A novel bridge-type compliant displacement amplification mechanism under compound loads based on the topology optimisation of flexure hinge and its application in micro-force sensing

The design and modelling of bridge-type compliant displacement amplification mechanisms (CDAMs) are key components in precision engineering. In this study, a bridge-type CDAM under compound loads with an optimum flexure hinge configuration is designed, analysed, and tested. For the case when the flexure hinge configuration is unknown, the internal force distribution for a bridge-type CDAM under compound loads is analysed, and the topology of the flexure hinge is optimised. By applying different volume constraints, the optimised flexure hinge configurations are all V-shaped. Subsequently, a static model of the V-shaped flexure hinge is established. For a bridge-type CDAM with V-shaped flexure hinges, the compliance matrix of the flexure hinge is combined with the relationship among the local compliance matrices in a serial mechanism; consequently, the analytical relationship between the output displacement, output force, and input force is derived. The CDAM is parametrically optimised to further improve the output performance. Simulations and experiments verify the topology optimisation result, static model, and parametric optimisation result. Finally, the CDAM and its static model are applied to the tensile manipulation and micro-force sensing in a microfiber tensile test.

The design and modelling of bridge-type compliant displacement amplification mechanisms (CDAMs) are key components in precision engineering.In this study, a bridge-type CDAM under compound loads with an optimum flexure hinge configuration is designed, analysed, and tested.For the case when the flexure hinge configuration is unknown, the internal force distribution for a bridge-type CDAM under compound loads is analysed, and the topology of the flexure hinge is optimised.By applying different volume constraints, the optimised flexure hinge configurations are all V-shaped.Subsequently, a static model of the V-shaped flexure hinge is established.For a bridge-type CDAM with V-shaped flexure hinges, the compliance matrix of the flexure hinge is combined with the relationship among the local compliance matrices in a serial mechanism; consequently, the analytical relationship between the output displacement, output force, and input force is derived.The CDAM is parametrically optimised to further improve the output performance.Simulations and experiments verify the topology optimisation result, static model, and parametric optimisation result.Finally, the CDAM and its static model are applied to the tensile manipulation and micro-force sensing in a microfiber tensile test.
The design and modelling of bridge-type CDAMs are key to optimising their performance and sensing external microforces, and numerous studies have focused on these aspects.For example, Lobontiu and Garcia [13] utilised Castigliano's second theorem to model the displacement amplification ratio of a lumped bridge-type CDAM.Ma et al [14] formulated the displacement amplification ratio of a lumped bridge-type CDAM based on the instantaneous centre method and pseudorigid-body model, and derived the spring stiffness in the pseudo-rigid-body model according to the linear elastic beam theory.Qi et al [15] derived the average ideal displacement amplification ratio of a lumped bridge-type CDAM during the deformation process based on the improved instantaneous centre method, and subsequently adopted the linear elastic beam theory to analyse the deformation of flexure hinges, which was used to formulate the displacement amplification ratio in a concise form.Ling et al [16] used empirical formulas to describe the influence of stress concentration on the axial and rotational stiffness of the flexure hinge, thereby improving the accuracy of the displacement amplification ratio model.Liu and Yan [17], by incorporating the effect of external loads, established a theoretical model to predict the displacement amplification ratio based on the linear elastic beam theory.Wei et al [18] established a general linear elastic model for lumped and distributed bridge-type CDAMs that can describe static properties.Choi et al [19] proposed a linear elastic model for the displacement amplification ratio of a lumped bridge-type CDAM, wherein all members are assumed to be elastic.Kim et al [20] designed a three-dimensional bridge-type CDAM with uniform flexure hinges, and obtained a high theoretical displacement amplification ratio for a small size.Xu and Li [21] designed a compound bridge-type CDAM to improve the output stiffness.Using topology optimisation, Clark et al [22] developed bridge-type structures to maximise the output displacement, whereas the study made the unrealistic assumption that the subsequent mechanism, after the bridge structure, had a predetermined stiffness, in order to simplify the analysis.Huo et al [23] proposed a modified bridge-type mechanism with asymmetric stiffness, which can obtain coupling driving motion easily.Ling [24] proposed a general two-port dynamic stiffness model for lumped and distributed bridgetype CDAMs that can accurately predict the former three-order natural frequencies.Dong et al [25] designed a highly efficient bridge-type CDAM based on negative stiffness.Chen et al [26] proposed two semi-analytical models considering the shearing effect and geometric nonlinearity to predict the output displacements of a bridge-type CDAM with a single input force and to perform nonlinear optimisation.Finally, Li et al [27] established a nonlinear analytical model of the displacement amplification ratio of a bridge-type CDAM based on the beam constraint model.
The researches presented above focus on two aspects: (1) enhancing the prediction accuracy of performance indexes by accurately establishing kinematic, static, or dynamic models of bridge-type CDAMs, and (2) improving the input/output performance of bridge-type CDAMs through proper configuration design.In these studies, most of the CDAMs belonged to lumped compliant mechanisms, wherein the flexure hinges were the primary elastic members.The performance of the lumped bridge-type CDAM can be effectively improved by properly designing the topological configuration of the flexure hinges, and some studies have attempted this approach.For example, Dong et al [28] utilised various flexure hinges in a lumped bridge-leverage CDAM to reduce displacement loss and improve load capacity.In addition, Chen et al [29] designed a bridge-type CDAM with circular axis leaf-type flexure hinges by considering the axis curvature of the flexure hinge and the dominant internal force; consequently, their designed bridge-type CDAM had an increased output displacement compared with the traditional bridge-type CDAM with the largest stroke.
Traditional researches on the design of flexure hinges in bridge-type CDAMs rely on empirical design or existing topological configurations, such as circular and leaf-type flexure hinges [30][31][32].However, these studies do not consider whether the topological configuration is optimal.Topology optimisation is an approach that determines the best material distribution in a given design domain that can minimise a given cost function while satisfying a series of constraints [33,34].A few researchers have utilised topology optimisation to design flexure hinges.For example, Liu et al [35] proposed a quasi-V flexure hinge via implementation of a topology optimisation method and using the rotation accuracy as the objective function.This quasi-V flexure hinge had a higher rotation accuracy compared with the filleted-V flexure hinge.Similarly, Qiu et al [36] designed a single-axis flexure hinge with a novel threedimensional configuration by applying the three-dimensional continuum topology optimisation theory and taking the maximum compliance along the deflection as the objective function; compared with the leaf-type flexure hinge, the rotation range herein was increased by 300%.
By using a topology optimisation method to design flexure hinges, empirical dependence on the existing topological configuration can be avoided.Nonetheless, existing researches on the topology optimisation of flexure hinges generally do not consider the actual internal force distribution of compliant mechanisms, which affects the effectiveness of topology optimisation results in compliant mechanisms.In bridgetype CDAMs, the general load boundary conditions are that, the input and output ports are subjected to forces, namely, compound loads.Therefore, to overcome the aforementioned shortcomings, the aim of this study is to design, analyse, and test a bridge-type CDAM under compound loads based on the topology optimisation of the flexure hinge considering the actual internal force distribution, with which the output performance of bridge-type CDAM can be maximised under compound loads to improve the manipulation flexibility in micronano-manipulation effectively.
The remainder of the study is organised as follows.In section 2, the internal force distribution for a bridge-type CDAM under compound loads is analysed, and topology optimisation of a flexure hinge in the CDAM is performed.In section 3, the static model is established for the topology optimisation result; the model is further used to derive the input-output analytical relationship of the bridge-type CDAM with compound loads based on the topology optimisation of the flexure hinge.In section 4, parametric optimisation of the CDAM is performed to further improve the output stroke.In sections 5 and 6, the topology optimisation result, static model, and parametric optimisation result are verified by finite element simulations and experiments, respectively.In section 7, the proposed bridge-type CDAM and its static model are applied to a microfiber test.Finally, in section 8, the conclusions are presented.

Internal force analysis
To obtain the boundary conditions for the topology optimisation of a flexure hinge in a lumped bridge-type CDAM under compound loads, the internal force distribution of the CDAM must be analysed.The lumped bridge-type CDAM is symmetric with the input force F in and output displacement e out simultaneously, as shown in figure 1(a), where F out denotes the external force acting on the output port.Because an internal force analysis is used to construct the boundary conditions, all the elastic members in the CDAM are defined as undetermined.A quarter of the CDAM can be used to conduct the internal force analysis, wherein the undetermined flexure hinges A and B, and the intermediate beam between the two flexure hinges form a variable cross-sectional beam A 1 B 1 , as shown in figure 1(b).For a lumped bridge-type CDAM under compound loads, only the in-plane degrees of freedom are considered; therefore, the corresponding displacement boundary conditions of beam A 1 B 1 are θ A1 = 0, y A1 = 0, θ B1 = 0, and x B1 = 0.For beam A 1 B 1 , θ A1 = 0.According to Castigliano's second theorem, The strain energy U A1B1 is expressed as a combination of the axial, bending, and shearing terms, as follows.
where l j (j = A, B); and L, E, G, A, I, and α are the length of flexure hinge j, the length of intermediate beam, the Young's modulus, the shear modulus, the cross-sectional area, the cross-sectional moment of inertia, and the shear coefficient, respectively.The axial force F is related only to F xB1 , and the shearing force N is only related to F yB1 , and the bending moment M is a function of F xB1 , F yB1 , and M B1 .Equation ( 2) is substituted into equation (1) as follows.

Topology optimisation of the flexure hinge
Generally, in a lumped bridge-type CDAM under compound loads, all flexure hinges are the same.Thus, flexure hinge A can be selected as an example to conduct the topology optimisation, for which the design domain is defined as a rectangular area, as shown in figure 2. The left side of the design domain is a fixed non-design domain, and the right midpoint is subjected to a horizontal force F x and vertical force F y , corresponding to the axial and lateral forces acting on A, respectively.In addition, two forces, F 1 and F 2 , act on two nodes near the right midpoint along the y-axis.Defining that F 1 = F 2 , and the directions of F 1 and F 2 are inverse, a moment acting on the right midpoint is generated, corresponding to the moment acting on A.
The design domain is discretised into m × n planar four-node rectangular elements.According to the variable density method, the topology optimisation model of A is as follows.
find ρ = (ρ 1,1 , ρ 1,2 , . . ., ρ i,j ) where ρ i,j is the density of element, V is the volume fraction of remaining material to the design domain, and V * is the upper limit of V.The compliance C is formulated as: where F, U, u, K, k e , and p are the global node force vector, global node displacement vector, element node displacement vector, global stiffness matrix, element stiffness matrix, and penalty factor, respectively.To enhance the compliance of the flexure hinge, the objective function of the topology optimisation model in equation ( 8) maximises C. With a large compliance of the flexure hinge, large output stroke can be obtained using the CDAM, which achieves high manipulation flexibility.Equation ( 8) is a single-object topology optimisation problem, which can be solved using the optimality criteria method [37].To make the contour of the optimisation results smooth, the contour plot matrix, generated by the MATLAB 'Contour' function, is used to output the optimal result.In addition, in post-processing, the MATLAB 'Heaviside' function is used to suppress the unclear edge in the optimal result caused by grey elements and is defined as follows.
where the filter threshold λ = 0.According to figure 3, under different volume constraints, the topology optimisation results of flexure hinge A are all V-shaped, with two beams in parallel.Figure 4 indicates that under different V * values, when the number of iteration steps increases, the objective function gradually decreases to a convergent value.The topology optimisation results are effective in terms of convergence.
It was found that when V * = 0.4, and F x and F y were set to 1 and 0.1, respectively, the topology optimisation results for different F 1 and F 2 values were all V-shaped.The topology optimisation results for different F y values when V * = 0.4, and F x , F 1 , and F 2 are set to 1, are shown in figure 5.It can be seen that as F y increased, the two beams forming  the V-shaped structure in the optimisation result gradually move closer together and tend toward a leaf-type flexure hinge.Similar results can be obtained for the other values of V * .In summary, the topology optimisation results indicate that in a bridge-type CDAM under compound loads, V-shaped flexure hinges are optimum.

V-shaped flexure hinge
The V-shaped flexure hinge is shown in figure 6(a), where the out-of-plane thickness is denoted by h.If only the in-plane deformation is considered, the generalised forces acting on the right midpoint G are F xG , F yG , and M zG .Essentially, By denoting the rotation of the flexure hinge at point G as θ G , the generalised displacement column vector at point G is: Further, using the compliance matrix method, the relationship between X G and F G can be modelled as: The active generalised forces and local coordinate systems for beams 1 and 2 in figure 6 are shown in figure 7.
Because the V-shaped flexure hinge is composed of beams 1 and 2 in parallel, the compliance matrix of the flexure hinge at point G, C VG , can be formulated as:  where C 1M is the compliance matrix of beam 1 at its right midpoint, C 2N is the compliance matrix of beam 2 at its right midpoint, R 1 and R 2 are the coordinate rotation transformation matrices from coordinate systems x 1 y 1 and x 2 y 2 to coordinate system x G y G , respectively, and T 1 and T 2 are the corresponding coordinate translation transformation matrices.
Beam 1 can be modelled according to Castigliano's second theorem: When the shearing effect is ignored, the strain energy U 1 can be expressed as: where I 1 and A 1 are the cross-sectional moment of inertia and the cross-sectional area, respectively.For beam 1, the displacement-load relation at the right midpoint is: Combining equation (15) with equation ( 16), the compliance coefficients in equation ( 17) can be derived as: Similarly, C 2N can be expressed as: where the compliance coefficients are: For the compliance matrix C VG , the element in the first row and first column C VG1-1 represents the axial compliance of a V-shaped flexure hinge.For a V-shape flexure hinge, figure 6(b) illustrates the corresponding leaf-type flexure hinge, for which the axial compliance C leaf can be formulated as a/Eh(b 1 cos θ 1 +b 2 cos θ 2 ).Numerical analysis shows that, using different values of a, E, h, b 1 , θ 1 , b 2 , and θ 2 , the ratio of C VG1-1 and C leaf is always smaller than 1, which infers that compared with the traditional leaf-type flexure hinge, Vshaped flexure hinge is of higher axial stiffness to reduce the parasitic axial deformation.

Input-output relationship of the CDAM
A quarter of a bridge-type CDAM based on V-shaped flexure hinges is illustrated in figure 8.
According to the force analysis shown in figure 8, for beam A 1 B 1 , the relationship between the generalised displacements at point B 1 and the generalised forces can be expressed as: By referring to equation ( 14), the compliance matrix C A of flexure hinge A at point A 2 can be derived.Similar to the derivation of equation ( 14), the compliance matrix C B of flexure hinge B at point B 1 is obtained, as follows.
where the coordinate translation transformation matrices T 3 and T 4 are: Beam A 1 B 1 is composed of flexure hinge A, an intermediate beam, and flexure hinge B in series.The compliance matrix of beam A 1 B 1 at point B 1 is: where T is the coordinate translation transformation matrix from point A 2 to point B 1 .
If F out = 0, then F yB1 = 0. Substituting F xB1 = 1/2F in and F yB1 = 0 into equation (21), and considering the central symmetry of the 1/4 structure and its moment equilibrium equation, the analytical relation between x A1 and F in as well as that between y B1 and F in is derived as: If F in = 0, then F xB1 = 0. Substituting F yB1 = -1/2F out and F xB1 = 0 into equation (21), similarly, the relation between x A1 and F out as well as that between y B1 and F out can be formulated as: Because the works done by the input force W in and output force W out are transformed into the total elastic potential energy V of CDAM, where W in = 2F in x A1 = 2F in (x A1-Fin +x A1-Fout ).Substituting equations ( 25) and ( 27) into the formulation of W in gives: The parameter W out is expressed as: For flexure hinge A, when F out = 0, the elastic potential energy V A-Fin , which includes the axial and rotational components, can be expressed by equation (32) as: where K A is the stiffness matrix of flexure hinge A at point A 2 , K Aij represents the elements of the ith row and jth column in K A (i = 1, 2, 3; j = 1, 2, 3), and K A and C A are the reciprocal matrices.For flexure hinge A, the moment acting on point A 2 is M A2-Fin = t r F xB1 .When F in = 0, the elastic potential energy V A-Fout is: where the moment acting on point A 2 is M A2-Fout = (a + L) F yB1 .
For flexure hinge B, when F out = 0, the elastic potential energy V B-Fin can be expressed as: where K B is the stiffness matrix of flexure hinge B at point B 1 , K Bij represents the elements of the ith row and jth column in the stiffness matrix K B of flexure hinge B at point B 1 , and K B and C B are reciprocal matrices.For flexure hinge B, the moment acting on point B 1 is M B1-Fin = F xB1 t r /2.When F in = 0, the elastic potential energy V B-Fout is expressed as: where the moment acting on point B 1 is M B1-Fout = F yB1 (2a+L)/2.From the above, V is formulated as: According to equations ( 29)- (36), the coupling relationship between F in , F out , and e out can be obtained as: where α, β, and δ are:

Parametric optimisation
In section 2, the topological configuration of the flexure hinges that enables lumped bridge-type CDAMs under compound loads to obtain the maximum output stroke was discussed.However, the output stroke of a lumped compliant mechanism depends not only on the topological configuration of the elastic members but also on the structural parameters.Therefore, parametric optimisation of a bridge-type CDAM based on Vshaped flexure hinges is presented in this section.
To maximise the output displacement of the CDAM, the objective function is defined as minimise -e out .The output displacement e out can be derived from the input-output relationship shown in equation (37), where F in > F out .When modelling the input-output relationship, the intermediate beams were regarded as rigid bodies, and the beams in the V-shaped flexure hinges were viewed as Euler-Bernoulli beams.Thus, two constraints are obtained: (1) the bending stiffness of the intermediate beams should be much greater than that of the V-shaped flexure hinges; and (2) the lower limits of the length-diameter ratio of the beams in the V-shaped flexure hinges should not be smaller than 5. Furthermore, both beams in the flexure hinge were viewed as the beam structure; therefore, the ratio of h to the beam width and length should be limited.In addition, the lower limits of b 1 and b 2 are limited by fabrication capacity.To further improve manipulation flexibility, the CDAM should be compact.Hence, the upper limits of a, L, and t r are constrained.From the above, the parametric optimisation model obtained is as follows.
Equation ( 41) is a constrained optimisation problem, which can be solved by the MATLAB 'fmincon' function.If the limits b min , a max , L max , and t rmax are designed to be 1 mm, 6 mm, 15 mm, and 5 mm, respectively, and the material chosen is aluminium alloy (Young's modulus: 71 GPa), the optimisation results are the same under different initial groups and for different F in and F out values, as listed in table 1.In the MATLAB 'fmincon' function, an output variable 'exitflag' can reflect the convergence of the optimisation.Under different initial groups and for different F in and F out values, the corresponding values of 'exitflag' are all 1 or 3, indicating the convergence of the optimisation.

Finite element simulation
In this section, small deflection-based static finite element simulations, which can be conducted using the commercial finite element analysis (FEA) software ANSYS Workbench, are used to verify the topology optimisation result, inputoutput relationship model, and parametric optimisation result.Besides, when comparing the bridge-type CDAM based on V-shaped flexure hinge with the traditional one, modal FEA is utilised to test the dynamic performance.In the FEA, free meshing is utilised, and the mesh size is default.The type of elements is Solid 186.

Topology optimisation results
In existing studies, the bridge-type CDAM with leaf-type flexure hinges is generally viewed to have the maximum output stroke.To verify the effectiveness of the topology optimisation result, a bridge-type CDAM with leaf-type flexure hinges was adopted as the comparison group.For two types of bridgetype CDAMs, the width of the leaf-type flexure hinges is equal to b 1 cos θ 1 + b 2 cos θ 2 , which is the equivalent width of the V-shaped flexure hinges, as shown in figure 6.In both bridge-type CDAMs, for the structure except for the flexure hinges, the dimensions are the same correspondingly.The boundary conditions in the static FEA for the initial and comparison group 1 are presented in figure 9(a).In the static FEA, the material is set as aluminium alloy, and the corresponding FEA results for the initial and comparison group 1 are shown in figures 9(b)-(e).Compared with the comparison group 1, the output displacement of initial group 1 is improved by 30.42%, while the maximum Von-Mises stress σ max is improved by 77.34%.For initial and comparison group 1, stress concentration occurs at all the flexure hinges, which is consistent with the general performance of lumped compliant mechanisms.In the modal FEA, the corresponding simulation results are shown in figures 9(f) and (g).Compared with the comparison group 1, the first-order natural frequency f 1 of initial group 1 is decreased by 16.89%.
The quantitative FEA results of initial groups 1-3 and their comparison groups are listed in table 2. Compared with the comparison groups, the output displacements of the initial groups 1-3 increase by 26.49% on average.Therefore, the topology optimisation result of flexure hinges based on the internal force analysis of a bridge-type CDAM under compound loads is verified.Under a certain driving force, the bridge-type CDAM with V-shaped flexure hinges achieves larger output displacement, in comparison with the CDAM with leaf-type flexure hinges.The proposed CDAM benefits to the application with the output stroke mainly limited by the maximum driving force.Besides, the stress analysis and modal analysis show that, if the output stroke is mainly limited by the allowable stress or the operating bandwidth range is crucial, the bridge-type CDAM with leaf-type flexure hinges is still a better choice, in comparison with the CDAM with Vshaped flexure hinges.

Input-output relationship model
To verify the input-output relationship model in equation (37), two design examples of the CDAM with different geometric parameters were used, as summarised in table 3, wherein the material is set as aluminium alloy.Ten groups of F in and F out values were applied in the static FEA.For the design examples, a comparison between the FEA results and theoretical values is presented in tables 4 and 5, where the theoretical values of e out were calculated using equation (37).For design example 1, when loading group 7 is applied, the boundary conditions are as shown in figure 10(a), and the FEA result is as shown in figure 10(b).Compared with the FEA results, the maximum  relative error of the theoretical values is less than 13.80%, and the average relative error is 6.47%.Therefore, the input-output relationship model of the bridge-type CDAM with V-shaped flexure hinges is found to be acceptable.

Parametric optimisation result
In the static FEA, for the initial groups and parametric optimisation result summarised in table 1, the boundary conditions     the parametric optimisation result.Therefore, the optimisation model in equation ( 41) is verified.Equation ( 37) was used to calculate e out for the initial groups and optimisation result, as presented in table 6.Compared with the FEA results, the maximum relative error is 10.21%.Hence, the effectiveness of the input-output relationship model is further verified.

Experimental verification
Based on the parametric optimisation results in section 4, an experimental prototype is designed and analysed in this section.Simultaneously, a bridge-type CDAM prototype with leaf-type flexure hinges is fabricated for comparison.The topology optimisation result is verified by testing the performance of the two prototypes.Furthermore, the validity of inputoutput relationship model is tested by comparing the theoretical and experimental values.

Design and analysis of the experimental prototype
Generally, bridge-type CDAMs are driven by piezoelectric stack actuators.When mounting the actuators in the CDAMs, the mounting errors will lead to the direction error of F in .For the prototype of the parametric optimisation result, two fixedguided beams are added to both sides of the input structure, as shown in figure 12, with which the out-of-plane deformation induced by the direction error of F in can be reduced.Besides, with the guiding beams, the structural stability is improved, which benefits reducing the fabrication error.
For fixed-guided beam CD, the force analysis is illustrated in figure 13, where y D = 0 and the rotation θ D = 0. Using Castigliano's second theorem, the linear elastic relation between the generalised displacements at point D and the generalised loads is formulated as follows.The strain energy U CD consists of the axial deformation and bending deformation terms: where A CD = ht CD is the cross-sectional area and I CD = th 3 CD /12 is the moment of inertia.Substituting equation (43) into equation (42), Furthermore, by substituting θ D = 0 into equation (45), By combining equation (44) with equation ( 46), the analytical relationship between x D and F xD is derived as: According to the force analysis in figure 13(b) and the static equilibrium equation, The analytical relationship between the vertical displacement at point A 1 x A1 , F xB1 , and F yB1 is deduced by the superposition of x A1-Fin -F xB1 and x A1-Fout -F out relation, which can be formulated by referring to equations ( 25) and ( 27), as: where Assuming that the input structure is rigid, x D is equal to x A1 .Using a combination of equations ( 47) and (49), and x D = x A1 , the following is obtained.
Therefore, for the experimental prototype, equation ( 30) is modified as: Using a combination of equations ( 54), ( 29) and ( 31)-( 36), the coefficients α, β, and δ in equation ( 37) are modified; the modified formulations for α, β, and δ of the bridgetype CDAM prototype with leaf-type flexure hinges can be obtained in a similar manner.

Experimental setup
The 7075 aluminium alloy was selected as the material, and two experimental prototypes (designed in section 6.1) were fabricated by electrical discharge machining.Considering the parametric optimisation result as an example, the experimental prototype is shown in figure 14.A piezoelectric stack actuator (40Vs15, Coremorrow, China) provides F in for the prototype.The piezoelectric controller E00.D6, Coremorrow, China) supplies a high-precision input voltage to the actuator.Preload bolt 1 (M2×0.4) is used to adjust the preload between the actuator and the CDAM, thus eliminating the clearance between the actuator and input structure.Preload bolt 2 (M3×0.4) is used to eliminate the clearance between the left sides of output structure and auxiliary part.Besides, F out is applied when preload bolt 2 is adjusted.To avoid friction between the auxiliary part and experimental prototype, a clearance is ensured between the lower surface of the auxiliary part, upper surface of the framework, and upper surface of the output structure.When a certain voltage is applied to the actuator, the auxiliary  part moves towards left with the output structure.A marked point exists on the upper surface of the auxiliary part, and the point position is in the red circle in figure 14.Therefore, the displacement of the marked point is equal to that of the output structure.To measure F in , an input force sensor (MD-3, Bengbu Sensor, China) is mounted between the actuator and preload Bolt 1.The accuracy is 0.3%.An output force sensor (XMT-808, Bengbu Ocean Sensor, China), which measures F out according to the static equilibrium relationship, is mounted between the right surface of the framework and preload bolt 2 with an accuracy of 0.1% of the range.
In this study, optical micro-metrology, a technique widely used in precision engineering owing to its advantages of contactless measurements and low cost [39][40][41], was implemented to measure e out .An overview of the experimental setup is presented in figure 15(a).The assembly is fixed on a three-degrees-of-freedom (3-DOF) micro-motion platform consisting of a z-axis motion stage (ZM07A-S3K, KOHZU, Japan) and an xy-axes motion stage (YM10A-C3-CL, KOHZU, Japan).For the z-axis motion stage, the maximum motion resolution is 0.25 µm, which enables precise focusing.For the xy-axes motion stage, the maximum motion resolution is 0.5 µm, which can precisely adjust the in-plane position of the prototype.The hardware of the optical micro-metrology system is composed of a charge-coupled device camera (PL-D755, PixeLINK, Canada), z-axis adjustment stage, zoom lens (1-60 191, Navitar, USA), objective lens (1-60 227, Mitutoyo, Japan), fibre-optic power supply (1-60 563, Navitar, USA), flexible light pipe, and PC.Both the 3-DOF micro-motion platform and the hardware of the optical micro-metrology system are fixed on a vibration isolation table (Zolix, China).The Pixelink Capture software package is  used to capture the images corresponding to the deformation of the CDAM and determine the pixel coordinates of the marked point.Figure 15(b) presents the marked point in the field of view, which is a black circle.In a 2D image measuring system, pixel equivalent is an important parameter to describe the relationship between pixels of digital images and actual size of measured piece.When the zoom is set to 2.5×, using the Pixelink Capture software package and a calibration board (R1L1S1P, THORLABS, USA), the pixel equivalent of the optical micro-metrology system was calibrated to 0.293 µm/pixel, as shown in figure 16, in which the reciprocal of calibrated pixel equivalent is presented, namely 3.415 pixels µm −1 .

Experimental process and results
For both prototypes, the preload between the actuator and CDAM, called 'input preload' hereinafter, was set as 10 N, and the preload between the auxiliary part and the output end, called 'output preload' hereinafter, was set as 1 N. Next, a voltage starting from 0 V and ending at 60 V was applied to the actuator in increments of 5 V.For each voltage, the stable pixel coordinates of the marked point were recorded.By multiplying the pixel equivalent with the pixel coordinate difference, e out was obtained experimentally.The experimental results are presented in tables 7 and 8.The average experimental results of the prototype for the parametric optimisation result under three groups of preload (table 9) are summarised in tables 10-12.Based on the experimental results listed in tables 7 and 10-12, the experimental relationship between F in , F out , and e out was fitted using the Matlab 'Fitting Toolbox', as follows.
It is found that when F in = 2.33 N and F out = 0.24 N, the output displacement of bridge-type CDAM based on V-shaped    Furthermore, according to tables 7 and 10-12, when the applied voltage is 15-60 V, the average relative error of the theoretical e out calculated using equation (37) is 7.92% compared with the experimental e out .Therefore, the effectiveness of the input-output relationship model is verified.In addition, tables 7 and 10-12 infers that, using the proposed CDAM and derived model, F out with relatively small value can be estimated by F in with relatively large value and e out , which benefits the micro-force sensing.

Application of the bridge-type CDAM with V-shaped flexure hinges and its static model in a microfiber test
When a bridge-type CDAM is applied to conduct microfiber tensile test, the mechanism is subjected to loads both at the input and output ends.The input force F in is supplied by an actuator, and F out is exerted by the tensile test.Correspondingly, in this section, the bridge-type CDAM under compound loads with an optimum flexure hinge configuration, V-shaped flexure hinge, is applied in the tensile test of a nylon microfiber (WEGO, China), wherein the derived static model is utilised.The application is used to further validate the inputoutput relationship model.
The test platform is illustrated in figure 17, wherein each end of the microfiber is bonded to an aluminium alloy sheet.
One sheet is fixed to the output structure of the experimental prototype for the parametric optimisation result, and the other is fixed to an auxiliary component.The experimental prototype is assembled with a piezoelectric stack actuator, an input force sensor, and preload bolt 1, and mounted on a 3-DOF micro-motion platform.The auxiliary component is fixed on an XY-axes micro-motion stage.With the assistance of optical micro-metrology, the 3-DOF micro-motion platform and XYaxes micro-motion stage are adjusted such that both ends of the microfiber are kept at the same height, and the microfiber is kept straight.In addition, both the micro-motion platform and stage are fixed on the bases, which are fixed on a vibration isolation table.
The output displacement e out , which is equal to the elongation of the microfiber, was measured through the experimental process described in section 6.3 Additionally, the diameter of the microfiber can be measured by optical micrometrology.According to the input-output relationship model as well as the measured F in and e out , the value of F out , which is equal to the tensile force, was obtained.For the nylon microfiber with a diameter of 51.32 µm, the stress-strain curve was further obtained, as shown in figure 18.
The stress-strain curve is nearly linear, corresponding to the linear elastic stage; hence, the Young's modulus can be determined as the slope.For the nylon microfiber, the Young's modulus is estimated to 1.304 GPa on average.Besides, the tensile test of the microfiber was conducted by a  commercial platform with direct force sensing (Suzhou Intellrising Technology Co., Ltd China, as shown in figure 19(a)).In figure 19(b), the blue solid line presents the calibrated stress-strain curve, and the red dash line is the initial tangent of the loading segment in the curve; therefore, the slope of the red dash line equals the calibrated Young's modulus, namely 1.176 GPa.The estimated value approaches the calibrated value.Therefore, the Young's modulus of a microfiber can be accurately measured by the tensile test platform based on the bridge-type CDAM with V-shaped flexure hinges, and the derived input-output relationship model is further validated.

Conclusion
In this study, a bridge-type CDAM under compound loads with an optimum flexure hinge configuration was designed, analysed, and tested.The internal force distribution property of a bridge-type CDAM under compound loads was analysed and combined with the variable density method, by taking the maximum compliance as the objective function, to conduct topology optimisation of the flexure hinge.Under different volume constraints, the topology optimisation results of the flexure hinges were found to be all V-shaped.Based on the matrix displacement method, Castigliano's second theorem, and the relation among the local compliance matrices of a parallel mechanism, a static model of the V-shaped flexure hinge was established.Using the compliance matrix of the V-shaped flexure hinge, internal force distribution property, and relation among the local compliance matrices of a serial mechanism, the analytical relationship between the input force, output force, and output displacement of a bridge-type CDAM with V-shaped flexure hinges was derived.Parametric optimisation of the CDAM was performed to further improve the output stroke.
In the FEA and experimental verification, a bridge-type CDAM with leaf-type flexure hinges was used as the comparison group.The FEA results show that compared with the comparison groups, the output displacements of the bridge-type CDAMs with V-shaped flexure hinges increased by 26.49% on average when using the same input and output force.Therefore, the topology optimisation result of the flexure hinge was validated.Additionally, the maximum relative error of the input-output relationship model, when compared with the simulation values, was found to be less than 13.80% for two groups of bridge-type CDAMs based on V-shaped flexure hinges with different dimensions.The simulation results show that the output displacement of each initial group was smaller than that of the parametric optimisation result when the same input and output forces were applied.Hence, the parametric optimisation model was verified.
It was also found that compared with the comparison groups, a bridge-type CDAM with V-shaped flexure hinges can generate a larger output displacement with an average increase of 25.43% when using the same input and output forces in the experiments.These results further verified the validity of the topological-optimisation result.Furthermore, the average relative error of the theoretical output displacement calculated by the input-output relationship model, when compared with the experimental results, was 7.92%, thus verifying the accuracy of the input-output relationship model.Experimental results revealed that, using the proposed CDAM and derived model, the external force acting on the output port with relatively small value could be estimated by the input force with relatively large value and output displacement, which benefits the micro-force sensing.
The bridge-type CDAM with V-shaped flexure hinges and its static model were further applied to the tensile manipulation and micro-force sensing in a microfiber tensile test.The test results showed that, the Young's modulus of the nylon microfiber was found to be 1.304 GPa on average, approximating to the calibrated value (1.176 GPa).Therefore, the derived input-output relationship model was further validated.
The limitation of this research is that, small deflection assumption was applied in the topology optimisation and static models, whereas geometrical nonlinearity was not considered.Besides, Euler-Bernoulli beam assumption was utilised in the static models.In our future work, the geometrically nonlinear topology optimisation of the flexure hinges in the bridgetype CDAM under compound loads will be conducted, and the input-output relationship will be formulated considering geometrical nonlinearity and Timoshenko beam equations, which benefits the miniaturisation of bridge-type CDAM under compound loads.

Figure 1 .
Figure 1.Bridge-type CDAM under compound loads with undetermined elastic members and force analysis of beam A 1 B 1 .

Figure 2 .
Figure 2. Design domain and boundary conditions of the topology optimisation.

3 .
Setting m = 40, n = 40, Young's modulus E ijkl = 1, and Poisson's ratio µ = 0.3, the topology optimisation results when F x = 1, F y = 0.1, and F 1 = F 2 = 1, are shown in figure 3, and the corresponding iteration values of C are shown in figure 4. In general, the volume constraints influence the topology optimisation results of flexure hinges [38].Therefore, in figure 3, the topology optimisation results under different volume constraints are presented.

Figure 4 .
Figure 4. Iteration values of C during the topology optimisation.

Figure 6 .
Figure 6.Schematics of a V-shaped flexure hinge and the corresponding leaf-type flexure hinge.

Figure 7 .
Figure 7. Active generalised forces, dimensions, and local coordinate systems of beams 1 and 2.

Figure 9 .
Figure 9. FEA of initial and comparison group 1.

Figure 10 .
Figure 10.FEA of design example 1 with loading group 7.

Figure 11 .
Figure 11.FEA of initial group 2, initial group 3 and the parametric optimisation result.

Figure 12 .
Figure 12.Experimental prototype of the bridge-type CDAM with V-shaped flexure hinges.

Figure 13 .
Figure 13.Force analysis of the local structure in the experimental prototype.

Figure 14 .
Figure 14.Assembly of the experimental prototype, piezoelectric stack actuator, force sensors, and auxiliary part.

Figure 15 .
Figure 15.Experimental setup and its field of view.

Figure 17 .
Figure 17.Microfiber tensile test platform based on the bridge-type CDAM with V-shaped flexure hinges.

Figure 18 .
Figure 18.Stress-strain curve obtained by the platform in figure 18 and input-output relationship model.

Figure 19 .
Figure 19.Commercial microfiber tensile test platform with direct force sensing and calibrated stress-strain curve of the nylon microfiber.

Table 1 .
Initial groups and optimisation result (the units of θ 1 and θ 2 are '•', and the units of other parameters are 'mm').

Table 2 .
Quantitative FEA results of initial groups 1-3 and their comparison groups.

Table 3 .
Design examples of the bridge-type CDAM with V-shaped flexure hinges (the units of θ 1 and θ 2 are '•', and the units of other parameters are 'mm').

Table 4 .
Comparison between the FEA results and the theoretical values of eout for design example 1.

Table 5 .
Comparison between the FEA results and the theoretical values of eout for design example 2.

Table 6 .
Theoretical and simulation eout of the initial groups and parametric optimisation result (F in = 1 N, Fout = 0.1 N).

Table 7 .
Experimental and theoretical eout of the prototype for the parametric optimisation result (input preload: 10 N, output preload: 1 N).

Table 9 .
Input preload and output preload.

Table 10 .
Experimental and theoretical eout of the prototype for the parametric optimisation result using group 1 in table9

Table 11 .
Experimental and theoretical eout of the prototype for the parametric optimisation result using group 2 in table9 flexure hinges is improved by 24.63% compared with the bridge-type CDAM with leaf-type flexure hinges.For the other groups of F in and F out in table 8, the average increase is 25.43%.Therefore, the topology optimisation result of the flexure hinges based on the internal force analysis of bridgetype CDAM under compound loads is verified.

Table 12 .
Experimental and theoretical eout of the prototype for the parametric optimisation result using group 3 in table 9.