Dimensional Synthesis of Watt II and Stephenson III Six-Bar Slider-Crank Function Generators for Nine Prescribed Positions

: This paper proposes an efﬁcient exact dimensional synthesis method for ﬁnding all the link lengths of the Watt II and Stephenson III six-bar slider-crank function generators, satisfying nine prescribed precision points using the homotopy continuation method. The synthesis equations of each mechanism are initially constructed as a system of 56 quadratic polynomials whose B é zout number, which represents the maximum number of solutions, is 2 56 ∼ = 7.21 × 10 16 . In order to reduce the size of the system, multi-homogeneous formulation is applied to transform the system into 12 equations in 12 unknowns, and the multi-homogeneous B é zout number of the system is 286,720. The Bertini solver, based on the homotopy continuation method, is used to solve the synthesis equations to obtain the dimensions of the two mechanisms. For the arbitrarily given nine precision points, the proposed method yields 37 and 31 defect-free solutions of Watt II and Stephenson III six-bar slider-crank mechanisms, respectively, and it is conﬁrmed that they pass through the prescribed positions.


Introduction
The dimensional synthesis of a mechanism determines the dimensions of the individual links that generate a desired motion. Since the development of an analytical approach to the design of four-bar linkages introduced by Freudenstein [1], numerous studies have been conducted on the dimensional synthesis, which can be classified into two categories. One is the exact synthesis method, known as the precision point approach, in which a finite number of exact positions are to be satisfied. The other is the approximate synthesis, which seeks the dimensions that approximately satisfy prescribed positions.
In general, dimensional synthesis problems are formulated as a system of nonlinear equations and require heavy computation. For this reason, various approximate synthesis approaches using numerical methods [2,3], optimization methods such as least square techniques [4][5][6], genetic algorithms [7][8][9][10], and evolutionary algorithms [11][12][13][14][15] have been used widely in recent times. However, the approximate synthesis methods require appropriate initial values that are close to the actual solutions of the synthesis equations. When unsuitable initial values are assumed, the solutions may not be found. In addition, the methods may not find all the solutions of the synthesis equations.
In contrast, since the exact synthesis method can find all the solutions without estimating an initial value, it yields diverse mechanism dimensions that satisfy a desired motion. Among the studies on the precision point approach, Huang et al. [16] and Almandeel et al. [17] used elimination methods to perform the dimensional synthesis of four-bar path generators for five precision points and slider-crank function generators for four precision points, respectively. Subbian and Flugrad [18] and Wampler et al. [19] carried out the dimensional synthesis of four-bar path generators for five precision points and nine precision points, respectively. For the dimensional synthesis of six-bar linkages with revolute joints only, Dhingra et al. [20] and Plecnik and McCarthy [21][22][23] applied the the solution could increase tremendously. Hence, in order to carry out the dimensional synthesis efficiently, it is necessary to reduce the size of the equation system. In this section, the synthesis equations for the two mechanisms are derived and simplified by eliminating some unknowns.

Synthesis Equations of Watt II Six-Bar Slider-Crank Function Generator
A Watt II six-bar slider-crank linkage is shown in Figure 1a, in which the input link r 1 rotates about the fixed pivot O, and the output slider translates vertically. A fixed frame is positioned so that its origin is at O and oriented so that its x-axis is perpendicular to the direction of slide. The horizontal distance from O to the line of path of the moving pivot E is the offset denoted by h. In the fixed frame, the input angle θ is measured from the positive x-axis and the output displacement of the slider p is the vertical distance from the x-axis to E, the center of the revolute joint attached to the slider. The angle of link r 0 is η, and the angle between links r 3 and r 4 is β. The angular displacements of links r 2 , r 3 , r 4 , and r 5 are denoted by φ, α, α + β, and δ, respectively. fied by eliminating some unknowns.

Synthesis Equations of Watt II Six-Bar Slider-Crank Function Generator
A Watt II six-bar slider-crank linkage is shown in Figure 1a, in which the input link r1 rotates about the fixed pivot O, and the output slider translates vertically. A fixed frame is positioned so that its origin is at O and oriented so that its x-axis is perpendicular to the direction of slide. The horizontal distance from O to the line of path of the moving pivot E is the offset denoted by h. In the fixed frame, the input angle θ is measured from the positive x-axis and the output displacement of the slider p is the vertical distance from the x-axis to E, the center of the revolute joint attached to the slider. The angle of link r0 is η, and the angle between links r3 and r4 is β. The angular displacements of links r2, r3, r4, and r5 are denoted by ϕ, α, α + β, and δ, respectively. Figure 1b shows the mechanism in its first and jth positions. In the Figure, Δθj, Δϕj, Δαj, and Δδj denote the rotation angles of links r1 to r5, respectively, and Δpj represents the displacement of the slider, all measured from the initial position to jth position of the mechanism. Let θ1 and p1 denote the angle of input link r1 and the position of output slider in the initial position, respectively. Then, the prescribed positions of the function generator to be synthesized are given pairwise as where θj and pj are the input angle and the position of the slider in jth position, respectively, and n is the number of precision points to be satisfied.  Figure 1b shows the mechanism in its first and jth positions. In the Figure, ∆θ j , ∆φ j , ∆α j , and ∆δ j denote the rotation angles of links r 1 to r 5 , respectively, and ∆p j represents the displacement of the slider, all measured from the initial position to jth position of the mechanism. Let θ 1 and p 1 denote the angle of input link r 1 and the position of output slider in the initial position, respectively. Then, the prescribed positions of the function generator to be synthesized are given pairwise as where θ j and p j are the input angle and the position of the slider in jth position, respectively, and n is the number of precision points to be satisfied. The dimensional synthesis equations of the Watt II six-bar slider-crank function generator can be constructed by the closure equations of the two loops OABC and OCDE shown in Figure 1 as follows. Loop OABC: For the initial and jth positions, the loop closure equations can be written as and r 1 e i∆θ j + r 2 e i∆φ j = r 0 + r 3 e i∆α j , j = 2, . . . , n, respectively. Subtracting Equation (2) from Equation (3) gives Loop OCDE: For the initial and jth positions, the loop closure equations can be written as and respectively. Subtracting Equation (5) from Equation (6) gives Separating the real and imaginary parts of Equations (4) and (7) leads to and r 4 cos α 1 + β + ∆α j − cos(α 1 + β) = r 5 cos δ 1 + ∆δ j − cos δ 1 , j = 2, . . . , n, (10) respectively. Note that θ 1 , φ 1 , α 1 , and δ 1 denote the angles of the corresponding links shown in Figure 1a in the initial position. Among the unknowns in Equations (8) through (11), ∆φ j , ∆α j , and ∆δ j should satisfy trigonometric identities cos 2 ∆φ j + sin 2 ∆φ j = 1, cos 2 ∆α j + sin 2 ∆α j = 1, where j = 2, . . . , n.
The unknowns in the synthesis equations, Equations (8)- (12), are r 1 , r 2 , r 3 , r 4 , r 5 , θ 1 , φ 1 , α 1 , β, δ 1 , cos∆φ j , sin∆φ j , cos∆α j , sin∆α j , cos∆δ j , and sin∆δ j , where j = 2, . . . , n. Hence, the problem of the dimensional synthesis for the function generator that satisfies n prescribed positions is formulated as 7(n − 1) equations in 10 + 6(n − 1) unknowns. For 9 prescribed positions, n is equal to 9 and the synthesis equations consist of 56 quadratic polynomials with respect to 58 unknowns. Therefore, to match the number of unknowns and the number of equations, two free choices can be made. The total degree of a system of m polynomial equations, which is referred to as the Bézout number, is defined as ∏ m i=1 d i , where d i is the degree of the ith polynomial [35]. Hence, the total degree of the synthesis equations for 9 precision points is 2 56 ∼ = 7.21 × 10 16 . Since the system size is too large to solve, it is necessary to reduce the size of the synthesis equations as explained in the following sections.

Synthesis Equations of Stephenson III Six-Bar Slider-Crank Function Generator
A Stephenson III slider-crank linkage is shown in Figure 2a. The input link r 1 rotates about the fixed pivot O, and the output slider translates along a line parallel to y-axis. The angle between links r 2 and r 4 is β. The angular displacements of links r 2 , r 3 , r 4 , and r 5 are denoted by φ, α, φ + β, and δ, respectively. As shown in Figure 2b, ∆θ j , ∆φ j , ∆α j , and ∆δ j represent the rotation angles of links r 1 to r 5 , measured from their initial positions to jth positions, respectively. The prescribed precision points are given as Equation (1).
Squaring both sides of Equation (18), adding the result, and simplifying using the trigonometric identity cos 2 (∆αj) + sin 2 (∆αj) = 1 gives Equation (19), the system of synthesis equations for n prescribed positions, has n−1 eighth-degree polynomials in 10 unknowns, rix and riy, i = 1,…, 5, given in Equation (15). Hence, for 9 position synthesis, n is equal to 9 and there are 8 equations in 10 unknowns, and two unknowns can be selected as free choices. Among the 10 unknowns, r1x and r1y are chosen as the free choices in this study to reduce the calculation time as explained in Section 3.1. Now, the total degree of the 9 precision point synthesis equations is 8 8 = 16,777,216, which is much smaller than that of the synthesis equations derived in Section 2.1. However, the system size of the simplified synthesis equations can be further reduced by using auxiliary equations and applying the multi-homogeneous Bézout theorem. This process is described in Section 3.

Synthesis Equations of Stephenson III Six-Bar Slider-Crank Function Generator
A Stephenson III slider-crank linkage is shown in Figure 2a. The input link r1 rotates about the fixed pivot O, and the output slider translates along a line parallel to y-axis. The angle between links r2 and r4 is β. The angular displacements of links r2, r3, r4, and r5 are denoted by ϕ, α, ϕ + β, and δ, respectively. As shown in Figure 2b, Δθj, Δϕj, Δαj, and Δδj represent the rotation angles of links r1 to r5, measured from their initial positions to jth positions, respectively. The prescribed precision points are given as Equation (1). The dimensional synthesis equations of the Stephenson III six-bar slider-crank function generator can be derived using the closure equations of the two loops, OABC and OADE, shown in Figure 2 as follows.
As in case of the Watt II slider-crank mechanism, the dimensional synthesis equations, Equations (26)- (30), for the Stephenson III function generator that satisfies n prescribed positions are formulated as a system of 7(n − 1) equations in 10 + 6(n − 1) unknowns which are r 1 , r 2 , r 3 , r 4 , r 5 , θ 1 , φ 1 , α 1 , β, δ 1 , cos∆φ j , sin∆φ j , cos∆α j , sin∆α j , cos∆δ j , and sin∆δ j , with j = 2, . . . , n. For 9 prescribed positions, n is equal to 9 and there are 56 equations in 58 unknowns. Hence, two unknowns need to be assumed as free choices in order to obtain the solutions. The total degree of the synthesis equations is 2 56 ∼ = 7.21 × 10 16 . In the following section, the synthesis equations for n precision points are simplified by eliminating some of the unknowns.

Additional Degree Reduction with the Multi-Homogeneous Theorem
The Bézout number of a polynomial system represents the largest number of solutions that the system can have, and is also the number of paths to be tracked to find all isolated solutions in the continuation method. According to the multi-homogeneous Bézout theorem, the number of solution paths and thus the computing time can be significantly reduced by dividing the unknowns into appropriate homogeneous groups by the multi-homogeneous formulation [35,36].
As the last step for reducing the system size, the multi-homogeneous Bézout theorem is applied to the synthesis equations. Then, the synthesis equations can be solved by the tracking of only 286,720 paths as explained in Sections 3.1 and 3.2.

2-Homogeneous Formulation for Watt II Six-Bar Slider-Crank Mechanism
For the multi-homogeneous formulation of the synthesis equations for the Watt II slider-crank mechanism, the 12 unknowns are arranged into two groups λ 1 and λ 2 that constitute the loop OABC and loop OCDE shown in Figure 1 as follows. λ 1 : r 2x , r 2y , r 3x , r 3y , M 1 , M 2 , λ 2 : r 4x , r 4y , r 5x , r 5y , M 3 , M 4 .
In order to determine the multi-homogenous Bézout number for a polynomial system whose unknowns are arranged into m homogeneous groups λ 1 to λ m , let l be the number of polynomial equations to solve, d i,j be the degree of the ith equation with respect to the unknowns in group λ j , and k j be the number of unknowns in group λ j . Then, the multi-homogenous Bézout number of the system is defined as the coefficient of the term ∏ m j=1 λ k j j of the following equation [35].
Note that λ j in Equation (40) is used as a null value for calculating the multi-homogenous Bézout number.
Since the system of 12 synthesis equations for the Watt II slider-crank function generator, Equations (19) and (37) with Equation (39), is a 2-homogeneous system, substituting l = 12, m = 2, k 1 = k 2 = 6, and d i,j shown in Table 1 into Equation (40) gives Hence, the multi-homogenous Bézout number of the system, the coefficient of the term λ 6 1 λ 6 2 in Equation (41), can be obtained by using the binomial theorem as (2 8 8!/4!4!)2 2 2 2 = 286,720, which agrees with the results reported by Dhingra et al. [20] for the synthesis of the Watt II six-link function generator with revolute joints only for 9 precision positions.
If other unknowns than r 1x and r 1y are selected as the free choices, the number of auxiliary equations to convert the system into a homogenous structure increases and 2homogenous Bézout number becomes higher. For example, if r 2x and r 2y are assumed as the free choices, it is necessary to define 6 auxiliary equations and the 2-homogeneous Bézout number is (2 8 8!/4!4!)2 4 2 2 = 1,146,880. Therefore, it is important to select the unknown that minimize the number of auxiliary equations as the free choices. In this case, it is efficient to assume r 1x and r 1y as the free choices to reduce the 2-homogeneous Bézout number in consideration of the calculation time.
As shown in Table 2, the degrees of the individual equations with respect to the unknowns in each group are the same as those of the Watt II six-bar slider-crank function generator. Therefore, the 2-homogeneous Bézout number of the system is also 286,720, which agrees with the results of Dhingra et al. [20] for the synthesis of the Stephenson III six-bar function generator with revolute joints only for 9 precision points. For the same reason as described in Section 3.1, r 1x and r 1y are assumed to be the free choices in this study to reduce the 2-homogeneous Bézout number.
Consequently, the 2-homogeneous Bézout number of the synthesis equations for each mechanisms is about 1/58 of the total degree of the polynomial systems that are derived by eliminating some unknowns in Sections 2.2 and 2.4. Hence, the process proposed in this section can significantly reduce the time to solve the synthesis equations.
In this research, the calculation of the synthesis equations is carried out by using the Bertini software program [37], which is based on the homotopy continuation method, for obtaining all the solutions to the Watt II and Stephenson III six-bar slider-crank synthesis equations that satisfy 9 prescribed precision points. Among the real solutions, however, some may have kinematic defects. The following section describes the process of sorting out feasible solutions.

Sorting Out Feasible Solutions
In order for the synthesized mechanisms to function as six-link slider-crank linkages, they should satisfy the following two conditions. The first is that the input link needs to be capable of complete rotation, and the second is the prescribed positions should lie on a single stroke of the slider. Hence, the solutions that do not satisfy these two conditions should be excluded. This section describes the process for screening the feasible mechanisms by means of the full rotation condition of the input link and the displacement analysis of synthesized mechanisms to check whether prescribed positions are present on a single stroke of the slider.

Full Rotatability of Input Link
The input link r 1 of each mechanism is in the four-bar loop OABC shown in Figures 1  and 2. Hence, it is required for the four-bar to have a full rotatable crank as the input link. Among the four-bar linkages, those that allow complete rotation of the input link are the crank-rocker and double-crank, and each mechanism satisfies the following condition, respectively [38].
Crank-rocker mechanism : where The value of r 0 in Equation (44) can be determined by the loop closure equation for the four-bar in each mechanism. From Figure 1 and Equation (2) for the Watt II slider-crank and Figure 2 and Equation (20) for the Stephenson III slider-crank, the xand y-components of r 0 can be written as r 0 cosη = r 1x + r 2x − r 3x and r 0 sinη = r 1y + r 2y − r 3y , respectively, where r 1x and r 1y are the prescribed values as free choices, and r 2x , r 2y , r 3x , and r 3y are obtained from the solution of the synthesis equations. Then, r 0 = (r 0 2 cos 2 η + r 0 2 sin 2 η) 1/2 . Similarly, since r ix and r iy (i = 1, 2, 3) represent the xand y-components of r i as defined in Equations (15) and (33), r i in Equation (44) can be determined by r i = (r ix 2 + r iy 2 ) 1/2 . Based on the condition provided in Equation (43), the solutions that allow the full rotation of the input link can be found from among all the solutions of the synthesis equations.
The next section describes the displacement analysis for further discerning the mechanisms in which all the prescribed precision points are present on a single stroke of the output slider.

Displacement Analysis
If all link lengths r i , i = 0, . . . , 5, and the offset h are given, for a given value of the input crank angle, the Watt II and Stephenson III six-bar slider-crank mechanisms can each be assembled into four different configurations called assembly modes or branches, and some of the prescribed positions of a synthesized mechanism may lie on different assembly modes. If all the prescribed positions do not lie on a single stroke of the slider on one branch, the solution is unacceptable. Therefore, it is necessary to select the solution mechanisms that satisfy this condition by analyzing the displacement of each mechanism.
For the analysis of the six-bar slider-crank mechanism, the four-bar linkages commonly used in the two mechanisms need to be analyzed first. The angular displacements α of link r 3 and φ of link r 2 shown in Figures 1 and 2 can be determined as [39] where A = 2r 0 r 3 cos η − 2r 1 r 3 cos θ, B = 2r 0 r 3 sin η − 2r 1 r 3 sin θ, C = r 2 0 + r 3 1 + r 2 3 − r 2 2 − 2r 0 r 1 (cos η cos θ + sin η sin θ), and φ = tan −1 r 0 sin η + r 3 sin α − r 1 sin θ r 0 cos η + r 3 cos α − r 1 cos θ , respectively. Notice that when only the link lengths of the four-bar mechanism and the angle of the ground link η shown in Figures 1 and 2 are given, there are two sets of α and φ for a given crank angle θ due to the positive (α + ) or negative (α − ) sign in Equation (45), which correspond to the two assembly modes. However, since the solution of the synthesis equations in this study yields the xand y-components of each link in its initial position, the sign to be used in Equation (45) and the assembly mode of the four-bar can be readily identified.

Watt II Six-Bar Slider-Crank Mechanism
In order to analyze the slider displacement of the Watt II slider-crank mechanism, the position of the joint E in Figure 1 needs to be determined. To find the coordinates of E, the position of joint D can be expressed as where α is calculated from Equation (45), and β is the angle between link r 3 and link r 4 that can be determined using the solution of the synthesis equations as Now, the position of the moving pivot E is where h = r 0 cos η + r 4x − r 5x .
Equation (50) indicates that there are two solutions for the position of the joint E. That is, the slider and link r 5 can be assembled above (with the positive sign: E y + ) or below (with the negative sign: E y − ) the moving pivot D. As mentioned in Section 4.1, however, since the result of this research gives the orientation of each link in its initial position, the sign to be used in Equation (50) and the assembly mode of a synthesized mechanism can be identified.

Stephenson III Six-Bar Slider-Crank Mechanism
The position of the moving pivot D in Figure 2 can be determined by where β is the angle between link r 2 and link r 4 that can be determined using the solution of the synthesis equations as Then, the coordinates of the moving pivot E are where As in the case with the Watt II slider-crank mechanism, the ± sign to be used in Equation (54) can be determined by considering the xand y-components of link r 5 of the synthesized function generator.
By applying the solution screening process explained in this section, the Watt II and Stephenson III six-bar slider-cranks without kinematic defects can be finally selected from among all the solutions of the synthesis equations. The next section provides examples of the dimensional synthesis method proposed in this research.

Numerical Examples
In this section, examples are given for the dimensional synthesis of the Watt II and Stephenson III six-bar slider-crank function generators that satisfy 9 precision points. The synthesis equations of the two mechanisms derived in this study were calculated by using the homotopy continuation method, through which all the solutions were obtained. Then, the feasible solutions were selected based on the two criteria described in Section 4.

Dimensional Synthesis of Watt II Six-Bar Slider-Crank Function Generator
The prescribed positions that the Watt II six-bar slider-crank function generator should satisfy are given in Table 3, which shows the relative displacement of the output slider, ∆p j = p j − p 1 , with respect to the relative input crank angle, ∆θ j = θ j − θ 1 , for i = 2, . . . , 9, each measured from the initial positions of the slider and input crank, respectively. The free choices were assumed as r 1x = 0.12268 and r 1y = 0.87294. For the given precision points, Equations (19) and (37) are solved by using the Bertini software package. The computation time was 3.42 h on a single node of Intel ® Xeon ® W-2245 CPU@3.90 GHz. Among 25,630 nonsingular solutions obtained, those that have less than 0.01% structural error are selected as feasible solutions. As a result, there were 37 mechanisms that have completely rotatable cranks and pass all the prescribed positions. Table 4 shows the number of the feasible mechanisms for each assembly configuration. The kinematic diagrams in their initial positions and slider displacements of two synthesized mechanisms are shown in Figure 3 and the corresponding solutions are listed in Table 5.    Figure 3.

Dimensional Synthesis of Stephenson III Six-Bar Slider-Crank Function Generator
The prescribed positions for the Stephenson III six-bar slider-crank function generator are given in Table 6 and the free choices were assumed as r 1x = 0.12859 and r 1y = 1.0473. Using the Bertini software package, the computation time to solve the synthesis equations given in Equations (36) and (38) was 4.38 h on the same work station referred to in Section 5.1, and a total of 36,061 nonsingular solutions were obtained. Among them, 31 Stephenson III six-bar slider-cranks that have less than 0.01% structural error were found, and the number of the feasible mechanisms for each assembly configuration is tabulated in Table 7. The kinematic diagrams in the initial positions and slider displacements of two synthesized mechanisms are shown in Figure 4 and the corresponding solutions are listed in Table 8. Table 7. The number of feasible mechanisms for each configuration.

Solutions
Link Lengths Orientations (Deg)

Conclusions
This paper presents the dimensional synthesis of the Watt II and Stephenson III six-bar slider-crank function generators that satisfy nine precision points of the output slider for the prescribed rotational angles of the input link. In this study, the system of synthesis equations for each mechanism initially derived by 56 quadratic polynomials is simplified to eight eighth-degree polynomial equations by eliminating some unknowns. Then, by defining four auxiliary quadratic polynomials as new unknowns, the synthesis equations are converted into eight quartic and four quadratic homogeneous polynomials in 12 unknowns. By this process, the Bézout number of the system that represents the maximum number of isolated solutions is reduced from 2 56 ∼ = 7.21 × 10 16 to 1,048,576. In order to decrease the computation time of the synthesis equations further, 2-homogeneous formulation is applied by arranging the unknowns of the system into two groups and the multi-homogeneous Bézout number of the system is 286,720. In this research, the synthesis equations are solved by using the Bertini solver, which uses the homotopy continuation method.
As is known, all the solutions obtained by solving the synthesis equations do not yield feasible six-link slider-crank mechanisms. For the solution mechanism to function properly as desired without any defects, it must have a full rotatable crank and the prescribed precision points should lie on a single stroke of the slider. The screening process to select suitable linkages among the solutions is explained in detail.
The proposed method is verified by carrying out synthesis examples. For arbitrarily given nine precision points, the methods determined 37 feasible Watt II slider-crank mechanisms among 25,630 nonsingular solutions and 31 Stephenson III six-bdar slider-crank linkages out of 36,061 solutions. The structural errors of the synthesized mechanisms determined in this research are ranged from 5.42 × 10 −3 % and 7.95 × 10 −12 %.
In order to synthesize six-link slider-crank function generators for other numbers of precision points than nine, the n-1 polynomials in 10 unknowns given in Equation (19) for the Watt II and Equation (36) for the Stephenson III slider-crank mechanism would be the starting point for the synthesis. Once the number of precision points to be synthesized for is chosen, the number of free choices is determined. When selecting free choices among the 10 unknowns in Equations (15) or (33), it is efficient to choose the unknowns that can minimize the number of auxiliary equations to obtain the lower value of the multi-homogeneous Bézout number of the system. Then, define auxiliary equations that can convert the system into a homogeneous structure and apply multi-homogeneous formulation by arranging the variables into homogeneous groups. By solving the resulting system of equations and sorting out feasible solutions, the dimensions of the six-link slider-crank function generators can be determined.
In order to reduce the computation time to solve the system of synthesis equations, the use of newly developed homotopy continuation methods [40][41][42][43] can be considered. Among them, the Diagonal homotopy method [40] seeks only the nonsingular solutions for a system of nonlinear equations. Since there exist many degenerate solutions in a system of non-linear equations, using this method to solve the system may reduce the calculation time significantly.