This paper presents a geometric synthesis method for compliant mechanisms based on similarity transformation of pole maps. Motion generation is a typical and common mechanism synthesis task, so this study takes it as the design requirement to expound the proposed method. Most of the current research work relies on numerical solution of the nonlinear Bernoulli–Euler beam model, numerical simulations or physical experiments to study the synthesis method of compliant mechanisms. There is a lack of simpler and more efficient methods to achieve motion generation of compliant mechanisms with various topologies. This study is based on pole map which is a geometric tool to describe the motion of rigid-body mechanisms. In this paper, we first demonstrate the feasibility of applying the similarity transformation of pole map to compliant mechanisms. It is proved that the pole map of compliant mechanisms has the same characteristic as rigid-body mechanisms during similarity transformation. Then we present the procedure of synthesis method in detail and expound the establishment method of function module which can avoid the functional defects of the final designed mechanism. At last, we take the compliant geared linkages and compliant four-bar linkage as examples to illustrate the novel synthesis approach. The result is an applicable and effective synthesis method for motion generation of compliant mechanisms.

The compliant mechanism can be defined as a mechanism that obtains some of its motion by means of the deformation of elastic elements (Howell, 2001). If the length of the compliant section is similar to the length of the rigid section, the geometric nonlinearity caused by large bending displacements must be considered (Kimball and Tsai, 2002). At present, researchers have developed many methods for analyzing the large deflection of flexural beams, such as elliptical integral method, beam constraint model, Adomian decomposition method, pseudo-rigid-body model (PRBM), and non-linear finite element analysis method (FEA). Based on these studies, researchers can design compliant mechanisms. Midha et al. (1994) introduced the kinematics-based design method. The theoretical approach using the numerical solution of the nonlinear Bernoulli–Euler beam model is one of the most common analysis and design methods, but its calculation process is very complicated (DeBona and Zelenika, 1997). Howell and Midha (1994) created the synthesis approach based on PRBM, and it can provide a practical means for analyzing and designing the compliant mechanism. Li et al. (2017) proposed 3R PRBM method and used it to analyze and design compliant mechanisms. In the field of compliant linkage synthesis, Bagivalu Prasanna et al. (2020) introduced a methodology to determine the possible deflected configuration of a compliant four-bar mechanism for a given set of load or displacement boundary conditions. Valentini and Pennestri (2018) presented an original methodology for addressing the kinematic synthesis of a compliant four-bar mechanism by using PRBM. Alqasimi et al. (2016) designed a linear bistable compliant crank–slider mechanism by using PRBM to solve the kinetic and kinematic equations. Hanke et al. (2015) presented a graphical synthesis algorithm and used this method to solve two-position synthesis of planar compliant mechanisms. Huang and Schimmels (2019, 2020) researched the synthesis method of compliant five-bar linkages and presented a geometric approach to the passive realization of any given planar compliance with a six-joint fully serial or six-spring fully parallel mechanism. Topology optimization method has been widely used in the design of distributed compliant mechanisms (Zhou and Mandala, 2012; Patiballa and Krishnan, 2017; Liu et al., 2017), but it commonly leads to complex structures that may cause higher manufacturing costs using elastic material. Diab and Smaili (2017) presented an ants-search-based method for optimum synthesis of compliant mechanisms under various design criteria. Choi and Cho (2020) presented a configuration and sizing design optimization method for large deformation planar compliant mechanisms.

Since it is difficult to analyze the displacement of elastic elements or flexural beams during the deformation, the synthesis method of compliant mechanisms is usually not as systematic as rigid-body mechanisms. Therefore, the novel efficient and simple approach is the focus in the research field of compliant mechanism synthesis. The motion generation of compliant mechanisms can be used to solve some practical engineering problems, such as using compliant mechanisms to replace rigid-body mechanisms for avoiding disadvantages of rigid-body mechanisms or achieving the guidance task of some light-weight products. Thus, the goal of this study is to explore a new method for motion generation of compliant mechanisms. Lin et al. (2018) proposed a method to describe the finitely separated positions by using the pole map. Taking this method as the fundamental principle, Lin et al. (2018) use pole maps as unified geometric identification between the guidance mechanisms and given positions. By performing similarity transformation on a certain basic mechanism, a geometric synthesis approach for planar motion generation is proposed. This method can be applied to the design of rigid-body mechanisms with various types. However, this geometric synthesis approach has not been used in the design of compliant mechanisms, and it lacks practical and effective calculation procedures for the synthesis of more than three positions. Therefore, still using the pole maps as geometric tool to describe planar positions, this paper will study the feasibility of the similarity transformation of compliant mechanisms and propose a novel method for motion generation of compliant mechanisms.

Although the principle of similarity transformation of pole map has been applied to the motion analysis of rigid-body mechanisms, it is not clear whether it can be applied to compliant mechanisms. Hence this paper first demonstrates the feasibility of applying this principle to compliant mechanisms. The geometric similarity of deformation curves, which are formed by flexural beams with different lengths, is discussed, and it is proved that the pole map of compliant mechanisms has the same characteristic as rigid-body mechanisms during similarity transformation. Then this paper presents the detailed procedure of synthesis method. Next, the function module is introduced as the basis of mechanism synthesis, which can achieve the functional characteristics required by the design. Finally, this study illustrates the synthesis approach with two examples of three-position and four-position motion generation. Compared with other methods, the synthesis method proposed in this paper is based on geometric similarity transformation, and hence it can avoid the overly complicated mathematical calculation. Function module is introduced at the beginning of the design process, so this method can ensure that the final solution mechanism will 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. In addition, this method is not limited to the design of a specific type of compliant mechanism, and it can be applied to compliant mechanisms with various topological structure.

Prescribed positions

Motion positions and pole map.

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

Position description by the pole map.

For any rigid-body mechanism, the position

Geometric similarity transformation of rigid-body mechanism.

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

Large deflection of a flexural beam subject to arbitrary end loads.

Scale transformation of the flexural beam.

If this flexural beam is scaled by the scaling factor

Subject to the end load

Substituting the above calculation results and Eq. (10) into Eq. (9), we
have

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 linkage

Geometric similarity transformation of the compliant mechanism.

According to static analysis of link

Therefore, for the compliant mechanism

According to Eq. (13), the driving force of the transformed mechanism can be
calculated, i.e.,

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

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.

Analyzing the motion generation task specified by the design
requirement, the given finitely separated positions of the final solution
mechanism can be obtained so that the pole map

By geometric similarity transforming

Main contents of the geometric approach.

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.,

Geometric transformation of pole map.

This paper establishes the function module of compliant mechanisms and takes it as the basic object of geometric similarity transformation. At the beginning, we need to choose a compliant mechanism as a normalized function module that is likely to achieve the functional characteristics required by the design. Then we will analyze the normalized function module and obtain its kinematic equations expressed by variables. Finally, by using the kinematic equation, dimension parameters of the mechanism are adjusted and determined so that the function module can satisfy the functional characteristics of design requirements.

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.,

3R pseudo-rigid-body model.

The nondimensional forward kinematic equations are

The nondimensional statics equations are

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

Compliant geared linkage.

Closed-loop equation of compliant four-bar mechanism can be decomposed into

The force and moment that link

The formula of guidance angle

The trajectory equation of guidance point

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

The effect of parameters on rotation angle of the guidance link.

The effect of parameters on trajectory of the guidance point.

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

In this section, two applications of exact three-position synthesis and fuzzy four-position synthesis are presented to illustrate how the proposed approach is applied.

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

Parameters of the design.

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:

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

The pole map of the given separated position of the design task can be
calculated by substituting the data of Table 1 into Eq. (1). Taking

Through geometric transformation, pole map

Function module and pole map.

Solution mechanism.

Deformation of the flexure beam.

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

Parameters of the design.

Function module and pole map.

Solution mechanism.

Deformation of the flexure beam.

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

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

The pole map of the given separated position of the design task can be
calculated by substituting the data of Table 2 into Eq. (1). Taking

Through geometric transformation, pole map

Relationship between input angle and pole order.

Relationship between guidance order and pole order.

Pole maps are a geometric tool that can accurately describe multiple planar positions, and it can reveal the relationship between guidance mechanisms and given design tasks. Based on similarity transformation of pole maps, Lin et al. (2018) proposed a new approach to rigid-body mechanism synthesis and used geometric way to find the approximate solution of multi-position motion generation. Also based on similarity transformation of pole maps, this paper expands its application field and proposes a novel synthesis method for compliant mechanisms. In addition, this paper illustrates the establishment and adjustment method of the function module of compliant mechanisms and presents a practical calculation procedure for four-position synthesis through numerical example.

The first example is a practical design task of three-position motion generation, and the second example is four-position synthesis. As shown in Sect. 5.1, this method can be used to solve some practical engineering problems, such as using compliant mechanisms to replace rigid-body mechanisms for avoiding disadvantages of rigid-body mechanisms or achieving the guidance task of some light-weight products.

Compared with other methods, the synthesis method proposed in this paper is
based on similarity transformation of pole map, so it is unique and has the
following features.

Function module is introduced at the beginning of the design process, and hence this method can ensure that the final solution mechanism will satisfy the functional characteristics and transmission characteristics of design requirements. As long as we can complete kinematic and static analysis of a certain type of compliant mechanism, this type of mechanism can be chosen as a function module. Therefore, this method is not limited to the design of a specific type of compliant mechanism, and it can be applied to compliant mechanisms with various topological structure.

This method is based on similarity transformation of pole map, so it can avoid defects by traditional synthesis methods based on Burmester kinematic geometry theory, such as order defect. The following is the detailed discussion.

Suppose there are four motion positions:

If there are two guidance tasks which have the same given positions and
different guidance order, as shown in Fig. 20, obviously the shapes of the
two pole maps are the same, but the pole order is different. Usually, we
label motion positions and poles in the guidance order, and these two pole
maps

This paper proposes a novel geometrical approach to compliant mechanism synthesis based on similarity transformation of pole maps. The study demonstrates the feasibility of applying the geometric similarity transformation to the compliant mechanism, and it is proved that the pole map of compliant mechanisms has the same characteristic as rigid-body mechanisms during similarity transformation. Then this paper proposes the procedure of synthesis method and expounds the establishment method of function module. In addition, this work illustrates the synthesis approach with two examples.

All the code used in this paper can be obtained from the corresponding author upon request.

The data are available upon request from the corresponding author.

SL and YZ proposed the methodology. YZ and HW wrote the paper. JJ and NM took part in the discussion of the paper.

The authors declare that they have no conflict of interest.

This paper was edited by Engin Tanık and reviewed by two anonymous referees.