to avoid the vibration of the microgripper at any of its natural frequencies.  Figure 14. The first to sixth modal forms of IMG2 (a-f).
From modal forms given in Figure 14 and Table 10, it is recommended that the microgripper operates at frequencies lower to 31.633 Hz, corresponding to first modal frequency, which coincides with the expected performance of the microgripper. Regarding the analysis confidence, in [44], after analysis of other two methods, the authors concluded that their results confirmed Finite Elements Analysis as the most recommended method and used it to study complex structures.  From modal forms given in Figure 14 and Table 10, it is recommended that the microgripper operates at frequencies lower to 31.633 Hz, corresponding to first modal frequency, which coincides with the expected performance of the microgripper. Regarding the analysis confidence, in [44], after analysis of other two methods, the authors concluded that their results confirmed Finite Elements Analysis as the most recommended method and used it to study complex structures.
3.9. Holding Microparticles with Silicon and Polysilicon IMG2 Figure 15 shows results of simulation, by FEA, of the IMG2 implemented with silicon, showing it holding a microparticle. Simulation was also made with polysilicon. Results of both cases are given in Table 11, which shows the generated stress and the reaction force into microparticles of Au, glass ceramic, polycarbonate and carbon fiber (with Young's modulus, in GPa, of 78.5, 91, 0.238, and 230, respectively). In general, microparticles produces slightly lower values of stress, and total reaction forces into microparticles for the case of silicon IMG2. The lowest values of stress and reaction force correspond to the polycarbonate particle, and the largest values of both parameters correspond to carbon fiber particles. From these results, all particles under holding could be manipulated without damaging them.  Simulation conditions for the manipulation of the microparticles shown in Figure 15, performed with Ansys Workbench, are given in Table 12, providing a description of the complete performed process.    Simulation conditions for the manipulation of the microparticles shown in Figure 15, performed with Ansys Workbench, are given in Table 12, providing a description of the complete performed process.

Conclusions
An important feature of the proposed microgripper is its configuration based on the arrangement of two simple compliant rhombus frame used at the base of each of the jaws. To improve its response, parameterization process was performed for chevron actuator and the microgripper, in both cases, without convection. With a thickness of 70 µm, the reaction force of the microgripper is compensated, considering the other design elements selection, which was focused on displacement.
Parameterization processes are very useful to improve geometries. It provides graphical information, which aid the user's decision making. It is also necessary, to corroborate the complete device performance in order to have a final decision.
To analyze the main performance parameters of the microgripper, convection was considered. Based on the initial microgripper, two improved microgrippers were developed (IMG1 and IMG2), simulated with silicon and polysilicon. In the last case, microgrippers have bigger values of displacement and force, even though they are feed with lower voltage. The comparison of IMG2 with a more complex microgripper found in literature, shows a larger value of displacement (50%), but a lower value of force (30%).
From the jaw's displacement and reaction force, the diameters allowed for the subjection objects are found between 84.64 µm and 108 µm, using IMG2, with weights lower than 612.2 µg. Some tests of microparticles holding were performed with IMG2, using as subjection objects to microcylinders of Au, glass ceramic, polycarbonate and carbon fiber, showing the stress on them, as well as the total reaction force induced, with permissible values in accordance to their mechanical properties, avoiding generating a damage on them.
It is important to note that the temperature in the jaws of IMG2, implemented with Polysilicon, applying 0.6 V to chevron actuator, is of 231.76 • C, but when convection and a feed of 2 V, are considered, a temperature in jaws is reduced to 25.21 • C. In both cases, these temperatures are not extreme, and are suitable for several kind of objects.
As future work, it is considered the fabrication of IMG2 using silicon or polysilicon. It is also possible to improve the microgrippers performance using other materials or reinforce the design, implementing additional structural elements. Funding: This research was funded by Consejo Nacional de Ciencia y Tecnologia, CONACyT, grant reference number A1-S-33433. "Proyecto Apoyado por el Fondo Sectorial de Investigación para la Educación".