These authors contributed equally to this work.

In this study, the authors propose a

The positional synthesis of the 4R linkage is the inverse calculation of its
kinematic analysis, i.e., to calculate the parameters of linkage
according to several specified positions of the coupler. It is widely used
in engineering. The Burmester theory (Cera and Pennestrì, 2018, 2019; Shirazi, 2007) points out that at most, five accurate
positions can be specified in positional synthesis. Because the number of
candidate solutions for a five-position synthesis problem is no more than
six, this can easily lead to a situation where no viable solution is
available, and this renders four-position synthesis more practical for use
in engineering. According to the Burmester theory, any two center points yield a linkage solution such that the four-position synthesis problem has

It is easy to determine that three factors affect the synthesis efficiency of the 4R linkage: (i) the lack of effective methods for identifying defective solutions, (ii) the blindness of the selection of points on the center curve, and (iii) the absence of a viable solution owing to the small number of available solutions in the domain. Previous studies have thus focused on these issues. To identify defective solutions, Filemon (1972) used the intersection of opposite sides of opposite pole quadrilaterals to divide the center curve into six segments, and found that the 4R linkage formed by the two center points in the same segment had the same order of couplers. Similarly, Waldron and Strong (1978), and Gupta and Beloiu (1998) used region segmentation technology with the Burmester curve to identify defects in the branch and circuit of the linkage. Based on the same technology, Sun and Waldron (1981) controlled the transmission angle of the linkage within a prescribed range while realizing defect-free linkage synthesis. The most effective method to mitigate the blindness of selection of points on the center curve involves visualizing the solution domain. McCarthy (2013) and Ruth and McCarthy (1999) proposed the

The region-based segmentation technology for the Burmester curve attempts to remove the points on the center curve or those in the circle curve that may cause defects before obtaining the linkage solution. However, this technology cannot accurately distinguish between defective and non-defective solutions, and excludes part of the reasonable linkage solutions while excluding defective solutions. This significantly reduces the chance of finding feasible solutions. By contrast, the type map method carries out defect discrimination once the parameters of linkage have been determined; thus, it has a higher accuracy than region-based segmentation technology. Moreover, type map can visualize the solution domain so that engineers can select the center points according to the type of linkage, thereby avoiding the blindness of the process of linkage synthesis. However, the number of candidate solutions provided by type map is usually too small to meet engineering needs. Approximate and accurate position synthesis are subjects in different fields of research. They can be used to solve the four-position synthesis problem and usually obtain only one linkage solution per calculation, but this solution is obtained by active search in the solution domain through an optimization algorithm. Theoretically, a satisfactory solution can be obtained directly by adding a sufficient number of constraints to the optimization model, but this is not feasible in many cases: a mathematical model with many constraints is too complex to establish, and strong constraints may also cause the optimization algorithm to fail to converge. To sum up, the type map method can completely replace the traditional synthesis method based on region segmentation technology. Therefore, only the possibility of its integration with the problem of approximate-motion synthesis needs to be considered. To solve the above three problems that affect the efficiency of synthesis, the concept of error in approximate-motion synthesis is introduced to accurate-position synthesis in this study to simultaneously expand and visualize the solution domain. Specifically, a solutions map method based on a telomere genetic algorithm (TGA) that can analyze and expand the solution domain is proposed in this study.

As shown in Fig. 1, the purpose of this study is to design a 4R linkage,

Motion of planar 4R linkage.

For any two positions of the coupler

In this case, the shape of the center curve depends on the value of

Note that the order of tan(

To facilitate programming, the range of values of

Generation of center curve

Center curves generated by a computer.

Considering that it is difficult to generate the circle curve from the
center curve by using the geometric method, the traditional vector
elimination method (Liang and Chen, 1993) is improved so that it can
generate the circle curve according to the coordinates of the points on the
center curve. According to this method, the RR links (i.e.,

By substituting the above formulae into Eq. (8), the following equation is
obtained after simplification:

After the numerical computation of matrix inversion, the above equation is
transformed into

In previous research (Liang and Chen, 1993; Zhao, 2009), Eq. (11) has been
decomposed into several sub-formulae that can be merged into a high-order
equation with only one variable by elimination:

After solving the equation by dichotomy or Newton's iterative method, the values of all variables can be obtained in a step-by-step manner based on the relationships between them. Note that because it is difficult to unify the symbolic definition in Fig. 1 and the algebraic definition in Eq. (9), there are significant differences between Eqs. (11) and (12) as derived in the literature, but the principle of vector elimination considered is consistent. Although this method can offer a direct solution for the coordinates of points on the center curve and the circle curve, rectangular coordinates that cannot reflect the trend of the Burmester curve are used to generate the points on the center curve. For example, this method needs to first specify the abscissa of the point on the center curve, where this coordinate can be taken arbitrarily on the plane and its value is independent of the shape of the center curve. It is thus difficult to parameterize the calculation process and avoid jumps when the point on the center curve is generated. This is also why we use the projective geometry method to calculate the coordinates of the point on the center curve. As the center curve has been generated, the vector elimination method is modified in this study. The improved solution process is as follows (Eqs. 13–16):

The following equation is equivalent to Eq. (11):

According to the second and third sub-formulae of Eq. (13), the expressions
containing

Further,

Once the coordinates of point

When the coordinates of points

Note that the variable

According to the method proposed by Martin and Murray (2002), the 4R linkage
can be classified into eight types based on the signs of

Classification of 4R linkages.

As a discontinuous attribute, the linkage type is often not the only
objective of linkage synthesis. It is thus also necessary to consider the
values of continuous attributes, such as the link length, transmission
angle, acceleration, and force, or their weighted values in the design of
the linkage mechanism (Jia et al., 2021; Trejo et al., 2015; Wilhelm et al.,
2017). For convenience of understanding, this study takes the transmission
angle as the index to assess the performance of the linkage. As shown in
Fig. 3, if the mass of each link and the friction of the revolute joint
are neglected, the coupler

Transmission angle of the 4R linkage.

For each 4R linkage synthesized according to the Burmester theory, we need to
discriminate among three kinds of defects, i.e., defects in the circuit,
branch, and order, as shown in Fig. 4.

Because the domain of definition of the arccosine function is

Only if all the four components of the

Linkage defects in the circuit, branch, and order.

The principle of the solutions map proposed in this paper is to map all
linkage solutions to a 3D color-coded surface based on linkage type to
visualize the distribution of the solutions. There are three steps to
generate the solutions map:

The solutions map consists of the following parts:

The method proposed here, based on the solutions map, has the characteristics of a geographical map. The concept of the

Color encoding according to linkage type.

Position synthesis task.

Solutions map for the position synthesis of the 4R linkage.

Model of the raster of the solutions map.

Linkage solutions obtained from the solutions map.

Coordinate-related information of the linkage solutions.

The flowchart of the solutions map method is shown in Fig. 8. In view of the limitations of space, the generation of the coupler curve and the kinematic and dynamic analyses of the linkage are not the main contents of this paper, and are thus not discussed here.

Flowchart of the solutions map method.

Although the solutions map can quickly locate the required types of linkage solutions and greatly improve the efficiency of synthesis, most 4R linkages synthesized based on the Burmester theory have kinematic defects, where this limits the number of available solutions. In many extreme cases, the given task of position synthesis has no solution. Consider the synthesis task given in Table 4 as an example. After excluding various defective solutions, the number of available solutions in the solutions map is 5694, accounting for only 10.86 % of the total number of solutions, 52 441, as shown in Fig. 9. Note that the solutions in the same region of the solutions map are usually highly similar, which renders options that are very limited. Therefore, it is necessary to expand the solution domain to improve the probability of obtaining feasible solutions.

Distribution of solutions without kinematic defects in the solutions map.

In the context of the problem of approximate motion synthesis, the coupler
curve of the 4R linkage only needs to pass approximately through the given
four positions. Similarly, although accurate positional information needs to
be specified in position synthesis, it is only needed for calculation. In
engineering applications, errors are usually allowed in some given positions

Allowable errors in the position synthesis task.

When the positional information changes within the allowable range of error,
a solutions map with different proportions of the feasible solutions can be
obtained to expand the solution domain. This is a typical optimization
problem, and its mathematical model can be described as follows:

The larger the objective function

The genetic algorithm (Nachaoui et al., 2021; Oliveira et al., 2022) is an
optimization method to simulate the mechanism of natural evolution. The
standard genetic algorithm consists of three basic operators: selection,
crossover, and mutation. However, practical applications have shown that
when the population is small, such an algorithm is premature. Although
increasing the population can yield better results, the required population
is different in different problems, and the calculation time is directly
proportional to the size of the population. In previous work, catastrophe
strategy has often been used to avoid the premature convergence of the
genetic algorithm. This method simulates a major destruction event in
natural evolution, eliminates all solutions except the optimal solution, and
restarts the algorithm. Catastrophe strategy is effective for specific
problems but its computational efficiency is very low. To improve the global
search ability of the genetic algorithm, a new operator called the

As shown in Fig. 10, telomeres are DNA-protein complexes at the end of
eukaryotic chromosomes. Their function is to protect the chromosomal
structure and control the cell-division cycle. Every time a cell divides,
the length of telomeres decreases. When telomeres are exhausted, the cells
will gradually stop dividing due to the destruction of the DNA structure.
Therefore, the telomere is also known as the mitotic clock. In the standard
genetic algorithm, not all individuals participate in the crossover and
mutation operations. Their participation depends on the probabilities of
crossover and mutation. Some old individuals thus have no chance of
producing new individuals in successive generations of evolution. The core
idea of the

A variable

The initial length of the telomere is

If the crossover or mutation operation can produce a new individual that is different from the parent, the length of the telomere of the new individual is reset to

Biological structure and algorithmic simulation of the telomere.

In applications, the performance of the telomere operator can be further
improved by the following methods:

The initial length of the telomere is variable. If the fitness of the
optimal individual in the current generation does not change compared with
that in the previous generation,

When the old individual is eliminated, a new individual is produced by it through non-uniform mutation (Chauhan et al., 2021; Ma, 2021). This method has a strong capability of global search in the early stage, and changes to local fine search in the later stage. The position of mutation

Genetic algorithm-based optimization of solutions map based on the telomere operator.

To further verify the effectiveness of the telomere operator, the results of
optimization of the standard genetic algorithm (GA) and the telomere genetic
algorithm (TGA) were compared with populations of

Comparison of results of optimization of the GA and TGA.

The typical fitness curves of the GA and TGA are shown in Fig. 12. Under normal circumstances, the optimal fitness value of the population rapidly improves in the initial stage of calculation of the genetic algorithm. Consider the curves of convergence “GA (1)” of the standard genetic algorithm and “TGA (1)” of the telomere genetic algorithm as examples: the initial fitness values of GA (1) and TGA (1) were below 0.2, but both increased to more than 0.7 within the first three generations. However, another form of the curve of convergence is also very common, where the GA prematurely falls into the local optimal solution. Consider the curves of convergence “GA (2)” of the GA and “TGA (2)” of the TGA as examples: when the GA falls into the local optimal solution, all individuals in the population are highly similar and the crossover operation can no longer produce a new solution. Although GA (1) jumped out of the local optimal solution through the mutation operation in the 20th generation, it fell into another local optimal solution and finally failed to achieve global convergence. By sharp contrast, TGA (2) could still generate new solutions with the help of the telomere operator after falling into the local optimal solution, jumping out of the local optimal solution many times until it achieved global convergence. We can glean important information from TGA (2): a large number of local optimal solutions may be distributed in the solution domain of optimization problems of the solutions map, where this is similar to the domain of solutions of multi-modal functions. Based on this assumption, we think that increasing the population size helps to expand the search scope of the algorithm. However, the telomere operator, which can enable the algorithm to jump repeatedly between local optimal solutions, can achieve a similar effect and is clearly more efficient.

Typical convergence curves of the GA and TGA.

In general, the GA can only obtain a solution with the highest
fitness after convergence. However, a solutions map with a high fitness does
not necessarily mean that a satisfactory solution can be obtained. In the
problem of synthesis of linkage positions, the evaluation of the solution is
multi-faceted. In addition to the absence of kinematic defects, the linkage
solution needs to satisfy geometric constraints, such as the length of the
linkage and position of installation, as well as the requirements of
mechanical performances, such as kinematics and dynamics. Therefore, even if
the proportion of available solutions of the optimized solutions map is very
high, the solutions it provides may still be completely rejected in the
subsequent analysis due to limitations imposed by other conditions of
evaluation. A more feasible method is to generate multiple solutions maps to
provide designers with more candidate solutions. If the optimal solutions map cannot provide a satisfactory solution, we can continue to search for
feasible solutions in other solutions maps.

The modified fitness function

In Eq. (27), the measurement of similarity by the sharing function depends
on two distance functions,

In this study,

By embedding the sharing function based on the pseudo-histogram into the fitness function of the TGA, multiple groups of solutions maps can be obtained in a single calculation. Consider once again the position synthesis task in Table 4 as an example. Four groups of typical solutions maps obtained based on the above algorithm are shown in Fig. 13. It is clear that the Bhattacharyya coefficient calculated from the pseudo-histogram can adequately reflect the similarity between different solutions maps. Moreover, the niche realized by the pseudo-histogram can avoid the convergence of the GA to the same solution.

Solutions maps generated by the niche genetic algorithm based on the pseudo-histogram method.

To further verify the influence of the niche technology based on the pseudo-histogram on the GA, the results of optimization of the GA and the TGA
were compared with populations of

Comparison of results of niche optimization of the GA and TGA.

The solutions map is a computer-aided method of synthesis. To improve the
efficiency of synthesis, the linkage synthesis software

Software used for the synthesis of the linkage position.

In addition to the calculation of points on the center and the circle, each
linkage solution in the solutions map requires the calculation and
visualization of the types of linkage and defect, and transmission-related
performance. If the time needed for the calculation is too long, the
synthesis efficiency of the software decreases. Most of the computation
time is spent on the visualization of the solutions map. To shorten this,
the Java3D API is used to generate the solutions map. Consider the solutions map shown in Fig. 5 as an example.

According to the analysis in Sect. 1, past methods can be divided into three categories: traditional method based on region-based segmentation technology, type map based on solution domain analysis, and approximate motion synthesis based on optimization. Both the type map and TGA-based solutions map can be used to analyze the solution domain; thus their synthesis efficiencies are much higher than that of the traditional, blind method. Compared with the type map, the TGA-based solutions map adds a dimension to visualize continuous attributes of the linkage, and can expand the solution domain through the TGA and niche technology. Although the TGA-based solutions map also uses the optimization algorithm, its objective of optimization is the solution domain and not a single linkage solution. In addition, it blurs the boundary between accurate position synthesis and approximate motion synthesis, which means that the two problems can be solved by a unified method in some cases. When the allowable errors in all guidance positions are zero, the TGA-based solutions map solves the problem of accurate position synthesis; when errors are introduced to part of the guidance positions, it solves the mixed problem of exact–approximate synthesis; when errors are introduced to all guidance positions, it solves the problem of approximate motion synthesis.

Compared with previous methods, the highlight of the proposed TGA-based
solutions map is that it can provide more candidate solutions for the
problem of the four-position synthesis of the 4R linkage to satisfy the
needs of subsequent design. Consider the synthesis task in Table 4 as an
example once again. Statistical results show that the number of candidate
solutions that can be obtained by the type map or the basic solutions map is
5694, whereas the number of candidate solutions that can be obtained by the TGA-based solutions map is 25 670. Considering that solutions in the area of the same linkage type of the type map or the solutions map are highly similar, it is not rigorous enough to compare the two methods in terms of the number of candidate solutions only. If the linkage needs to be free of motion-related defects and the minimum transmission angle at the four specified positions is greater than 30

Analysis of candidate regions.

In this paper, the authors examined the computer-aided synthesis of the
planar 4R linkage. The main conclusions can be summarized as follows:

A solutions map method was proposed. It can reveal the distribution of the linkage solutions of different attributes, including discontinuous attributes (such as linkage and defect types) and continuous attributes (such as link length, transmission angle, acceleration, and force, or their weighted values) so that the required linkage solutions can be quickly located.

A method of generating the solutions map was proposed. The center curve was first calculated by projective geometry and the circle curve was then obtained by vector elimination. A defect discrimination algorithm was proposed based on this which can quickly eliminate defective linkage solutions from the solutions map to improve the efficiency of linkage synthesis.

An improved genetic algorithm (GA) based on the telomere operator was proposed and used to expand the domain of solution of the solutions map. A niche construction method based on a pseudo-histogram was proposed based on this, such that more solutions maps and candidate solutions can be obtained after optimization. The results showed that the telomere genetic algorithm (TGA) significantly outperformed the traditional GA regarding the problem of expanding the domain of solutions of the solutions map.

The TGA-based solutions map was used to develop the software

Compared with previous research, the proposed TGA-based solutions map can provide more candidate solutions for the problem of the four-position synthesis of the 4R linkage. Moreover, it blurs the boundary between accurate position synthesis and approximate motion synthesis, which means that the two problems can be solved by a unified method in some cases.

The TGA-based solutions map can also be applied to the finite positional synthesis of any linkage with at least two free variables, including planar and spatial linkages. The specific method of operation is consistent with that in this study. It is first necessary to specify two free variables in the linkage synthesis model to form the solutions map and to then use the TGA to optimize it.

All the data used in this paper can be obtained by request from the corresponding author.

YZ, LX, and GW wrote the paper and participated in the
algorithm discussion of constructing the telomere genetic algorithm (TGA). YZ
wrote the fast validation program of TGA under MATLAB. GW proposed the
concept of TGA and solutions map, developed

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.

This research has been supported by the Shandong Provincial Natural Science Foundation (grant no. ZR2020QE163), the Program of Shandong Provincial Key Laboratory of Horticultural Machineries and Equipment (grant no. YYJX-2019-01), the Shandong Provincial Key Research and Development Program (grant no. 2018GNC112008) and the China Agriculture Research System of MOF and MARA (grant no. CARS-27).

This research has been supported by the Shandong Provincial Natural Science Foundation (grant no. ZR2020QE163), the Program of Shandong Provincial Key Laboratory of Horticultural Machineries and Equipment (grant no. YYJX-2019-01), the Shandong Provincial Key Research and Development Program (grant no. 2018GNC112008), and the China Agriculture Research System of MOF and MARA (grant no. CARS-27).

This paper was edited by Francisco Romero and reviewed by two anonymous referees.