In the present study, Fourier theory is applied to establish the expression of rigid-body poses of a spherical four-bar crank slider rigid-body guidance mechanism. According to an analysis of the harmonic components of the trajectory curve and rigid-body rotation angle, it has a certain relationship with the geometric parameters of the mechanism. On this basis, the rigid-body poses are normalized by preprocessing. Then, the rotation angle of the curve around the

With the constant advancement of mechanical manufacturing technology and the widespread application of linkage mechanisms, the dimensional synthesis of linkage mechanisms (especially spatial linkage mechanisms (Wei et al., 2013; Liu et al., 2023) becomes increasingly important. In general, the dimensional synthesis of linkage mechanisms entails motion generation, path generation, and function generation.

The purpose of motion generation or rigid-body guidance is to generate a linkage mechanism required to provide guidance via a prescribed sequence of poses (Sandor et al., 1984). Due to the motion generation required for both the position and the orientation, it is highly complex to conduct the process of motion generation. There are many studies conducted on how to achieve motion generation (Lin, 2013). By applying the Buchberger algorithm, Li and Chen (1996) proposed the synthesis of planar four-bar rigid-body guidance. Avilés et al. (1994) put forward an optimum method of synthesis for planar mechanisms. It is applicable to achieve dimensional synthesis for several types of mechanisms and kinematics such as function generation, path generation, motion generation, or a mix of the above types. Hayes and Zsombor-Murrary (2004) developed a general algorithm by means of kinematic mapping. As for rigid-body guidance, this general algorithm combines type and dimensional synthesis of planar mechanisms. Proposed by Yoon and Heo (2012), the constraint force design method enables topology optimization for a planar rigid-body guidance mechanism. With the advantage of pole, inversion, and overlay techniques (Bagci, 1984), a synthesis method for the function, path and motion generation of spherical mechanisms as indicated by Bagci (1984) can be applied in other mechanisms such as spherical slider crank, spherical crank rocker, and spherical cam follower. Lee and Russell (2007) and Lee et al. (2009) proposed a method that is applicable to describe the rigid-body positions of spherical four-bar mechanisms. Ruth and McCarthy (1999) developed a computer-aided design (CAD) system according to Burmester's planar theory and further proposed a motion generation method for the design of spherical linkages. Through the Burmester curve used in computer modeling and geometric construction for the synthesis of spherical mechanisms, Shirazi (2007) synthesized the 4R spherical linkage capable of guiding an antenna meeting four specified postures in a 3D workspace. Alizade et al. (2013) put forward a motion generation method for the synthesis of spherical linkages by providing rigid-body guidance for the spherical four-bar mechanism. Russell and Sodhi (2001) proposed a new method of synthesis for the multiphase motion generation of adjustable revolute–revolute–spherical–spherical (RRSS) mechanisms. Subsequently, the method was widely applied to the synthesis of a spatial RRSS mechanism for achieving phases of both precise and tolerances of rigid-body positions (Russell and Sodhi, 2002). Yu et al. (2007) presented a new computer method to approximately synthesize a four-bar path mechanism according to the coupler-angle function curve. Myszka et al. (2010) proposed a method to achieve dimensional synthesis for the motion generation of a linkage mechanism with five prescribed positions (Myszka and Murray, 2010). Subsequently, four-bar linkage rigid-body guidance synthesis was achieved by using the guidance-line rotation method (Wang et al., 2002). For planar mechanisms, a dimensional synthesis method was proposed by Peñuñuri et al. (2011) based on differential evolution.

Concerning the motion generation of linkage mechanisms, the outputs are periodic functions when the input link rotates continuously, which is irrelevant to the position or angle of the mechanism. The output of spatial mechanisms can be indicated by the Fourier series expression on the basis of Fourier transform. Fourier series theory was first introduced by Meyer zur Capellen (1954) to the analysis and synthesis of planar linkage mechanisms. Subsequently, Chu and Sun (2010) and Mullineux (2011) applied Fourier theory to explore the dimensional synthesis of a spherical four-bar mechanism, proposing an approach to the path generation of a spherical four-bar mechanism. In line with Fourier series theory, Sun et al. (2012) established the uniform model for the dimensional synthesis of linkage mechanisms including planar, spherical, and spatial mechanisms. They also illustrated the geometric significance attached to the harmonic characteristic parameter of the coupler curve and output function curve.

Currently, the Fourier series methods have been widely used in the function and path synthesis of spatial mechanisms and spherical mechanisms (Sun and Chu, 2010, 2008; Chu and Cao, 1993), not the motion generation of a spherical crank slider mechanism (Sun et al., 2012). Herein, the above theory is used to achieve the motion generation of spherical four-bar crank slider mechanism. The size of the target mechanism is optimized by using a genetic algorithm, which is capable of motion generation.

Motion generation or rigid-body guidance synthesis aims to obtain a series
of prescribed rigid-body poses, which involves quantitative or
qualitative design. There are two parts (orientation and position of
rigid body) involved in the rigid-body guidance-line output of linkage
mechanisms. As shown in Fig. 1, the spherical arc

The position and direction of the rigid body.

Figure 2 shows the geometrical model of mechanism in the Cartesian coordinates

As shown in Fig. 3b and c, the coordinate

As shown in Fig. 3b and c, the (

Model of spherical four-bar crank slider rigid-body guidance mechanism.

The transformation of the coordinate system.

The position of point

Therefore, the position of point

The position of point

Figure 4 shows the spherical mechanism in a general position of installation. The translation and rotation of the frame can be described by

Installation position of the spherical mechanism.

According to Eq. (17), the projection of the position of point

In the previous work, the dimensional synthesis of the spherical four-bar
slider crank mechanism was studied, and a numerical atlas method was
proposed to identify the basic dimensional types of the mechanism. Besides this,
the output of the linkage mechanism was described by harmonic parameters, and
the design results were obtained by comparing the prescribed design
requirements with that in the numerical atlas database. However, the feature
parameter extraction algorithm restricts the proposed method into solving
only the rigid-body guidance synthesis of spherical linkage mechanisms in
particular positions, where the rotation axis of the input component is
parallel to the

As shown in Fig. 5,

Schematic graph of the preprocessing.

Then, it is possible to calculate

Through rotation and transformation, the above method is used to eliminate
the influence rotation of the frame around the

From Fig. 6, we can get the orientation output as follows:

The rigid-body guidance line

According to Fourier series theory, the function of

Similarly, when the initial angle is denoted as

When the initial angle is indicated by

According to Eq. (29), the Fourier series expansion of

If the initial angle of the crank

Apparently Eqs. (33), (38), and (44) are implicitly related to Eqs. (37), (43), and (47). After the normalization of Eqs. (33), (38), (44), (37), (43), and (47), different factors such as the actual size, initial angle, and Installation position are eliminated. Besides this, the general law of the harmonic component of the orientation output and the mechanism parameters is determined.

After dividing Eq. (33) by

Similarly, after dividing Eq. (38) by

By comparing Eq. (50) with Eq. (51), it can be found out that they have the
same amplitude accordingly (except for item 0 and

Through the above analysis, the harmonic components of the position output are
expressed by Eqs. (48) and (50) (except for individual items). Also, the function

According to Eq. (31), the orientation output function

The translation of the orientation output function.

According to Eq. (47), the Fourier series

MOBDT involves all the parameters of MPBDT by comparing them. Thus, as the first step, a number atlas database of the orientation output is established by inputting both the OHCPDT and MOBDT. Then, fuzzy identification is performed to recognize several groups of MOBDT that satisfy the prescribed orientation output of the guidance mechanism as studied in this paper. Accordingly, a position problem of the rigid body can be transformed into a path generation problem when a reference point is chosen. According to Chu and Sun (2010), the characteristic parameters of both the design conditions and the MPBDT can be obtained by taking advantage of a fast Fourier transform (FFT). Subsequently, the actual length size and installation parameters can be calculated according to those parameters and theoretical formulas of the guiding mechanism as studied in this paper.

By comparing Eqs. (33) and (38) with (37) and (43), respectively, the
relationship between the amplitudes and phase angles can be determined, to
derive the theoretical formulas about actual installation size, coupler
point position, and installation size parameters. In this paper, the
prescribed position

The harmonic component of the angular characteristic function of the rigid body
is defined as

The actual dimension of the crank (

The initial angle

The rotation angle

The spherical radius of the mechanism:

The central angle

The translation distance

The translation distance

Flowchart of the synthesis process.

Continued.

It is widely known that the parameters of the MOBDT have different effects on
the output of the rigid body. Firstly, one group of MOBDT is chosen to
verify the influence of each parameter. By changing each parameter with the
same variable

The MOBDT with different parameters.

The step size of OHCPDT in the output properties database is denoted as

According to Chu and Sun (2010), the proper MOBDT can be obtained by the following equation:

Based on the fuzzy theory, the similarity function makes a survey of the
comparability.

Genetic algorithms represent one of the most adaptive methods to simulate the process of biological evolution on a computer. Its solution to the optimization problem starts by randomly generating the initial population that meets the constraints. Each individual in the population is taken as the first solution to the problem. Then, the selection, proliferation, crossover, mutation, and other genetic operations are conducted. By eliminating the basic size function which is clearly different from the objective function, a smaller basic size function is retained, and a new population is obtained. The new group members have less error than the previous generation group, which is significantly better than the previous generation. Through this continuous breeding evolution, the genetic algorithm is used to optimize the target mechanism.

The main steps are as follows:

Given the size type, frame length, two connecting rod lengths, and the crank length that need to be optimized, the initial population size rate, mutation rate, and maximum algebra are set.

Given the constraints and range (the corresponding ranges are given in Table 2) of the spherical four-bar rigid-body guidance, the constraints are

The size of the initial population of the spherical four-bar rigid-body guidance mechanism is set in random generation (Step 2).

According to the above Eqs. (10) and (11), the position of the trajectory

1D Fourier transform and 2D Fourier transform are performed on the solved

Equation (61) is used to calculate the similarity between the extracted characteristic parameters and the Fourier characteristic parameters of the objective function.

All sizes are arranged in the order of similarity, and the individuals with low similarity replace the individuals with high similarity.

Each of the two sizes forms a group and starts to cross-reproduce, with new sizes produced.

The random selection of certain generations is performed for mutation.

According to the results of Step (6), the similarity of all size types average, and the best individual similarities are expressed.

Length range of the rod.

Taking into account the characteristics of the output objective function of the spherical four-bar rigid-body guidance mechanism, the genetic algorithm is used in this paper to optimize the mechanism, as shown in Fig. 9. The population size is 500, and the crossover probability is 0.8. The two-point crossover operator is used, and the geometric programming sorting selection is carried out. The number of genetic iterations is 200.

Optimization flowchart.

The design requirements and reference point

Design requirements.

The prescribed position and orientation.

The amplitude and phase of the prescribed orientation.

Based on the above analysis, a variable step size output properties database
of the mechanism is first established. Through a 1D FFT and
normalization process, the OHCPDT of the design requirements can be
extracted. Table 4 shows four terms of amplitudes of the prescribed
orientation. The matching method is used to identify 10 groups of the MOBDT
from the database. According to each group of the MOBDT, we establish the
initial population, so as to obtain the MOBDT of the desired mechanism, as
shown in Table 5. Then, Eq. (52) is applied to obtain the actual dimensions
of the

The identified MOBDT (10 groups are chosen).

The amplitude and phase of the prescribed position.

Dimension and arrangement parameters.

The amplitude and phase of the RBPO of the 10th group of the MPBDT.

Finally, the comparison diagrams of three groups of design results and given design requirements are shown in Figs. 11, 12, and 13. The design requirements are shown in blue, and the design results are shown in red. It can be seen that the fitting curve overlaps completely. Part (b) in the figure is the comprehensive result error in azimuth. Part (c) in the figure is the comprehensive result error of the pitch angle. Part (d) in the figure is the comprehensive result error in the rigid-body guidance angle.

Comparison drawing of the first group of dimensions.

Comparison drawing of the second group of dimensions.

Comparison drawing of the third group of dimensions.

A new method is proposed to address the motion generation problem of spherical four-bar crank slider rigid-body guidance mechanisms. According to the corresponding parameters of the position output and direction output of the research institutions in this paper, a method is developed to establish the output properties database of the mechanism. Then, the solution to the dimension parameters is proposed as well. By using the output properties database and the solution method, motion generation is achieved.

The analytical methods are applicable only to deal with the problem of finitely divided positions. However, the method proposed in this paper can be used to effectively solve the problem of motion generation with infinite prescribed positions in theory. Through comparison with analytical methods, nonlinear equations are avoided in this paper, which makes the method much more efficient and simple.

The method combines the advantages of the analytical method and the atlas method characterized by precision and simple calculation. Through a computer, it is achievable to quickly and accurately identify various groups of OHCPDT which meet the design requirements. Besides this, the dimension and installation parameters can be calculated as well, with several optimal options available to users. At last, this method can be extended to the motion generation of other types of linkage mechanisms, such as a spherical four-bar mechanism and RCCC mechanisms.

The code is available upon request from the corresponding author.

The data are available upon request from the corresponding author.

All the authors contributed to the study conception and design. The mathematical model and design method were proposed by JS and WL, experimental design and analysis were performed by ZL and HL. The first draft of the paper was written by WZ, ZL and WL. All the authors commented on previous versions of the paper. All the authors read and approved the final paper.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors acknowledge financial support from the National Natural Science Foundation of China, Hubei Provincial Natural Science Foundation of China, the Science and Technology Research Project of the Jilin Provincial Department of Education, Central Government Guides Local Science and Technology Development Projects of Hubei Province, and Science and Technology Innovation Team of Hubei University of Arts and Science.

This research has been supported by the National Natural Science Foundation of China (grant no. 51775054), Hubei Provincial Natural Science Foundation of China (grant no. 2022CFC035), the Science and Technology Research Project of the Jilin Provincial Department of Education (grant no. JJKH20220672KJ), Central Government Guides Local Science and Technology Development Projects of Hubei Province (grant no. 2018ZYYD016), and Science and Technology Innovation Team of Hubei University of Arts and Science (grant no. 2022pytd01).

This paper was edited by Daniel Condurache and reviewed by five anonymous referees.