Nonlinear design, analysis, and testing of a single-stage compliant orthogonal displacement amplifier with a single input force for microgrippers

To achieve dexterous and stable micro/nanomanipulation, a large grasping stroke, compact design, and parallel grasping are required for microgrippers; thus, a single-stage compliant orthogonal displacement amplifier (CODA) with a single input force would be an ideal transmission mechanism. However, the existing small-deflection-based design schemes cannot adapt to large deflections or shearing effect, thereby affecting the orthogonal movement transformation accuracy. This study proposed, analyzed, and experimentally investigated a nonlinear design scheme for a single-stage CODA with a single input force. First, the nonlinear design principle is described qualitatively. By combining closed-form analytical modelling, finite element analysis, and numerical fitting, the nonlinear extent of a pre-set variable cross-sectional beam in the CODA is formulated. By utilizing the beam constraint model and small-deflection-based modelling, the nonlinear extent of the undetermined uniform straight beam in the CODA is derived. Based on the design principle and nonlinear models, a nonlinear design scheme is proposed quantitatively. Finite element simulations and experimental tests are conducted to verify the proposed scheme, and the limitations of our previous study are revealed.


Introduction
In recent decades, robotic micro/nanomanipulation has been developed for various applications in precision engineering, * Authors to whom any correspondence should be addressed.
Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.such as chip placement [1], teleoperated microsurgery [2], and microassembly [3].Microgrippers, which grasp, transport, place, and release micro/nanoscale objects, are typical micro/nanomanipulation end effectors [4][5][6].To achieve dexterous and stable manipulation, microgrippers require the following features: a large grasping stroke, compact design, and parallel grasping.
Utilizing hybrid drive systems and designing displacement amplifiers are two strategies to enlarge the grasping stroke.By accumulating the strokes of actuators, hybriddriven microgrippers [5,[7][8][9][10] can effectively increase the grasping movement at the cost of complicated driving and control systems.Compliant mechanisms that do not introduce clearance and friction issues are commonly used to construct displacement amplifiers, in which single-stage displacement amplifiers, e.g. the lever-type mechanism [11], triangulation-amplification-based mechanism [12,13], Scott-Russell amplifier [14], and topology-optimized amplifier [15], are more compact than the multistage ones [16,17].
In our previous study, a single-stage CODA with a single input force was proposed, and bidirectional symmetric input forces/displacements were not required [13], which can match the output properties of piezoelectric stack actuators for micro/nanomanipulation, electrostatic drivers and chevron electrothermal actuators.However, the design equations were based on the small-deflection assumption, and the allowable input force was not indicated.Besides, the actual displacement amplification ratio was limited by stress stiffening.Subsequently, a geometrically nonlinear analysis of the proposed CODA was conducted, and a static constraint restricting a large deflection degree was established [40].This constraint indicates the relationships between the allowable input force, material, and dimensions.In addition, a compressionbased design was used to eliminate the stress stiffening within the proposed CODA.In the compression-based design, the displacement amplification ratio increases with the input force [24].Furthermore, the input-output displacements, input stiffness, fundamental frequencies, and frequency response of the proposed CODA were calculated using a graphic transfer matrix method [41].However, the static constraint, structural optimization, and graphic transfer matrix methods did not alter the small-deflection-based design schemes.Moreover, the design schemes employed in previous studies disregarded the shearing effect, which can be induced by flexible elements that are relatively short compared to their width [42].
Consequently, in scenarios involving large deflections or miniaturization designs, the orthogonal movement transformation accuracy is affected.
In this study, a nonlinear design scheme for a single-stage CODA with a single input force is proposed, analyzed, and experimentally tested.This study breaks through the smalldeflection-based design schemes existing in previous studies, which affect the orthogonal movement transformation accuracy.The remainder of this paper is organized as follows: section 2 describes a nonlinear design principle qualitatively.In section 3, the nonlinear extents of the pre-set variable cross-sectional beam and undetermined uniform straight beam in the CODA are formulated.In section 4, a nonlinear design scheme is quantitatively proposed, based on the design principle and nonlinear models.In section 5, the proposed scheme is verified using finite element analysis (FEA).In section 6, an experiment is conducted on a microgripper that incorporates a nonlinear design result, in order to further validate the proposed scheme.Finally, section 7 concludes the study.

Nonlinear design principle
The proposed single-stage CODA is driven by a single input force, i.e.F in (figure 1), and comprises an input structure, two variable cross-sectional beams, two output structures, and two undetermined uniform straight beams with uniform thickness h.The single-stage CODA with a single input force is symmetric about F in ; therefore, for the purpose of a simplified analysis, the left part of the CODA is considered.The variable cross-sectional beam AB on the left comprises two flexure hinges, i.e. a and b, and a uniform segment.Essentially, an arbitrary type of flexure hinge can be utilized for beam AB.In figure 1, leaf-type flexure hinges are used to enhance the deformation ability.Triangulation amplification can be achieved by properly pre-setting the dimensions of beam AB.The undetermined uniform straight beam CD with a fixed lower end was introduced, in which three undetermined parameters (u = [L d , B, γ]) exist, as shown in figure 1.Using certain design equations to determine u, orthogonal movement transformation can be obtained.Consequently, this enables the establishment of a displacement boundary condition, X B = X C .In our previous study, small-deflection-based design equations were derived as [13]: where Γ XC-vC (v = FZ, FX, M) denotes the compliance coefficient of beam CD.Using Castigliano's second theorem, Γ XC-vC is formulated as a function of u.The parameters f 1 , f 2 , and η are related to Γ XC-vC (u).Similarly, the compliance coefficients of beam AB S can be derived.The parameters q, λ, κ 1 , and κ 2 are functions of S. Compliance formulations are not included in this study.The lengths of lines BC1 and CC1, denoted as l BC1 and l CC1 , respectively, characterize the in-plane size of the output structures.
The length l BC1 is designed for specific applications.By adjusting l CC1 , different values of u can be calculated using equations ( 1)-(3); thus, multiple solutions exist for these equations.If the geometrical nonlinearity and shearing effect are neglected, the displacement X j (j = B, C) remains equivalent to its small-deflection value without considering shearing effect.However, when both geometrical nonlinearity and shearing effect are introduced, X j deviates from its original small-deflection value.The deviation extent was defined as the nonlinear extent, which can be divided into a geometrically nonlinear component and a shearing component.Owing to different values of L d and B, the nonlinear extents of beam CD under certain loads differ.For beam AB, the dimensions were pre-set; thus, under certain loads, the nonlinear extent is unique.If the value of l cc1 is selected such that it equalizes the nonlinear extents of the two beams, the displacement boundary condition X B = X C can be satisfied even in scenarios involving large deflections or miniaturization designs.In this case, the parasitic rotational movement of the output structures is restricted under different values of F in and the dimensions.

Nonlinear modelling
According to section 2, the key point of the nonlinear design principle is the formulation of the nonlinear extents.In this section, the nonlinear extents of beams AB and CD are formulated.Section 3.1 derives the nonlinear extent of variable cross-sectional beam AB, and section 3.2 derives the nonlinear extent of undetermined beam CD.

Nonlinear modelling of the variable cross-sectional beam
The variable cross-sectional beam AB is presented in figure 2, in which the displacement boundary conditions at point A are: X A = 0 and the rotation θ A = 0.The nonlinear modelling process of beam AB is divided into two steps.First, considering shearing deformation, small-deflection-based modelling was conducted.Second, the extent of geometrical nonlinearity was derived.Based on steps 1 and 2, the nonlinear extent of beam AB can be formulated.

3.1.1.
Step 1: small-deflection-based modelling considering the shearing effect.In Step 1, beam AB is assumed to be linearly elastic.As X A = 0, Castigliano's second theorem describes the generalized displacement of point B in terms of strain energy First, the shearing deformation in beam AB is ignored; thus, V AB comprises axial and bending deformation terms.
where L AB = 2l+L 2 and E denotes Young's Modulus.The cross-sectional area (A AB ), distance between a point in beam AB and the axis of flexure hinge B (λ), and cross-sectional moment of inertia (I AB ) are functions of p v .Substituting equation ( 5) into equation ( 4), the relationship between X B and the generalized loads at point B can be derived as where the compliance coefficients S XB-FXB , S XB-FZB , and S XB-MB are The value of X B calculated by equation ( 6) is denoted as X BsE .For the derivation above, the shearing effect and geometrical nonlinearity are neglected.Hereafter, by introducing a correction factor, the shearing effect is considered, i.e. the correction factor for the shearing effect (CFSE).The CFSE of X B , denoted by a dimensionless coefficient α, is defined as where X Bs denotes the small-deflection value of X B .Smalldeflection FEA utilizing free meshing is employed to obtain X Bs .Specifically, in the static analysis module of ANSYS Workbench, disabling the 'Large Deflection' option allows for small-deflection FEA.
If the bending stiffness of the uniform segment is considerably larger than that of flexure hinges a and b, for beam AB, only a and b are viewed as elastic elements.It is noted that shearing deformation occurs only in elastic elements; thus, α is related to l, t, and h, which are dimensions of flexure hinges.As the slenderness ratio l/t increases, the shearing effect in the flexure hinges decreases [43], thereby leading to a reduction in α.For an arbitrary material, L/t r , and F in , the relationship between α and l/t is illustrated in figure 3, where diverse slenderness ratios are generated by keeping l constant and varying t.If diverse slenderness ratios are generated by keeping t constant and varying l, the corresponding CFSEs differ from those shown in figure 3.
The abovementioned single-factor analysis indicates that the CFSE is negatively correlated with l/t, which is similar to the correlation between the CFSE and h.In addition, the slenderness ratio cannot fully characterize the effects of l and t on the CFSE.Therefore, a dimensionless parameter φ is defined as shown in equation ( 9), where the denominator includes l/t and h to characterize the negative correlation.
where the positive weights a s , b s , and c s were determined by trial and error.The expected weights satisfy the following requirements: (1) a s , b s , and c s should make φ dimensionless; and (2) the weights should make the relationship between α and φ approach a mapping relationship.
Based on the requirements, a s = 3.2, b s = 2, and c s = 3.2 are chosen.For each discrete example in figure 3, φ can be calculated using equation (9).Combining the calculated φ with the values of α in figure 3 generates 36 discrete results, as shown in figure 4. Using a least-squares fitting procedure, the relation between α and φ is derived as α = 0.462φ 0.165 + 0.897 , (10) where the constants are not related to the material properties, load, or dimensions.

3.1.2.
Step 2: Formulating the extent of geometrical nonlinearity.Owing to geometrical nonlinearity, X B deviates from X Bs .To characterize the deviation, the correction factor for the geometrical nonlinearity (CFGN) of X B , denoted by a dimensionless coefficient β, is introduced; The CFGN is defined as To obtain X B , geometrically nonlinear static FEA with free meshing can be implemented.Specifically, in the static analysis module of ANSYS Workbench, activating the 'Large Deflection' option enables geometrically nonlinear static FEA.
As β decreases, the deviation between X B and X Bs increases; thus, 1-β is utilized to characterize the extent of geometrical nonlinearity.For beam AB, when the axial load F ZB increases, the extent of geometrical nonlinearity increases.Hence, 1-β is positively correlated with F ZB .In terms of the structural parameters, the dimensions of the flexure hinges and material properties influence 1-β.
Furthermore, considering F ZB = 10 N and five kinds of typical engineering materials, i.e. stainless steel (E = 193 GPa), copper alloy (E = 110 GPa), titanium alloy (E = 96 GPa), aluminium alloy (E = 71 GPa), and magnesium alloy (E = 45 GPa), the relationship between 1-β, l/t, and h is presented in figure 5, in which diverse slenderness ratios are generated by keeping l constant and varying t.If diverse slenderness ratios are generated by keeping t constant and varying l, the corresponding extents of geometrical nonlinearity differ from those shown in figure 5.When a value ranging from 1 N to 10 N, incremented by 1 N intervals, is assigned to F ZB , a similar relationship between 1-β, l/t, and h can be obtained.
The single-factor analysis above infers three qualitative results: (1) the extent of geometrical nonlinearity is negatively correlated with h, which is the same as the correlation between the extent of geometrical nonlinearity and E; (2) the extent of geometrical nonlinearity is positively correlated with l/t and F ZB ; and (3) the slenderness ratio cannot fully characterize the l and t effects on the extent of geometrical nonlinearity.
A dimensionless factor ω is defined (equation ( 12)), where E and h are considered denominators to characterize the negative correlation, and F ZB and l/t are considered numerators to characterize the positive correlation For the 180 discrete examples presented in figure 5, as the value of F ZB is assigned from 1 N to 10 N, incremented by 1 N intervals, the corresponding values of ω can be computed individually using equation (12), resulting in a total of 1800 ω values.Combining the calculated ω with the corresponding values of 1-β generates 1800 discrete results.Similar to the requirements outlined in step 1, the positive weights a g , b g , c g , d g , and e g in equation (12) were determined through trial and error.The determined weights are , and e g = 1, which render ω dimensionless and enable the 1800 discrete results to approach a mapping relationship, as shown in figure 6. Utilizing least-squares fitting, the quantitative relationship between 1-β and ω is derived as where the constants are not related to the material properties, load, or dimensions.Using equations ( 10) and ( 13), the nonlinear extent of beam AB can be formulated as follows: ) . (14)

Nonlinear modelling of the undetermined beam
The undetermined beam CD is used to guide the movement of the output structure; thus, the slenderness ratio L d /B is large, and the shearing effect can be neglected.Therefore, the nonlinear extent of beam CD, which is denoted as ∆ε CD , equals the extent of geometrical nonlinearity β CD .Figure 7 depicts the deformed and undeformed undetermined beams, in which point C moves along the X C axis without rotation, corresponding to Z C = 0 and θ C = 0.In the coordinate system PDQ, the values of P and Q characterize the position of a cross section in beam CD and the deflection, respectively.Considering geometrical nonlinearity, the bending moment of beam CD is expressed as  For transverse deflection within 10% of the beam length, the beam curvature can be linearized by assuming the linear slope with a small prediction error [44,45]; thus, the Euler-Bernoulli beam equation is simplified as where I CD denotes the cross-sectional moment of inertia for beam CD.We define F Q = F XC cos γ-F ZC sin γ and F P = F ZC cos γ+F XC sin γ.The loads, position coordinates, and deflections are normalized with respect to the beam geometry and material parameters According to equations ( 15)-( 17), we obtain the following equation: By considering the second derivatives of both sides of equation ( 18) with respect to p, a fourth-order differential equation is derived as For equation (19), the general solution is where b = f 0.5 p .The boundary conditions of beam CD are defined as Substituting equation ( 21) into equation ( 20), the integration constant C i (i = 1,2,3,4) can be determined as follows: • f y , Using equations ( 20) and ( 22), the normalized deflection at point C is formulated as follows: Using equations ( 17) and ( 23), and the geometric relationship, X C is derived as follows: Equation ( 24) is the geometrically nonlinear model of X C .Similar to the modelling process shown in the first stage of section 3.1.1,the small-deflection-based formulation of X C is expressed as where X Cs denotes the small-deflection value of X C , and A CD denotes the cross-sectional area.Furthermore, the nonlinear extent of beam CD is

Nonlinear design scheme
Section 2 indicates that if the value of l cc1 that derives ∆ε AB = ∆ε CD is selected, the displacement boundary condition X B = X C can be satisfied under a large deflection.According to the nonlinear modelling described in section 3, ∆ε AB is a function of F ZB , and ∆ε CD is a function of F XC , F ZC , and M C .By utilizing the loading equilibrium equations and the displacement boundary conditions of the orthogonal movement transformation, the loads acting on points B and C can be formulated as [13] A nonlinear design scheme for a single-stage CODA with a single input force is illustrated in figure 8, where l CC10 denotes the initial value of l CC1 and ∆l CC1 denotes the increment of l CC1 .The lower limit of the input force F in-min and the upper limit F in-max are customer-specified parameters; ∆F in denotes the increment in F in .Theoretically, the error ε should be as small as possible, whereas considering that the modelling error cannot be avoided in Sections 3.1 and 3.2, the value of ε is increased in practice, e.g.ε = 10%.

FEA verification
Geometrically nonlinear static FEA was used to verify the nonlinear design scheme, and ANSYS Workbench was used to conduct the FEA.The FEA settings are analogous to those outlined in section 3.1.2.Six simulation examples have been designed.The pre-set structural parameters of each example are listed in table 1.To verify the scheme from a general perspective, four crucial factors were considered in the selection of pre-set structural parameters: (1) different types of materials have been used in the examples, and (2) the pre-set dimensions of the examples are varied significantly, especially for examples V and VI, and (3) in example VI, the material chosen is silicon (Young's modulus: 187 GPa), which was not used while formulating the CFGN in section 3.1, and ( 4) the examples do not include the 36 discrete cases in figure 3 and the 180 discrete cases in figure 5. Setting ∆l CC1 , ∆F in , and F in-max to 0.01 mm, 5 N, and 20 N, respectively, based on the nonlinear design scheme illustrated in figure 8, l CC1 and u are determined, as listed in table 2. Although a smaller value of ∆l CC1 would yield a more precise nonlinear design result, ∆l CC1 = 0.01 mm was chosen considering the fabrication limits.
The parasitic rotational movement of the output structure BC1 can be characterized using equation (28), and the Z-axis parasitic movement is evaluated using equation ( 29) [40].
In [40], the orthogonal movement transformation was realized based on the optimal design result.Comparing example II with the optimal design result, the values of the pre-set structural parameters are the same, whereas the values of l CC1 differ.For the optimal design result, l CC1 was set as 0.21 mm, and the selection process was arbitrary.According to the geometrically nonlinear static FEA, when F in = 10 N, the ς value of example II is 2.11µrad/µm and that of the optimal design result is 7.49µrad/µm (figure 9).Compared to the latter, ς of example II is reduced by 71.83%.Furthermore, regarding different input forces, the ς and δ values of example II and the optimal design result are illustrated in figure 10.Compared to the latter, the ς value of example II decreases by 76.69% on average.In addition, for both CODAs, the values of δ are less than 1.80%.Thus, when geometrical nonlinearity is considered, the orthogonal movement transformation accuracy of example II is far higher than that of the optimal design result mentioned in [40].
Comparing example I with the initial group in [40], we found that the pre-set structural parameters remain the same, whereas the l CC1 values differ.For the initial group, l CC1 was set to 0.13 mm, and the selection process was arbitrary.Based on [40], owing to the considerably large deflection degree of beam AB, both example I and the initial group cannot transmit the movement orthogonally, considering geometrical nonlinearity.The geometrically nonlinear static FEA indicates that when F in = 10 N, the ς value of example I is 0.40µrad/µm, whereas that of the initial group is 62.14 µrad/µm (figure 11); therefore, the parasitic rotational movement can be almost eliminated in example I. Furthermore, regarding different input forces, the ς and δ values of example I and the initial group are illustrated in figure 12, which indicates that the ς     Values of ς and δ in example II and the optimal design result in [40] regarding different input forces (using geometrically nonlinear static FEA).and δ values of example I under different forces are smaller than 5.90 µrad/µm and 1.20%, respectively.Compared to the initial group, ς and δ are reduced by 96.07%and 98.54%, respectively.Thus, when geometrical nonlinearity is considered, example I can be opted for transmitting the movement orthogonally, and the orthogonal movement transformation accuracy of example I is far higher than that of the initial group mentioned in [40].Thus, the limitations of [40]     of ς and δ are 4.29 µ rad/µm and 0.43%, respectively, thereby verifying the orthogonal movement transformation when geometrical nonlinearity is considered.
The dimensions of example VI are significantly smaller than those of the other examples; therefore, considering the same load/displacement boundary conditions, a large deflection is evident in example VI.Using small-deflection static FEA, the ς and δ values are 40.91 µrad/µm and 1.35%, respectively.The corresponding geometrically nonlinear static FEA results are illustrated in figure 14.The average values of ς and δ are 39.52µrad/µm and 0.81%, respectively, which approximate the small-deflection FEA results.Therefore, in example VI, by utilizing the nonlinear design scheme, the adverse impact of a large deflection on the orthogonal movement transformation accuracy is suppressed.
From above, the effectiveness and advantages of the nonlinear design scheme are verified.

Experimental test
In this section, a microgripper is designed based on the nonlinear design results of example II.Optical micrometrology, a technique widely used in precision engineering owing to its advantages of contactless measurements, high precision, and low cost [46][47][48], was implemented to test the microgripper prototype, which verifies the nonlinear design scheme.

Prototype design, analysis and fabrication
The transmission mechanism of the designed microgripper is shown in example II while considering nonlinear design results.The gripper arms are connected to the output structures of the CODA, and two fixed-guided flexible segments are added to both sides of the input structure (figure 15).The out-of-plane deformation induced by the direction error of F in in the experiment can be reduced using fixed-guided flexible segments.In addition, the structural stability of the CODA with guiding beams is improved, which reduces the fabrication error.
The grasping and Z-axis parasitic movements of the jaw are influenced by X B and X C . Figure 16 illustrates the movement of point E, i.e. the midpoint of the executive port in the left jaw.In figure 16, B', C', and E' denote the positions of B, C, and E after the deformation of the CODA, respectively.
Based on the geometrical relationship shown in figure 16, we derive   where l CE and l BE denote the lengths of lines CE and BE, respectively; θ out denotes the rotation of the left output structure in the CODA; and θ 2 , θ 3 , and l BE are formulated as follows: If the grasping (X E ) and Z-axis parasitic (Z E ) movements are measured, based on equations ( 30) and (31), X B and θ out can be obtained.Subsequently, the experimental value of X C is determined utilizing the geometrical relationship: A 7075 aluminium alloy was considered, and a microgripper prototype was fabricated using electrical discharge machining.A piezoelectric stack actuator (PSt150/2×3/7, Coremorrow, China) was used to provide F in (figure 17).The stiffness of the piezoelectric stack k PSA is 25 N/µm.A preload bolt (M1.2) was employed to adjust the preload between the actuator and CODA, thus eliminating the clearance between the two components.

Experimental setup
Figure 18 illustrates the experimental setup.A piezoelectric controller (E00.D6; Coremorrow, China) supplied a highprecision input voltage to the actuator.The assembly was mounted on a three-degree-of-freedom (3-DOF) micromotion platform comprising a Y-axis stage (ZM07A-S3K, KOHZU, Japan) and an X-and Z-axis stage (YM10A-C3-CL, KOHZU, Japan).The Y-axis stage has a maximum motion resolution of 0.25 µm, which enables the acquisition of clearly focused images.For the X and Z-axis stages, the maximum motion resolution is 0.5 µm, which precisely adjusts the in-plane position Piotr's Computer Vision MATLAB Toolbox [49] and Pixelink Capture were used.The 3-DOF micromotion platform and optical micrometrology hardware tool were installed on a vibration isolation table (Zolix, China).

Experimental process and results
When the zoom was set to 4.5×, the pixel equivalent of the optical micrometrology system was calibrated to 0.162 µm/pixel using Pixelink Capture and a calibration plate (R1L1S1P, THORLABS, USA).A certain preload F p was applied between the actuator and prototype.Next, a voltage starting from 0 V and ending at 150 V was applied to  the actuator at intervals of 10 V.For each applied voltage, the stable pixel coordinate of the marked point on the left or right jaw was recorded.Figure 19 shows the marked points.When determining the marked points, two conditions should be satisfied: (1) the marked points should be close to the midpoints of the executive ports E and Er; (2) the marked points should be prominent in their respective vicinity.By multiplying the pixel equivalent with the pixel coordinate difference, the X-and Z-axis displacements of point E, denoted as X E and Z E , respectively, were obtained experimentally (figure 20(a)).For the right jaw, the corresponding midpoint displacements X Er and Z Er are illustrated in figure 20(b).
According to figure 20, certain differences exist between X E and X Er , and similar differences exist between Z E and Z Er owing to the misalignment between the ideal F in and the axis of the piezoelectric stack actuator in the assembly.As the X-axis displacement of the jaw represents the grasping movement and the Z-axis displacement represents the parasitic movement, the ratio of Z E to X E or the ratio of Z Er to X Er reflects the parallel grasping error.For the left jaw, the average parallel grasping error was 3.05%.For the right jaw, the corresponding error was 8.01%.The primary cause of the difference between the two errors is the directional error of F in .Furthermore, by setting the zoom to 1.5 ×, the initial grey images of the two jaws and corresponding images after applying a voltage (150 V) were acquired using Pixelink Capture (figures 21(a)-(d)).In this case, the pixel equivalent was calibrated to 0.478 µm/pixel.Using Piotr's Computer Vision MATLAB Toolbox, an optical flow analysis of the two jaws was conducted, as shown in  figures 21(e) and (f), in which the positive direction of the Z g axis is opposite to that of the Z axis.Figures 21(e) and (f) implied that the X-axis grasping movements of the two jaws were nearly symmetric to F in , and the Z-axis parasitic movements approached zero.Thus, the parallel movement was realized by the microgripper.To reduce the systematic error arising from the directional error of F in , the average results of figures 20(a) and (b) were utilized.The piezoelectric stack actuator was calibrated using an inductive gauge (E75 LVDT, Coremorrow, China) and voltmeter.The calibrated results are listed in table 3.For the actuator, the analytical relationship between the nominal output displacement ∆ n and actual displacement ∆ a is where K PSA and K in denote the stiffness of the actuator and input stiffness of the microgripper, respectively.The FEA indicates that K in is 1.91 N/µm.Therefore, the experimental value of F in can be estimated using the following equation: Using equation (37) and table 3, the average results of figure 20(a) and (b) were transformed into a figure illustrating the relation between the jaws displacements and F in (figure 22).Furthermore, utilizing the jaws displacements presented in figure 22, the average experimental values of ς for the output structures can be obtained using equations ( 28), (30), (31) and (35) as shown in figure 23.
Figure 23 indicates that with an increase in F in , the average experimental values of ς for the output structures initially decrease and then fluctuate within a narrow range without further increasing.The average value is 9.82 µrad/µm.Therefore, the effect of geometrical nonlinearity on the orthogonal movement transformation accuracy is effectively suppressed.

Conclusion
In this study, a nonlinear design scheme for a single-stage CODA with a single input force was proposed, analyzed, and experimentally tested.First, the nonlinear design principle was described qualitatively.By combining closed-form analytical modelling, FEA, and numerical fitting, the nonlinear extent of the pre-set variable cross-sectional beam in the CODA was formulated.By combining the beam constraint model  and small-deflection-based modelling, the nonlinear extent of the undetermined uniform straight beam in the CODA was derived.Based on the design principles and nonlinear models, an enumeration-based nonlinear design scheme was proposed.The FEA results verified the effectiveness of the proposed scheme.Furthermore, based on [40], considering the geometrical nonlinearity, if a considerably large deflection degree of the pre-set variable cross-sectional beam exists, the orthogonal movement transformation cannot be realized; however, the FEA results indicate that selecting the effective undetermined uniform straight beams in multiple solutions based on the nonlinear design scheme can help maintain the orthogonal movement transformation accuracy.Therefore, the limitations of [40] were revealed.The experimental results indicate that, with an increase in the input force, the microgripper based on the nonlinear design result realized parallel movement, and the parasitic rotational movement of the output structures initially decreased and then fluctuated within a narrow range without increasing; the average parasitic rotational movement was 9.82 µrad/µm, which is negligible.Hence, the effectiveness of the proposed scheme was experimentally verified.

Figure 1 .
Figure 1.Single-stage CODA with a single input force.

Figure 2 .
Figure 2. Dimensions and force analysis of beam AB.

Figure 3 .
Figure 3. Illustration of the relationship between α and l/t for arbitrary material, input force F in , and L/tr (generating diverse slenderness ratios by keeping l constant and varying t).

Figure 5 .
Figure 5. Relationship between 1-β and l/t (F in = 10 N, generating diverse slenderness ratios by keeping l constant and varying t).

Figure 8 .
Figure 8. Nonlinear design scheme for a single-stage CODA with a single input force.

Figure 9 .
Figure 9.Geometrically nonlinear static FEA results of example II and the optimal design result in[40](F in = 10 (N).

Figure 10 .
Figure 10.Values of ς and δ in example II and the optimal design result in[40] regarding different input forces (using geometrically nonlinear static FEA).
are revealed.With respect to different input forces, the ς and δ values of examples III-V are shown in figure 13.The average values

Figure 11 .
Figure 11.Geometrically nonlinear static FEA results of example I and the initial group in [40](F in = 10 (N).

Figure 12 .
Figure 12. ς and δ values of example I and the initial group regarding different input forces (using geometrically nonlinear static FEA).

Figure 14 .
Figure 14.Geometrically nonlinear static FEA results of example VI (using geometrically nonlinear static FEA).

Figure 16 .
Figure 16.Grasping and Z-axis parasitic movements of point E.

Figure 19 .
Figure 19.Marked points on the left and right jaws.

Figure 20 .
Figure 20.X-axis and Z-axis displacements of the midpoints of the executive ports in the two jaws.

Figure 22 .
Figure 22.Relation between the jaws displacements and F in .

Figure 23 .
Figure 23.Average experimental values of ς for the output structures.

Table 1 .
Pre-set structural parameters of each simulation example.

Table 2 .
l CC1 and u values of each simulation example (utilizing the nonlinear design scheme).

Table 3 .
Calibrated results of the piezoelectric stack actuator.