Prescribed positions EiUi,βii=1,2,…,n can be described by n-1 poles Pjii=1,2,…,nandi≠j relative to the initial position EjUj,βj and their corresponding rotation angles (Lin et al., 2018). The shape formed by these poles is defined as pole map. As shown in Fig. 1, the initial position E1 can reach E2, E3 and E4 by rotating around P12, P13 and P14 respectively, so the triangle P12P13P14 forms the pole map. Given any two plane positions EjUj,βj and EkUk,βk, the pole Pjk can be determined by a complex number as (Lin et al., 2018) 1Pjkαjk=i2Uke-iαjk-Ujeiαjksin⁡αjk, where αjk=βjk/2=βk-βj/2.

The displacement of finitely separated positions can be represented by a pole map. If the two pole maps are geometrically identical (i.e., the pole coordinates and corresponding angles in the first pole map are equal to those in the second pole map respectively), then the two displacements represented by the two pole maps are the same. As shown in Fig. 2, the displacement of plane positions U1, U2 and U3 determines the pole map P12P13, and the displacement of plane positions R1, R2 and R3 determines the pole map Q12Q13. If P12=Q12, α12=γ12, P13=Q13, α13=γ13, then the two series of displacements are the same, so positions Ui and Ri can be considered in the same motion plane Ei. Therefore, the pole map can be regarded as the identification for motion planes.

For any rigid-body mechanism, the position EiUi,βi of a certain component during the motion can be described by the initial position E1U1,β1 and pole P1i, and the pole map determined by the continuous displacement of the component is a pole curve. If the mechanism is subjected to geometric similarity transformation including translating, rotating and scaling, to obtain a new mechanism, then the finitely separated positions E1E2…En of the original mechanism will be transformed to new positions E1′E2′…En′. The relationship between the original pole map P12P13…P1n and the new pole map P12′P13′…P1n′ is geometric similarity, which is the same as the similarity relationship between the two mechanisms. For example, in the planar mechanism A0A1B1B0 shown in Fig. 3, the initial angle is ϕ=π/2, and P1i is the pole of guidance link A1B1 in the motion position of ϕ=ϕi. Mechanism A0′A1′B1′B0′ is obtained by geometrically transforming the original mechanism A0A1B1B0, and P1i′ can be obtained by calculating the pole of link A1′B1′ in the motion position of ϕ′=ϕi. Then the pole curve Π formed by P1i is geometrically similar to the pole curve Π′ formed by P1i′.

As shown in Fig. 3, the rigid-body mechanism has geometric similarity during the geometric transformation process, while the pole maps before and after the transformation also have the same geometric similarity. However, because of the complexity of the force and deformation, the feature of compliant mechanisms during the geometric transformation cannot be directly analogized from the law of rigid-body mechanisms. In order to study the feasibility of applying the similarity transformation of pole maps to compliant mechanism, it is necessary to demonstrate that the compliant mechanism and its pole maps both have the geometric similarity in translating, rotating, and dimension scaling, just like the transformation feature of rigid-body mechanism shown in Fig. 3. First, this paper will analyze the deformation of the flexural beam which satisfies the Bernoulli–Euler beam equation.

Figure 4 shows the deformation of a cantilever beam subject to a combined end force and external bending moment. The coordinate frame is established by taking the fixed end as the coordinate origin and the direction of initially straight beam as the positive direction of x axis. The end force F can be decomposed into a horizontal force Fx and a vertical force Fy. The coordinates of the free end Q are (a,b). Point A, whose position is denoted by (x,y), is an arbitrary point on the beam, and the curvature at A is 2κ=MAEI=Fya-x-Fxb-y+M0EI. The curve equation of the beam deformation can be expressed by arc length s, that is, rs=xsi+y(s)j. Differentiating Eq. (2) with respect to s, we obtain 3κ′s=1EIFxy′-Fyx′. According to the differential geometry, the formula x′′2+y′′2=κ2 is known. Differentiating this formula with respect to s yields 4x′′x′′′+y′′y′′′=κκ′. Since x′2+y′2=1, it can be differentiated as x′x′′+y′y′′=0. We continue to differentiate this equation and get x′′2+x′x′′′+y′′2+y′y′′′=0. Substituting x′′2+y′′2=κ2 into the equation leads to 5x′x′′′+y′y′′′=-κ2. According to Eqs. (4) and (5), we have 6x′′′y′′′=x′y′x′′y′′-1-κ2κκ′=1x′y′′-x′′y′-κ2y′′-κκ′y′κ2x′′+κκ′x′. Next, we calculate x′y′′-x′′y′, the denominator of the right side of Eq. (6). Since x′2+y′2=1, the expression (x′y′′-x′′y′)2 can be expanded and derived as x′y′′-x′′y′2=x′2y′′2-2x′y′′x′′y′+x′′2y′2=y′′2-y′2y′′2-2x′y′′x′′y′+x′′2-x′′2x′2=x′′2+y′′2-x′x′′+y′y′′2. Substituting x′x′′+y′y′′=0 and x′′2+y′′2=κ2 into the above equation yields x′y′′-x′′y′2=κ2. Considering the direction of the tangent vector x′,y′ and the normal vector x′′,y′′ at point A, we obtain x′>0,y′>0,y′′>0,x′′<0,κ>0, and hence x′y′′-x′′y′=κ. Substituting the above formula into Eq. (6) gives 7x′′′=-κy′′-κ′y′y′′′=κx′′+κ′x′. As shown in Fig. 5, a flexible beam with a length of L is under the action of the end loads F1(Fx1Fy1) and M1. The position of the free end point is denoted by (a1,b1), and the curve equations of beam deformation are denoted by x1s and y1s. According to Eqs. (2), (3) and (7), the differential equations describing the curve can be calculated as shown in the following. 8x1′′′=-Fy1a1-x1-Fx1b1-y1+M1EIy1′′-Fx1y1′-Fy1x1′EIy1′y1′′′=Fy1a1-x1-Fx1b1-y1+M1EIx1′′+Fx1y1′-Fy1x1′EIx1′, where a1=x1L, b1=y1L. When 0<s<L, x1s and y1s satisfy the differential equations of Eq. (8).

If this flexural beam is scaled by the scaling factor λ while the material and section shape remain the same, the length of the new beam is λL and the flexural rigidity EI remains unchanged, as shown in Fig. 5.

Subject to the end load F2(Fx2,Fy2) and M2, the position of free end point is (a2,b2), and curve equations representing the new beam are determined by the following differential equations. 9x2′′′=-Fy2a2-x2-Fx2b2-y2+M2EIy2′′-Fx2y2′-Fy2x2′EIy2′y2′′′=Fy2a2-x2-Fx2b2-y2+M2EIx2′′+Fx2y2′-Fy2x2′EIx2′ Supposing 10x2s=λx1sλy2s=λy1sλ and then successively differentiating Eq. (10), we can obtain x2′s=x1′sλ, y2′s=y1′sλ, x2′′s=1λx1′′sλ, y2′′s=1λy1′′sλ, x2′′′s=1λ2x1′′′sλ, y2′′′s=1λ2y1′′′sλ. From Eq. (10), we can get a2=x2λL=λa1 and b2=y2λL=λb1.

Substituting the above calculation results and Eq. (10) into Eq. (9), we have x1′′′sλ=-λ2Fy2a1-x1sλ-λ2Fx2b1-y1sλ+λM2EI⋅y1′′sλ-λ2Fx2y1′sλ-Fy2x1′sλEIy1′sλy1′′′sλ=λ2Fy2a1-x1sλ-λ2Fx2b1-y1sλ+λM2EI⋅x1′′sλ+λ2Fx2y1′sλ-Fy2x1′sλEIx1′sλ, where 0<sλ<L. If the end load satisfies the functional relationship given by Eq. (11), the above equations are obviously the same as Eq. (8). 11Fx1=λ2Fx2Fy1=λ2Fy2M1=λM2 Therefore, under the condition that the free end load satisfies Eq. (11), if x1=x1s and y1=y1s are the solutions of differential equations in Eq. (8), then x2s=λx1sλ and y2s=λy1sλ are the solutions of differential equations in Eq. (9). That is, for two flexural beams whose lengths are L and λL respectively, when the free end load satisfies Eq. (11), the deformation curves of beams have geometric similarity which can be written as 12λr1s=r2λs. It is easy to prove that during the scaling process of the compliant beam, the position of the free end point corresponding to the same end angle is also scaled in the same way. Similarly, the end point trajectory before and after scaling is also scaled, as shown by Π1 and Π2 in Fig. 5.

The scaling of flexural beams is discussed in Sect. 3.1, and then this paper will study the geometric similarity of the whole compliant mechanism during geometric transformation. Taking a typical mechanism as an example, the planar linkageA0A1B1B0 is a compliant mechanism, as shown in Fig. 6. The initially straight flexural beam B0B1 is fixedly connected with rigid link A1B1, and C is an arbitrary point in the mechanism. It is assumed that this compliant linkage maintains a static balance throughout the entire range of motion. When the input angle ϕ=ϕ0, the force that link A0A1 applies to link A1B1 is denoted by FA, and the driving torque is denoted by M1. At this time, the moment and force which link A1B1 applies to link B0B1 are denoted by M0 and F, respectively.

According to static analysis of link A1B1 and A0A1, we obtain A1B1×-F-M0=0A0A1×-FA+M1=0. Using F=FA, the formula of M1 can be obtained by adding the above two equations, i.e., 13M1=M0+A0B1×F. If the initial compliant linkages A0A1B1B0 in the motion position of ϕ=ϕ0 are subjected to a geometric transformation which takes A0 as a base point, T as a translating factor, δ as a rotating factor and λ as a scaling factor, and at the same time the flexural rigidity EI remains unchanged, then the transformation will obtain a new compliant linkage A0′A1′B1′B0′. The relevant parameters are all shown in Fig. 6. After the transformation, the new position of any point C in the linkages is 14C′=A0+T+λC-A0eiδ. We assume that the compliant mechanism A0′A1′B1′B0′ shown in Fig. 6 is in static equilibrium. According to Eq. (14), if point C is on the flexural beam, then the deformation curve of B0′B1′ is B0′C′=B0C⋅λeiδ. Obviously, the deformation curves of flexural beam B0′B1′ and B0B1 have geometric similarity. Using Eq. (11) and geometric relationship illustrated by Fig. 6, we obtain 15F′=1λ2FeiδM0′=1λM0. Now we analyze the static balance of link A1′B1′. Using Eq. (14), we have A1′B1′=A1B1⋅λeiδ, and hence the moment of point A1′ can be calculated and written as A1′B1′×-FA′-M0′=A1B1⋅λeiδ×-1λ2Feiδ-M0′=1λM0-M0′. Since M0′=1λM0, the compliant mechanism A0′A1′B1′B0′ shown in Fig. 6 is in a state of static equilibrium.

Therefore, for the compliant mechanism A0A1B1B0 at any position in the motion range, if ϕ′=ϕ, then the transformed mechanism A0′A1′B1′B0′ and the original mechanism always have a geometric similarity relationship.

According to Eq. (13), the driving force of the transformed mechanism can be calculated, i.e., M1′=M0′+A0′B1′×F′. Substituting Eq. (15) into the expression of M1′ and simplifying, we obtain 16M1=λM1′. It can be seen that if the driving force satisfies the relationship of Eq. (16), the planar compliant mechanism has geometric similarity before and after the transformation of translating, rotation and scaling. The position coordinates of any point on the mechanism satisfy the transformation formula given by Eq. (14).

In the following, this paper will discuss the transformation rules of poles and pole maps during the geometric transformation of compliant mechanism. For any mechanism containing flexible component, which can be denoted by A0A1…An, the pole of two given motion position EjUj,βj and EkUk,βk of a link can be represented as Pjkαjk, and Eq. (1) is the formula for calculating the pole. If the initial compliant mechanism is subjected to a geometric transformation which takes A0 as a base point, T as a translating factor, δ as a rotating factor and λ as a scaling factor, then we will obtain a new compliant mechanism A0′A1′…An′. According to Eq. (14), two motion positions after geometric transformation can be written as Ej′Uj′,βj+δ and Ek′Uk′,βk+δ, where Uj,k′=A0+T+λUj,k-A0eiδ. The pole Pjk′ of motion positions Ej′ and Ek′ can be calculated by Eq. (1), and it can be written as Pjk′αjk′=i2A0+T+λUk-A0eiδe-iαjk′-A0+T+λUj-A0eiδeiαjk′sin⁡αjk′=i2Uke-iαjk′-Ujeiαjk′sin⁡αjk′λeiδ+i2A0+T-λA0eiδsin⁡αjk′(e-iαjk′-eiαjk′), where αjk′=[(βk+δ)-(βj+δ)]/2=αjk. The above equation can be simplified into the form 17Pjk′αjk′=A0+T+λPjk-A0eiδ. Eq. (17) shows that before and after the geometric transformation of any compliant mechanism, the poles are also subjected to the same geometric transformation, and obviously the pole maps of original mechanism and transformed mechanism always have a geometric similarity relationship. It can be concluded that if the flexural rigidity EI maintains unchanged, then the finitely separated motion positions, poles and pole maps are similar before and after the geometric transformation. This similarity law of compliant mechanisms is the same as that of rigid-body mechanisms shown in Fig. 3.

In Sect. 3.2, the geometric similarity of the pole map during the geometric similarity transformation of compliant mechanisms is demonstrated. Based on this theory, a novel geometric approach for compliant mechanism synthesis is proposed. Next, this paper will expound the new synthesis method in detail, which can be divided into three main contents as shown in Fig. 7.

According to the mechanism type or topological structure characteristics specified by design requirements, a certain type of nondimensional compliant mechanism is chosen. Then the dimension parameters of this mechanism are adjusted and determined so that the mechanism can satisfy the functional characteristics of design requirements, such as the trajectory shape of the guidance point, the rotation angle range of the guidance link, the workspace and the mounting position of the fixed frame, etc. This chosen nondimensional mechanism is defined as a function module, whose establishment will be discussed in Sect. 4.

The pole map of four-position motion generation is a triangle, as shown in Fig. 8, and the tolerance of the pole corresponding to the fuzzy position can be calculated by Eq. (1), i.e., 18ΔP=i2Ũke-iα1k-Uke-iα1ksin⁡α1kmax⁡=ΔU2sin⁡α1k. Assuming that given positions of the design task are E1, Ẽ2, E3 and E4, we can adjust the initial rotation angle ϕ1 of the driving link to obtain the appropriate pole map Q12Q13Q14, which is quasi-similar to the pole map P12P13P14. As shown in Fig. 8, after similarity transformation, Q12′Q13′Q14′ should be quasi-congruent to P12P13P14 while satisfying the condition P12-Q12′<ΔP.

This paper uses 3R PRBM to analyze the deformation of the flexural beam, as shown in Fig. 9. Su (2009) presents the optimized parameters, i.e., kΘ1=3.51, kΘ2=2.99, kΘ3=2.58, γ0=0.1, γ1=0.35, γ2=0.40, γ3=0.15.

The nondimensional forward kinematic equations are 19QxL=γ0+γ1c1+γ2c12+γ3c123QyL=γ1s1+γ2s12+γ3s123θ0=Θ1+Θ2+Θ3, where L is the actual length of the flexural beam, c1=cos(Θ1), c12=cos(Θ1+Θ2), c123=cos(Θ1+Θ2+Θ3), and s1, s12, s123 are also similar abbreviations.

The nondimensional statics equations are 20fxL2EIfyL2EIMLEI=kΘ1Θ1kΘ2Θ2kΘ3Θ3J-1, where [J]-1 is written as (Su, 2009) J-1=1γ1γ2s2⋅γ2c12γ2s12γ2r3s3-γ1c1-γ2c12-γ1s1-γ2s12-γ3γ2s3+γ1s23γ1c1γ1s1γ1γ2s2+γ3s23.

This paper will take the compliant geared linkage as an example to illustrate the establishment method of the function module. In the compliant four-bar mechanism A0ABB0, a pair of gears are mounted on the driving link A0A, as shown in Fig. 10. The rigid-body coupler AB is fixedly connected to the initially straight flexural beam BB0, and there is a constant angle π/2-β0 between these two links. Gear GR is fixedly connected to AB, and gear Gr is connected to the guidance link CU. All absolute dimensions should be converted into relative dimensions that are ratios of component dimensions to the frame length L0, so as to obtain the normalized function module. For example, x3=L3/L0, where L3 denotes the length of flexible beam BB0. All other parameters are labeled in Fig. 10.

Closed-loop equation of compliant four-bar mechanism can be decomposed into 21x1cos⁡ϕ+x2cos⁡β0+θ0+x3QyL3-1=0x1sin⁡ϕ+x2sin⁡β0+θ0-x3QxL3=0, where θ0, QxL3 and QyL3 can be calculated by Eq. (19)

The force and moment that link AB applies to link BB0 are denoted as Fx, Fy and M, as shown in Fig. 10. The compliant mechanism is usually applied to products with small external load or uses rigid parts as working component such as multistable compliant mechanisms, so the external load applied by the link CU can be set equal to 0. Then the moment balance equation is 22-AB×Fxi+Fyj-M=0, where the magnitude of AB is the actual length of link AB, namely L2. According to the angles and positions shown in Fig. 10, we can decompose vectors of Eq. (22) and obtain 23-L2cos⁡β0+θ0Fy+L2sin⁡β0+θ0Fx-M=0. According to the direction of forces in Figs. 9 and 10, we have Fx=-fy and Fy=fx, so Eq. (23) can be rewritten into the nondimensional form 24x2x3cos⁡β0+θ0fxL32EI+x2x3sin⁡β0+θ0fyL32EI+ML3EI=0. Substituting the results of Eq. (20) into the Eq. (24) and then combining with Eq. (21), we can get three equations of unknown variables Θ1, Θ2 and Θ3, which represent the three angles in PRBM. Given the crank angle ϕ, we can use Newton–Raphson method to calculate the numerical solution of Θ1, Θ2 and Θ3 and obtain the value of θ0 by substituting the calculated numerical solution into Eq. (19).

The formula of guidance angle βϕ can be derived, that is 25βϕ=1+ρρϕ-β0+θ0ρ+βr, where βr is a constant related to the initial position of the link CU, and ρ=r/R.

The trajectory equation of guidance point U can be written as 26Ux=x4cos⁡ϕ+δ+x5cos⁡βUy=x4sin⁡ϕ+δ+x5sin⁡β. The deformation of the function module should be within the possible deformation range of the flexural beam, otherwise the final solution will not meet the practical requirements.

In order to obtain the final function module of the compliant mechanism, we need to find the appropriate mechanism dimension according to the functional requirements of motion generation task. The general functional requirement is the guidance characteristic of the mechanism, such as the trajectory shape of the guidance point, the rotation angle range of the guidance link, the functional relationship between the input angle and the output angle, etc. Because the guidance characteristic changes with the dimension, it is necessary to analyze the effect of the dimension on the output motion of the guidance component. If the output motion is not sensitive to the dimensional changes of some components, then the dimensional constraints of these components can be appropriately reduced. On the contrary, if the output motion is sensitive to the dimensional changes of some components, the effect of these dimensions on guidance characteristics needs to be analyzed in detail. If the design task is too special or the functional requirements is too strict, we cannot find an appropriate function module that satisfies the functional requirements specified by the design task.

Through computer mathematical software or interactive geometry software, we can easily adjust dimension parameters of the function module and analyze the kinematics visually in real time so that we can intuitively understand the effect of different dimensions on the functional characteristics. Taking the compliant mechanism shown in Fig. 10 as an example, this paper will present the process of analyzing the effect of dimensions on the guidance characteristic. Now we set all dimensions of this mechanism as x1=0.6, x2=1.2, x3=1.4, x4=0.4, x5=0.4, β0=45∘, δ=90∘, ρ=1 and βr=0∘ and then adjust the parameters respectively while keeping other parameters unchanged. The functional relationship between the guidance rotation angle and the input angle can be calculated by Eq. (25), and the trajectory of guidance point U can be obtained from Eq. (26), whose results are shown in Figs. 11 and 12.

Obviously, the compliant geared linkage has the characteristic of a large range of guidance angle that is primarily linear with the input angle, and the change of main parameters does not affect the basic type of guidance characteristic. It can be seen from Fig. 11 that x1 and x3 have little effect on β, x4 does not affect β, and ρ has a large effect on β. It can be seen from Fig. 12 that x1 and x3 have little effect on the trajectory of the guidance point, x4 can scale the trajectory curve, and δ can rotate the trajectory. Therefore, the dimension of the basic compliant four-bar linkage has less effect on guidance characteristic. When selecting dimensions of the function module, it is necessary to preferentially adjust parameters of the gear part according to the given design requirements, so as to obtain a suitable mechanism. In addition, we should determine the appropriate initial position of the function module so that range of guidance angle must meet the condition 27Δβ≥βkmax⁡-βkmin⁡, where βkmax⁡ and βkmin⁡ denote the maximum and minimum angle of the design requirement, and Δβ is the range of guidance angle.

The task is to design a three-station transfer device, which is used to detect some light-weight products on the production line. These three stations are three fixed planar positions, and the device should make the product reach these positions accurately. At position E1, the product on the production line is automatically placed in the pallet of device. At position E2, the product is detected, scanned and recorded by other equipment. At position E3, the product is taken out. In order to avoid interference with the workspace of the detection, loading and unloading equipment, the design task requires that the mounting position of the transfer device should be far away from the three stations. The three given positions are shown in Table 1.

Considering that the overall size of the transfer device is small, in order to reduce the weight of the structure and the friction caused by some revolute joints, we will use compliant mechanism as the basis of this device. According to the design requirements, we can choose a compliant geared linkage to achieve three-position motion generation. The mechanism type is shown in Fig. 13: A0B0 is the driving link, CU is the guidance link installed with the pallet, and the gear ratio is 1. We will use aluminum alloy to make the flexible beam B0B, and the angle between flexible beam and the rigid-body coupler AB is 45∘. The three given positions E1, E2 and E3 are labeled in Fig. 13. The synthesis procedure can be divided into the following steps.

The compliant geared linkage is chosen as normalized function module. Its basic kinematic formulas are given by Eqs. (25) and (26), and relevant dimensional parameters are marked in Fig. 10. Following the method presented in Sect. 4.3, we can adjust the dimension parameters so that the function module will satisfy the functional requirement. Since the three guidance positions are required to be away from the flexural beam, we can adjust some parameters to change the position of the guidance trajectory, as shown in Fig. 12. The appropriate dimension can be determined as x0=100, x1=60, x2=120, x3=140, x4=40, x5=40, δ=60∘ and βr=0∘. According to Eq. (27), the initial rotation angle ϕ1 of the driving link can be set to ϕ=90∘.

The design task is four-position synthesis of compliant four-bar linkage. The four positions are demonstrated in Table 2, and these positions are marked in Fig. 16. The tolerance of fuzzy position Ẽ2 is given as ΔU=0.01.

The synthesis procedure can be divided into the following steps.

According to the mechanism type specified by design requirements, a compliant four-bar linkage shown in Fig. 16 is chosen as normalized function module, and coupler AB is the guidance link. x3 represents the length of initially straight flexural beam, x2=x3, β0=45∘. Given rotation angle ϕ, we can calculate the deflection angle θ0 of the flexural beam by following the method presented in Sect. 4.2, and hence we can get the guidance angle β=β0+θ0. Setting point B as the guidance point, its position is Bx=x1cos⁡ϕ+x2cos⁡β and By=x1sin⁡ϕ+x2sin⁡β.

Then we adjust the dimension parameters so that the function module will satisfy the angle range required by design task. The appropriate dimension can be determined as x0=1, x1=0.4, x2=2, x3=1.4.