# On the Solution of Region-Based Straight-Line Mechanism Design Optimization

Authors: Weixiang Qian; Yinshuo Yang; Xiaoyan Gong; Zhichao Jin; Zhuocheng Qu
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

A universal design method for synthesis problems of mechanisms that realize the approximate straight-line trajectory is presented in this paper. First, given the expected straight line and prescribed fixed pivots, a general mathematics model with angles as design parameters to determine the initial position of two side links is established, through which all infinite possible straight-line mechanisms are obtained. Then, kinematic constraints are imposed, including type, transmission angle, size, straightness, and defect. All feasible solution mechanisms that meet the constraints are calculated and can be expressed in solution regions. It is intuitive and comprehensive for designers to observe the distribution pattern of the solution. In the end, an optimal high-precision straight-line mechanism can be selected in the feasible solution regions by setting an optimization aim. The second-order osculating mechanism synthesis method can provide more solutions for designers, but designers can use the third-order osculating mechanism synthesis method when a higher straightness requirement is imposed. This method addresses the synthesis problem of this kind of mechanism for straight-line guidance and the problem of choosing an optimal solution from an infinite number of solutions.


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## Body

## 1. Introduction

An approximate straight-line mechanism can replace guide rails to make a frictionless motion where trajectory is exceptionally similar to a straight line. It possesses practical value in engineering and is used in various machinery applications, such as industrial robots, harvesting machines, lifting machines, and intermittent machines. The theory and method of approximate straight-line mechanism synthesis have a history of nearly a hundred years and are still a popular point for scholars, with many related monographs and papers published. Sandor [1] compared the fourth form of the ESE with a double-valued solution with an extension of Bobillier’s construction and obtained a double-valued solution. Lyndon [2] introduced a new graphical method of the Euler–Savary equation typically encountered in synthesis straight-line mechanisms. Barker [3] described the classification system and the geometry of the solution space, laying the foundation for future work to study the properties of planar four-bar mechanisms. Shen [4] introduced Bobillier’s graphical method of the inflection circle and its characteristics and used it to synthesize a four-bar lifting mechanism that implements an approximate straight-line trajectory. Li and Wang [5] established a Ball point theory to solve the synthesis problem of approximate straight-line mechanism for giving point position and straight-line direction. The method can obtain multiple sets of crank-rocker mechanisms to meet the requirements. Yu [6] proposed a numerical comparison synthesis method for single and double approximate straight-line mechanisms, which can solve the synthesis problem of approximate straight-line mechanisms more effectively. Chiang and Bulatović [7–9] used the variable controlled deviation method and differential evolution algorithm (DE) to achieve high precision in many given points along a straight line. Still, the Burmester curve does not have a single and orderly expression. In addition, it is also complicated to find the required mechanism quickly from the infinite number of solutions.After that, the theory made new progress in the four-bar mechanism. Hu [10] proposed a method for determining the size of a four-bar mechanism by specifying three precise symmetry points to determine the instantaneous center, using the instantaneous center position method. Han [11] gave a modern synthesis approach to the classical mechanism synthesis problem. Wang [12] proposed a procedure to synthesize mechanisms for mixed motion and function generation, which has not been settled before. SM [13] has presented an algorithm for path generation synthesis of the four-bar linkage with three precision points. An optimal synthesis method of the four-bar path generator is introduced by Romero [14]. The method uses a robust mathematical formulation and an optimization algorithm to certify the robustness of the formulation. Wu [15] developed an analytical method to find accurate solutions of coupler curve equations for planar four-bar linkages based on a neoteric algebraic formulation of the coefficient equation system. Bai and Angeles [16] used a new approach of combining numbers and graphics, which enabled us to use a simple equation to solve the problem of mechanism synthesis. Brake [17] gave the dimension and degree of the whole issues in the Alt–Burmester family and provided more details concerning zero- and one-dimensional cases. Wang [18] proposed the geometric properties of spherical coupler curves and provided a sound theoretical basis for synthesizing the four-bar linkages.At the same time, many design methods have been proposed. Liu [19] studied a novel design framework to complete the motion generation of the four-bar linkage. The aim is to improve the dynamic performance of the linkage. Kafash [20] designed an objective function for the optimization of path generation of four-bar linkages by differential evolution (DE). Sleesongsom [21] proposed a constraint handling technique for optimum path generation of four-bar linkages and presented a new technique to address the constraints of both the input crank rotation and the Grashof criterion. Wang [22] solved the synthesis problem of the approximate straight-line mechanism theoretically at this specific position when the instantaneous center is infinite and made some explorations to synthesize the second-order straight-line mechanism under particular conditions. Xu [23] proposed an approach to design large-displacement straight-line mechanisms with rotational flexural joints based on a viewpoint that the straight-line motion is regarded as a compromise of rigid and compliant parasitic motion of a rotational flexural joint. Wang [24] introduced the calculation of circle and center points by the Burmester theory and developed a program package to find a satisfying linkage. Pickard [25] proposed an appropriate synthesis method for uncertainties present in the geometric parameters of linkages during dimensional synthesis. Zimmerman [26] provided an accurate solution to satisfy any combination of these exact synthesis problems that was not over-constrained by using poles and rotation angles as constraints. Yin [27–29] deduced the formula of the synthetic crank-rocker straight-line mechanism and drew the crank-rocker mechanism solution region, and Yin proved that the straight-line accuracy of the mechanism with the Burmester points is generally better than that of the mechanism with Ball point. Then, a synthesis approach for selecting an optimal mechanism with a Ball-Burmester point is developed based on solution regions. Cui [30] extended a method for solving the synthesis problem of four-bar linkage with four specified positions. The method provides a reference for the synthesis of other multi-bar linkages.The above research contributed to the development of classical approximate straight-line mechanism synthesis theory. However, more attention is paid to establishing and solving the method of straight-line mechanism synthesis models, with less focus on the performance analysis and optimization of infinite number mechanisms. Next, the design process and the parameter adjustment of these researches are not intuitive; with no prediction and no forward-looking aspect, designer cannot grasp the distribution patterns and trends of the mechanism solutions. Thus, the current design is more or less done manually by experience through repeated attempts, making the design inefficient. Overall, no unified theory or methodology for the mechanism synthesis realizes approximate straight-line trajectories.With the rapid development and widespread use of computer technology, many mechanism synthesis problems are expected to make substantial progress. This paper proposes a unified design method in which computer programming can be quickly implemented to solve the synthesis problems of mechanisms that realize approximate straight-line trajectories. The second- or third-order osculation mechanism can be in a general or special configuration. This method can effectively solve the problem that the optimization of an infinite number of approximate straight-line mechanisms is complicated, and the optimization process is not intuitive.


## 2. Inflection Circle Generation Method

For a linkageA0ABB0 as depicted in Figure 1, the instantaneous center P can be obtained by extending A0A and B0B to intersect. Based on the Euler–Savary equation, the geometric relationships between the moving pivots A and B, the curvature center A0 and B0, the inflection poles JA and JB, and the instantaneous center P are as follows:(1)AA0×AJA=PA2,BB0×BJB=PB2.Figure 1
Relationships between the parameters.The inflection polesJA and JB can be calculated by (1). Inflection circle is made by instantaneous center P, and inflection polesJA,JB.Any moving pointCon the inflection circle are satisfied:(2)PC=dsinα1,where PC and the angle α1 represent the polar coordinates of the moving point C and d is the diameter of the inflection circle.Any moving pointC located on the inflection circle, with the corresponding center of curvature tending to infinity, has a trajectory containing an approximately straight section, and the direction of the velocity VC at this point is perpendicular to PC.


## 3. General Configuration Mechanism Synthesis Model

When the mechanism is in the initial configuration, there are no specific positional relationships between the four bars; it is called general configuration. The design requirements are as follows.This paper uses a pointC (xC, yC) and a direction angle β to describe an expected straight line, where β is specified to take a positive value when turned counterclockwise from the x-axis to the expected straight line. Conversely, the angle β takes a negative value, as shown in Figure 2.Figure 2
Synthesis mechanism model.The frame is described by the coordinates of the two fixed pivotsA0xA0,yA0 and B0xB0,yB0. It can also be characterized by a fixed pivot A0xA0,yA0 and a frame vector r. It is required that the moving pivots A and B are determined to guide point C along the given direction to walk an approximate straight line.


### 3.1. Second-Order Osculation Straight-Line Mechanism

The inflection circle is a set of points with the instantaneous radius of curvature that is infinite in the coupler plane, where the point is called the inflection point. The coupler curve has at least second-order osculation with the straight line at the inflection point, so pointC must be the inflection point. The second-order osculation straight-line mechanism synthesis method provides more design parameters compared with the third order in the design; as long as the approximate straight-line length and the straight-line deviation of the selected inflection point are within the allowable range, the point can be used as the working point on the coupler plane. Thus, the second-order method can provide designers with more options.According to the inflection circle generation technology, the process of obtaining the inflection point for the four pivots of the known mechanism is shown in Figure3.Figure 3
Inflection point generation process.The mechanism synthesis method proposed in this paper starts from the opposite of the problem using backward thinking to derive the straight-line mechanism under the condition that the straight-line point (inflection point) is known. The synthesis approach is as follows.The first design variable is introduced to determine the instantaneous centerP: direction angle θ of the side link A0A. When θ is reversed from the x-axis to A0A, it is taken as a positive value. Conversely, θ is taken as a negative value; θ∈−π/2,π/2. Then, the intersection of A0A with normal of the expected straight line through point C is instantaneous center P.The coordinate of pointP is as follows:(3)xP=cotβ⋅xC+yC+tanθ⋅xA0−yA0cotβ+tanθ,yP=tanθ⋅yC+cotβtanθxC−xA0+cotβ⋅yA0cotβ+tanθ.The pointsP and C are located on the inflection circle, and then the position angle γ of the inflection circle is introduced as the second design variable. γ is an angle between line PC and line PO, its value interval is −π/2,π/2, and the angle γ is positive when line PC is turned counterclockwise to line PO. Then, the diameter d of the inflection circle can be obtained:(4)d=PCcosγ.The coordinate of the center pointO of the inflection circle is as follows:(5)xO=xP+dcosαPC+γ2,yO=yP+dsinαPC+γ2,where αPC is the rotation angle of the vector PC to the x-axis, αPC∈0,2π, and its value is determined by tanαPC=yP−yC/xP−xC.The inflection polesJA and JB are the intersections of lines PA0 and PB0 with the inflection circle, respectively.The coordinate of pointJA is as follows:(6)xJA=xO+a1a4+a5,yJA=yO−a4+a1a5,where(7)a1=yP−yA0xP−xA0,a2=yA0−a1xA0,a3=a12+1,a4=−a1xO−yO+a2a3,a5=±d24a3−a42.Two sets of coordinate values can be obtained by changing the sign ofa5, and JAdoes not coincide with point P. When calculating the point JB, we only need to replace xA0 and yA0 in (6) with xB0 and yB0 of the point B0.According to (1), the vector JAA is calculated after the points A0, P, and JA are determined:(8)JAA=PJA2A0JA+2JAP.According to (8), we can determine the moving pivot A coordinate. In the same way, the moving pivot B can be determined by the points B0, P, and JB.Finally, the fixed pivots are connected to obtain the initial positionA0B0BAC of the mechanism. An infinite number of second-order osculation straight-line mechanisms can be chosen by the designer through varying the values of the angles θ and γ.


### 3.2. Third-Order Osculation Straight-Line Mechanism

The intersection of the inflection circle and the curvature-stationary point curve beyond the polar point is Ball point. There are four infinitely close points between the coupler curve and the expected straight line at the Ball point. Consequently, the third-order osculation straight-line mechanism synthesis method can be used for straight-line mechanism design with high straightness requirements.The coordinate of the instantaneous center is still calculated by (3). In particular, it is essential to note that the inflection circle should be calculated by the curvature-stationary point equation rather than chosen directly from the inflection circle bundle. The method to solve the inflection circle is as follows.Because pointsA, B, and C are all moving points on the coupler plane, according to the Euler–Savary equation, we have the following:(9)1PA=PA0+dsinαad⋅PA0sinαa,1PB=PB0+dsinαbd⋅PB0sinαb,1PC=1dsinα1,where PA and αa, PB and αb, PC and α1 represent the polar diameter and polar angle of moving points A, B, and C in the polar coordinate system, respectively, as shown in Figure 4.Figure 4
Third-order osculation mechanism synthesis model.The curvature-stationary point curve equation is as follows:(10)1R=1Msinα+1Ncosα,where R and α represent the polar position of an uncertain moving point, M and N are instantaneous invariants.PointC is Ball point. Points A and B always make arc motions around their fixed pivots, so points A, B, and C are curvature invariant points and the curvature-stationary point curve equation is satisfied:(11)1PA=1Msinαa+1Ncosαa,1PB=1Msinαb+1Ncosαb,1PC=1Msinα1+1Ncosα1.Combining (9) and (11), we have the following:(12)PA0+dsinαad⋅PA0sinαa=1Msinαa+1Ncosαa,PB0+dsinαbd⋅PB0sinαb=1Msinαb+1Ncosαb,1dsinα1=1Msinα1+1Ncosα1.CounteractingM and N in (12), we can obtain the following:(13)PB0sinαb−α1sinαacosαa+PA0sinα1−αasinαbcosαbPA0⋅PB0sinαacosαasinαbcosαbsinα1cosα1=0.The anglesαa, αb, and α1 are defined as follows:(14)αa=αa′−φ,αb=αb′−φ,α1=α1′−φ,where αa′, αb′, α1′, and φ are angles between the rays PA0, PB0, PC, and the polar axis Pt to the x-axis, respectively.It is easy to know that if we want to guarantee that (13) has solutions, the instantaneous center P must not coincide with the fixed pivots A0 and B0; the angles α1, αa, and αb are not equal to 0 and π/2; the cubic curvature-stationary point curve does not degenerate; and the mechanism is in general configuration.Substituting (14) into (13) and rearranging it, we have the following:(15)k1sinαa′−φcosαa′−φ−k2sinαb′−φcosαb′−φ=0,where(16)k1=PB0sinαb−α1,k2=PA0sinαa−α1.Only one unknown is direction angleφ of the polar axis Pt in (15). After rearranging (15), we can get(17)tan2ϕ=k1sin2αa′−k2sin2αb′k1cos2αa′−k2cos2αb′.From (17), the direction angles φ and φ′ of the polar axes Ptlie in the interval (0, π), and the difference between φ and φ′ is π/2 (as shown in Figure 5); then, the polar angle α1 can be calculated from the third formula of (13) and substituted into the third formula of (8) to calculate the inflection circle diameter d. Position angle γ of the inflection circle is as follows: γ=π/2−α1, γ′=−α1. The center O of the inflection circle is calculated by (5). The following steps are the same as the second-order osculation mechanism and will not be repeated here.Figure 5
Mechanism synthesis model when1/M = 0. (a) Special configuration with A0A and A0B0 being collinear. (b) Special configuration with A0A and AB being collinear.
(a)(b)


## 3.1. Second-Order Osculation Straight-Line Mechanism

The inflection circle is a set of points with the instantaneous radius of curvature that is infinite in the coupler plane, where the point is called the inflection point. The coupler curve has at least second-order osculation with the straight line at the inflection point, so pointC must be the inflection point. The second-order osculation straight-line mechanism synthesis method provides more design parameters compared with the third order in the design; as long as the approximate straight-line length and the straight-line deviation of the selected inflection point are within the allowable range, the point can be used as the working point on the coupler plane. Thus, the second-order method can provide designers with more options.According to the inflection circle generation technology, the process of obtaining the inflection point for the four pivots of the known mechanism is shown in Figure3.Figure 3
Inflection point generation process.The mechanism synthesis method proposed in this paper starts from the opposite of the problem using backward thinking to derive the straight-line mechanism under the condition that the straight-line point (inflection point) is known. The synthesis approach is as follows.The first design variable is introduced to determine the instantaneous centerP: direction angle θ of the side link A0A. When θ is reversed from the x-axis to A0A, it is taken as a positive value. Conversely, θ is taken as a negative value; θ∈−π/2,π/2. Then, the intersection of A0A with normal of the expected straight line through point C is instantaneous center P.The coordinate of pointP is as follows:(3)xP=cotβ⋅xC+yC+tanθ⋅xA0−yA0cotβ+tanθ,yP=tanθ⋅yC+cotβtanθxC−xA0+cotβ⋅yA0cotβ+tanθ.The pointsP and C are located on the inflection circle, and then the position angle γ of the inflection circle is introduced as the second design variable. γ is an angle between line PC and line PO, its value interval is −π/2,π/2, and the angle γ is positive when line PC is turned counterclockwise to line PO. Then, the diameter d of the inflection circle can be obtained:(4)d=PCcosγ.The coordinate of the center pointO of the inflection circle is as follows:(5)xO=xP+dcosαPC+γ2,yO=yP+dsinαPC+γ2,where αPC is the rotation angle of the vector PC to the x-axis, αPC∈0,2π, and its value is determined by tanαPC=yP−yC/xP−xC.The inflection polesJA and JB are the intersections of lines PA0 and PB0 with the inflection circle, respectively.The coordinate of pointJA is as follows:(6)xJA=xO+a1a4+a5,yJA=yO−a4+a1a5,where(7)a1=yP−yA0xP−xA0,a2=yA0−a1xA0,a3=a12+1,a4=−a1xO−yO+a2a3,a5=±d24a3−a42.Two sets of coordinate values can be obtained by changing the sign ofa5, and JAdoes not coincide with point P. When calculating the point JB, we only need to replace xA0 and yA0 in (6) with xB0 and yB0 of the point B0.According to (1), the vector JAA is calculated after the points A0, P, and JA are determined:(8)JAA=PJA2A0JA+2JAP.According to (8), we can determine the moving pivot A coordinate. In the same way, the moving pivot B can be determined by the points B0, P, and JB.Finally, the fixed pivots are connected to obtain the initial positionA0B0BAC of the mechanism. An infinite number of second-order osculation straight-line mechanisms can be chosen by the designer through varying the values of the angles θ and γ.


## 3.2. Third-Order Osculation Straight-Line Mechanism

The intersection of the inflection circle and the curvature-stationary point curve beyond the polar point is Ball point. There are four infinitely close points between the coupler curve and the expected straight line at the Ball point. Consequently, the third-order osculation straight-line mechanism synthesis method can be used for straight-line mechanism design with high straightness requirements.The coordinate of the instantaneous center is still calculated by (3). In particular, it is essential to note that the inflection circle should be calculated by the curvature-stationary point equation rather than chosen directly from the inflection circle bundle. The method to solve the inflection circle is as follows.Because pointsA, B, and C are all moving points on the coupler plane, according to the Euler–Savary equation, we have the following:(9)1PA=PA0+dsinαad⋅PA0sinαa,1PB=PB0+dsinαbd⋅PB0sinαb,1PC=1dsinα1,where PA and αa, PB and αb, PC and α1 represent the polar diameter and polar angle of moving points A, B, and C in the polar coordinate system, respectively, as shown in Figure 4.Figure 4
Third-order osculation mechanism synthesis model.The curvature-stationary point curve equation is as follows:(10)1R=1Msinα+1Ncosα,where R and α represent the polar position of an uncertain moving point, M and N are instantaneous invariants.PointC is Ball point. Points A and B always make arc motions around their fixed pivots, so points A, B, and C are curvature invariant points and the curvature-stationary point curve equation is satisfied:(11)1PA=1Msinαa+1Ncosαa,1PB=1Msinαb+1Ncosαb,1PC=1Msinα1+1Ncosα1.Combining (9) and (11), we have the following:(12)PA0+dsinαad⋅PA0sinαa=1Msinαa+1Ncosαa,PB0+dsinαbd⋅PB0sinαb=1Msinαb+1Ncosαb,1dsinα1=1Msinα1+1Ncosα1.CounteractingM and N in (12), we can obtain the following:(13)PB0sinαb−α1sinαacosαa+PA0sinα1−αasinαbcosαbPA0⋅PB0sinαacosαasinαbcosαbsinα1cosα1=0.The anglesαa, αb, and α1 are defined as follows:(14)αa=αa′−φ,αb=αb′−φ,α1=α1′−φ,where αa′, αb′, α1′, and φ are angles between the rays PA0, PB0, PC, and the polar axis Pt to the x-axis, respectively.It is easy to know that if we want to guarantee that (13) has solutions, the instantaneous center P must not coincide with the fixed pivots A0 and B0; the angles α1, αa, and αb are not equal to 0 and π/2; the cubic curvature-stationary point curve does not degenerate; and the mechanism is in general configuration.Substituting (14) into (13) and rearranging it, we have the following:(15)k1sinαa′−φcosαa′−φ−k2sinαb′−φcosαb′−φ=0,where(16)k1=PB0sinαb−α1,k2=PA0sinαa−α1.Only one unknown is direction angleφ of the polar axis Pt in (15). After rearranging (15), we can get(17)tan2ϕ=k1sin2αa′−k2sin2αb′k1cos2αa′−k2cos2αb′.From (17), the direction angles φ and φ′ of the polar axes Ptlie in the interval (0, π), and the difference between φ and φ′ is π/2 (as shown in Figure 5); then, the polar angle α1 can be calculated from the third formula of (13) and substituted into the third formula of (8) to calculate the inflection circle diameter d. Position angle γ of the inflection circle is as follows: γ=π/2−α1, γ′=−α1. The center O of the inflection circle is calculated by (5). The following steps are the same as the second-order osculation mechanism and will not be repeated here.Figure 5
Mechanism synthesis model when1/M = 0. (a) Special configuration with A0A and A0B0 being collinear. (b) Special configuration with A0A and AB being collinear.
(a)(b)


## 4. Special Configuration Mechanism Synthesis Model

The curvature-stationary point curve will degenerate into the second-order curve (circle) and straight line from the third order if any of the anglesα1, αa, and αb is equal to 0 or π/2. At the same time, the moving pivots A and B, the fixed pivots A0 and B0, the inflection poles JA and JB, the instantaneous center P, and the Ball point C are in a particular geometric position relationship.When1/M = 0, (10) can be simplified:(18)R=Ncosα,sinα=0.When1/N = 0,(19)R=Msinα,cosα=0.When1/M = 0 and 1/N = 0,(20)cosα=0,sinα=0.Correspondences between the curvature-stationary curve degradation rules and mechanism configurations are shown in Table1.Table 1
Correspondences between the curvature-stationary curve degradation rules and mechanism configurations.
ConditionEquationCurve formInitial configurationPosition of moving pivotsPolar angle valueRelationship between points1/M ≠ 0 and 1/N ≠ 0Equation (10)Cubic curveGeneral configurationGeneral positionEquation (14)P, A0, B0, A, B, JA, JB, and C do not coincide with each other1/M = 0Equation (18)A straight line (linePt) and a circle with its center on line PtA side link (A0A) and a frame (A0B0) are collinearPointsA and C on a circleαb=0P, B0, and JB are coincidentA side link (A0A) and a coupler (AB) are collinearPointB on line PtP, B, and JB are coincident1/N = 0Equation (19)A straight line (linePn) and a circle with its center on line PnA coupler (AB) is parallel to a frame (A0B0)PointsA and B on a circleα1=π/2The connection ofJA and JB is parallel to AB and A0B0PointC on line PnThe Ball point on a side link (B0B)PointA on a circleα1=π/2C and JB are coincidentPointsB and C on line Pnαb=π/2A coupler (AB) is perpendicular to a side link (B0B)PointsA and B on a circleα1=π/2The connection ofJA and JB is parallel to AB and A0B0; C and JB are coincidentPointsB and C on line Pnαb=π/21/M = 0 and 1/N = 0Equation (20)Two straight lines (linePt and Pn)Two side links (A0A and B0B) are perpendicularPointB on Ptα1=π/2P, B0, and JB are coincidentPointsA and C on line Pnαa=π/2,αb=0C and JA are coincidentThe knowns are pointC (xC, yC) on the expected straight line, direction angleβof the expected straight line, and fixed pivotA0 (xA0, yA0). It is essential to add the configuration with two side links (A0A and B0B) being perpendicular; only coordinate x of the fixed pivot A0 needs to be given. The unknowns are fixed pivotB0 (xB0, yB0) and the moving pivotsA and B. The direction angleθ of A0A is the first design variable. The instantaneous center P is determined by (3) according to the direction angle θ.


### 4.1. Special Configuration 1/M = 0

The mechanism is in the configuration in which a side link (A0A) and a frame (A0B0) are collinear, 1/M = 0, and the cubic stationary point curve degenerates into a circle m1 whose center is on the polar tangent Pt and a straight line m2 that is coincident with the polar tangent Pt, as shown in Figure 6(a). Because a side link (A0A) and a frame (A0B0) are collinear and two side links intersect in the instantaneous center P, the point P coincides with the point B0. Then, according to (1), we can infer that the moving pivot B selected randomly on the polar tangent Pt can satisfy the Euler–Savary equation, and the points A and C are located on the circle m1.Figure 6
Mechanism synthesis model when1/N = 0. (a) Special configuration with a coupler (AB) being parallel to a frame (A0B0). (b) Special configuration with the Ball point on a side link (B0B). (c) Special configuration with a coupler (AB) being perpendicular to a side link (B0B).
(a)(b)(c)The inflection circle is obtained by solving for the angleφ of the polar axis Pt.PointsAandCsatisfy the Euler–Savary equation. Thus, (9) is transformed into the following:(21)PA=d⋅PA0sinαaPA0+dsinαa,d=PCsinα1.At the same time, pointsA and C are located on the circle m1, which satisfies the curvature-stationary point curve equation. From (18), we can obtain the following:(22)PA=Ncosαa,PC=Ncosα1.We counteractN in (20) to get the following:(23)PA=PCcosαacosα1.Equations (21) and (23) are combined:(24)PCcosαacosα1=PA0sinαa⋅PC/sinα1PA0+sinαa⋅PC/sinα1.Substituting the first formula of (14) into (24) and rearranging it, we get the following:(25)sin2αa′−φ=2kPC,where k=PA0sinαa−α1,k=PA0sinαa−α1; it represents the distance from the fixed pivot A0 to the line PC.Two values ofφ in (0∼π) are calculated by (23); the inflection circle is then determined. The process that follows is the same as that for the general configuration mechanism.It is not hard to see that if we want (25) to have the solution, then it must satisfy the following:(26)2kPC≤1.The equation of the linePC is as follows:(27)y−ycx−xc=tanβ.Therefore, the following can be obtained:(28)k=xA0tanβ−yA0−xctanβ+yctan2β+1.Meanwhile,(29)PC=xP−xc2+yP−yc2.Get (xP, yP) from (3); the ranges of θ are calculated by bringing (28) and (29) into (26).When2k=PC, θ is the critical value. Suppose point P∗ is located on PC; the direct line of the link A0A cannot fall within the shaded area between A0P1∗ and A0P2∗, as shown in Figure 5.The coordinate of the critical pointP∗ is as follows:(30)xP∗=xC±2kcosβ−π2,yP∗=yC±2ksinβ−π2,where the two critical points P1∗ and P2∗ can be obtained by changing the plus and minus signs, and the direction angles of the corresponding link A0A are θ1∗ and θ2∗. The value range of the design variable is θ∈0,2π, but θ∉θ1∗,θ2∗ and θ∉θ1∗+π,θ2∗+π.Introducing the position angleγB of the moving pivot B as the second design variable, γB=∠POB, which takes the value interval (−π/2, π/2) and line OP reversed to line OB as positive:(31)PB=OPtanγB.The moving pivotB can be found by (31).When the fixed pivotB0is interchanged withB, the linkA0A is collinear with the linkAB. The mechanism can also guide coupler point C to walk an approximate straight line, as shown in Figure 5(b).


### 4.2. Special Configuration 1/N = 0

When1/N = 0, the cubic stationary point curve degenerates into a circle m1 whose center is on the polar normal Pn and a line m2 which is coincident with the polar normal Pn. The mechanism in three configurations can be designed according to this degradation law: the coupler is parallel to the frame (AB∥A0B0), the Ball point is located on a side link (B0B), and the coupler is perpendicular to a side link (AB⊥B0B). For the three configuration mechanisms, the common characteristic is that Ball point C is located on the polar normal line Pn and the moving pivot A is on the circle m1,m1; the difference is in the position of the moving pivot B, as shown in Figure 6.The moving pointCsatisfies both the Euler–Savary equation and the curvature-stationary point curve equation; thus,(32)d=PCsinα1,cosα1=0.We can obtain the inflection circle diameterd and direction angle φ of the polar axis Pt:(33)d=PC,φ=β.After finding the inflection circle, the inflection poleJA and the corresponding moving pivot A can be calculated according to (6) and (8). Then, the calculation steps of the moving pivot B and the fixed pivot B0 are as follows:(1)
For the configuration with a coupler (AB) parallel to a frame (A0B0), moving pivots A and B are on the circle m1 simultaneously. The second design variable is position angle γB of the moving pivot B. γB is defined as γB=∠CPB, which takes on (−π/2, π/2), and γB is positive when line PC is turned counterclockwise to line PB. We can obtain the following by (17):(34)PA=Msinαa,PB=Msinαb,whereαb=γB+π/2.So the solution forPB is as follows:(35)PB=PAcosγBsinαa.Find the coordinate of the moving pivotB by (35):(36)xB=xP−PBsinβ+γB,yB=yP+PBcosβ+γB.The coupler curve is symmetrical when the moving pivotsA and B are axisymmetric about line Pn. Coupler AB is parallel to the frame A0B0; thus,(37)PB0=PA0PAPB.We can obtain the fixed pivotB0 by (37).(2)
For the configuration with Ball point on a side link (B0B), the moving pivot B can be arbitrarily selected on the polar normal line Pn. Therefore, the reference point TB on the polar tangent line Pt is introduced, and PTB is equal to the radius of the inflection circle. The second design variable is position angle γB of the moving pivot B. γB is defined as γB=∠PTBB, which takes on (−π/2, π/2), and γB is positive when line TBP is turned counterclockwise to line TBB; thus,(38)PB=dtanγB2.Find the coordinate of the moving pivotB by (38):(39)xB=xP+PBsinβ,yB=yP−PBcosβ.After the moving pivotB and the inflection pole JB (coincident with Ball point C) are determined, the coordinate of the fixed pivot B0 is calculated by the second formula of (1).(3)
For the configuration with a coupler (AB) perpendicular to a side link (B0B), the moving pivot B is on the polar normal line Pn and also on the circle m1, which can be seen as a particular case of the configuration (1) and (2). At this moment, Ball point C is on the side link B0B.Since the moving pivotsA and B are located on the circle m1, rearranging (33), we get the following:(40)PB=PAsinαbsinαa.The moving pivotB is also on the polar normal line Pn; thus,(41)cosαb=0.(36) and (41) are combined to solve PB, and the coordinate of the moving pivot B is calculated by (39). Finally, the fixed pivot B0 coordinates are calculated by the second formula of (1).


### 4.3. Special Configuration 1/M = 0, 1/N = 0

For the configuration withA0A⊥B0B, the stagnation point curve degenerates into a straight line m1 coincident with the polar tangent Pt and a straight line m2 coincident with the polar normal line Pn if the three points A, B, and C are all on the curvature-stagnation point curve. At this moment, point B must lie on the polar tangent Pt if points A and C lie on the polar normal Pn, as shown in Figure 7.Figure 7
Special configuration with two side links (A0A and B0B) being perpendicular.In particular, the direction angleθ of the link A0A cannot be adjusted in the design; its relationship with the direction angle β of the expected straight line is θ=β−π/2. Link A0A coincides with the expected straight-line normal through point C, so the instant center P can be selected randomly on line Pn. Introduce the position angle γP of the instantaneous center P as the first design variable; γP is defined as γP=∠CTPP, which takes on (−π/2, π/2), where the point TP is a reference point on the expected straight line; let CTP=CA0,CTP=CA0; γP is positive when line TPC is turned counterclockwise to line TPP; thus,(42)CP=CA0tanγP.The instantaneous centerP (fixed pivot B0) can be calculated by (42), and the moving pivot A can be calculated by the first formula of (1).The moving pivotB can be selected randomly on line Pt. Introduce the position angle γB as the second design variable. γB is defined as γB=∠PCB, which takes on (−π/2, π/2) and is positive when line CP is turned counterclockwise to line CB; thus,(43)PB=CPtanγB.The moving pivotB is determined by (43).


## 4.1. Special Configuration 1/M = 0

The mechanism is in the configuration in which a side link (A0A) and a frame (A0B0) are collinear, 1/M = 0, and the cubic stationary point curve degenerates into a circle m1 whose center is on the polar tangent Pt and a straight line m2 that is coincident with the polar tangent Pt, as shown in Figure 6(a). Because a side link (A0A) and a frame (A0B0) are collinear and two side links intersect in the instantaneous center P, the point P coincides with the point B0. Then, according to (1), we can infer that the moving pivot B selected randomly on the polar tangent Pt can satisfy the Euler–Savary equation, and the points A and C are located on the circle m1.Figure 6
Mechanism synthesis model when1/N = 0. (a) Special configuration with a coupler (AB) being parallel to a frame (A0B0). (b) Special configuration with the Ball point on a side link (B0B). (c) Special configuration with a coupler (AB) being perpendicular to a side link (B0B).
(a)(b)(c)The inflection circle is obtained by solving for the angleφ of the polar axis Pt.PointsAandCsatisfy the Euler–Savary equation. Thus, (9) is transformed into the following:(21)PA=d⋅PA0sinαaPA0+dsinαa,d=PCsinα1.At the same time, pointsA and C are located on the circle m1, which satisfies the curvature-stationary point curve equation. From (18), we can obtain the following:(22)PA=Ncosαa,PC=Ncosα1.We counteractN in (20) to get the following:(23)PA=PCcosαacosα1.Equations (21) and (23) are combined:(24)PCcosαacosα1=PA0sinαa⋅PC/sinα1PA0+sinαa⋅PC/sinα1.Substituting the first formula of (14) into (24) and rearranging it, we get the following:(25)sin2αa′−φ=2kPC,where k=PA0sinαa−α1,k=PA0sinαa−α1; it represents the distance from the fixed pivot A0 to the line PC.Two values ofφ in (0∼π) are calculated by (23); the inflection circle is then determined. The process that follows is the same as that for the general configuration mechanism.It is not hard to see that if we want (25) to have the solution, then it must satisfy the following:(26)2kPC≤1.The equation of the linePC is as follows:(27)y−ycx−xc=tanβ.Therefore, the following can be obtained:(28)k=xA0tanβ−yA0−xctanβ+yctan2β+1.Meanwhile,(29)PC=xP−xc2+yP−yc2.Get (xP, yP) from (3); the ranges of θ are calculated by bringing (28) and (29) into (26).When2k=PC, θ is the critical value. Suppose point P∗ is located on PC; the direct line of the link A0A cannot fall within the shaded area between A0P1∗ and A0P2∗, as shown in Figure 5.The coordinate of the critical pointP∗ is as follows:(30)xP∗=xC±2kcosβ−π2,yP∗=yC±2ksinβ−π2,where the two critical points P1∗ and P2∗ can be obtained by changing the plus and minus signs, and the direction angles of the corresponding link A0A are θ1∗ and θ2∗. The value range of the design variable is θ∈0,2π, but θ∉θ1∗,θ2∗ and θ∉θ1∗+π,θ2∗+π.Introducing the position angleγB of the moving pivot B as the second design variable, γB=∠POB, which takes the value interval (−π/2, π/2) and line OP reversed to line OB as positive:(31)PB=OPtanγB.The moving pivotB can be found by (31).When the fixed pivotB0is interchanged withB, the linkA0A is collinear with the linkAB. The mechanism can also guide coupler point C to walk an approximate straight line, as shown in Figure 5(b).


## 4.2. Special Configuration 1/N = 0

When1/N = 0, the cubic stationary point curve degenerates into a circle m1 whose center is on the polar normal Pn and a line m2 which is coincident with the polar normal Pn. The mechanism in three configurations can be designed according to this degradation law: the coupler is parallel to the frame (AB∥A0B0), the Ball point is located on a side link (B0B), and the coupler is perpendicular to a side link (AB⊥B0B). For the three configuration mechanisms, the common characteristic is that Ball point C is located on the polar normal line Pn and the moving pivot A is on the circle m1,m1; the difference is in the position of the moving pivot B, as shown in Figure 6.The moving pointCsatisfies both the Euler–Savary equation and the curvature-stationary point curve equation; thus,(32)d=PCsinα1,cosα1=0.We can obtain the inflection circle diameterd and direction angle φ of the polar axis Pt:(33)d=PC,φ=β.After finding the inflection circle, the inflection poleJA and the corresponding moving pivot A can be calculated according to (6) and (8). Then, the calculation steps of the moving pivot B and the fixed pivot B0 are as follows:(1)
For the configuration with a coupler (AB) parallel to a frame (A0B0), moving pivots A and B are on the circle m1 simultaneously. The second design variable is position angle γB of the moving pivot B. γB is defined as γB=∠CPB, which takes on (−π/2, π/2), and γB is positive when line PC is turned counterclockwise to line PB. We can obtain the following by (17):(34)PA=Msinαa,PB=Msinαb,whereαb=γB+π/2.So the solution forPB is as follows:(35)PB=PAcosγBsinαa.Find the coordinate of the moving pivotB by (35):(36)xB=xP−PBsinβ+γB,yB=yP+PBcosβ+γB.The coupler curve is symmetrical when the moving pivotsA and B are axisymmetric about line Pn. Coupler AB is parallel to the frame A0B0; thus,(37)PB0=PA0PAPB.We can obtain the fixed pivotB0 by (37).(2)
For the configuration with Ball point on a side link (B0B), the moving pivot B can be arbitrarily selected on the polar normal line Pn. Therefore, the reference point TB on the polar tangent line Pt is introduced, and PTB is equal to the radius of the inflection circle. The second design variable is position angle γB of the moving pivot B. γB is defined as γB=∠PTBB, which takes on (−π/2, π/2), and γB is positive when line TBP is turned counterclockwise to line TBB; thus,(38)PB=dtanγB2.Find the coordinate of the moving pivotB by (38):(39)xB=xP+PBsinβ,yB=yP−PBcosβ.After the moving pivotB and the inflection pole JB (coincident with Ball point C) are determined, the coordinate of the fixed pivot B0 is calculated by the second formula of (1).(3)
For the configuration with a coupler (AB) perpendicular to a side link (B0B), the moving pivot B is on the polar normal line Pn and also on the circle m1, which can be seen as a particular case of the configuration (1) and (2). At this moment, Ball point C is on the side link B0B.Since the moving pivotsA and B are located on the circle m1, rearranging (33), we get the following:(40)PB=PAsinαbsinαa.The moving pivotB is also on the polar normal line Pn; thus,(41)cosαb=0.(36) and (41) are combined to solve PB, and the coordinate of the moving pivot B is calculated by (39). Finally, the fixed pivot B0 coordinates are calculated by the second formula of (1).


## 4.3. Special Configuration 1/M = 0, 1/N = 0

For the configuration withA0A⊥B0B, the stagnation point curve degenerates into a straight line m1 coincident with the polar tangent Pt and a straight line m2 coincident with the polar normal line Pn if the three points A, B, and C are all on the curvature-stagnation point curve. At this moment, point B must lie on the polar tangent Pt if points A and C lie on the polar normal Pn, as shown in Figure 7.Figure 7
Special configuration with two side links (A0A and B0B) being perpendicular.In particular, the direction angleθ of the link A0A cannot be adjusted in the design; its relationship with the direction angle β of the expected straight line is θ=β−π/2. Link A0A coincides with the expected straight-line normal through point C, so the instant center P can be selected randomly on line Pn. Introduce the position angle γP of the instantaneous center P as the first design variable; γP is defined as γP=∠CTPP, which takes on (−π/2, π/2), where the point TP is a reference point on the expected straight line; let CTP=CA0,CTP=CA0; γP is positive when line TPC is turned counterclockwise to line TPP; thus,(42)CP=CA0tanγP.The instantaneous centerP (fixed pivot B0) can be calculated by (42), and the moving pivot A can be calculated by the first formula of (1).The moving pivotB can be selected randomly on line Pt. Introduce the position angle γB as the second design variable. γB is defined as γB=∠PCB, which takes on (−π/2, π/2) and is positive when line CP is turned counterclockwise to line CB; thus,(43)PB=CPtanγB.The moving pivotB is determined by (43).


## 5. Visual Optimization Method

According to the synthesis model of the mechanism proposed in this paper, the design variables can be varied to obtain an infinite number of approximate straight-line mechanisms for the expected straight line and fixed pivots. The initial position of linkA0A is determined by the first design variable. The link B0B is then determined for the third-order osculation of the general configuration and the configuration with a coupler (AB) perpendicular to a side link (B0B). Nevertheless, link B0B needs to be determined by the second design variable for other configurations. Since both design variables take values in (−π/2, π/2), an infinite number of mechanisms can be expressed in a limited plane area and the area is defined mechanism solution region.In practice, maybe, designers put forward some demands. The link length maximum and the range of pivot coordinates can restrict mechanism within specific workspaces. Transmission angle is used to measure the performance of a mechanism, and the designer should set its limit to promote sound force transmission. There are other constraints, such as straightness, types, and branch defect. Whetheror not the constraints are considered and constraint scopes depend on the practical needs or designer's interests. A feasible solution region is defined as zones in which solutions satisfy all constraints above in the solution region.


### 5.1. Mechanism Optimization Steps

The main steps of the solution region optimization method can be summarized as below:Step 1. Input initial conditions: the fixed pivotA0 (xA0, yA0), point C (xC, yC) on the expected straight line, and direction angle β of the expected straight line.Step 2. Select the range of design variables and the calculation step.Step 3. Set geometric and kinematic constraints. Choose the optimization goal according to the design purpose, such as the longest approximate straight line and the minimum deviation of straight line.Step 4. Mechanism synthesis: based on the above mathematical model, get all solutions to form solution regions and figure their attributes.Step 5. Feasible solution regions are generated by screening out solutions which are satisfying all constraints.Step 6. An optimum solution is obtained by comparison with the value of the optimization goal of the mechanism in feasible solution regions.


### 5.2. Example of Mechanism Synthesis

The expected straight-line and the fixed points are given in Table2.Figure 8
Straight-line performance parameters.Figure 9
Contour sheets of the length. (a) Second-order osculation straight-line mechanism. (b) Third-order osculation straight-line mechanism. (c) Special configuration withAB⊥B0B. (d) Special configuration with A0A and A0B0 being collinear. (e) Special configuration with A0A and AB being collinear. (f) Special configuration with AB∥A0B0. (g) Special configuration with Ball point on B0B. (h) Special configuration with A0A⊥B0B.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 10
The optimal mechanism of different configurations. (a) Two-order osculation straight-line mechanism (M1). (b) Three-order osculation straight-line mechanism (M2). (c) Special configuration with A0A and A0B0 being collinear (M3). (d) Special configuration with A0A and AB being collinear (M4). (e) Special configuration with AB∥A0B0 (M5). (f) Special configuration with Ball point on link B0B (M6). (g) Special configuration with AB∥B0B (M7). (h) Special configuration with A0A⊥B0B (M8).
(a)(b)(c)(d)(e)(f)(g)(h)Table 2
Condition parameters.
Geometric featuresxA0yA0xB0yB0xCyCβ (°)General40108002050140SpecialA0A⊥B0B———Others10The set of kinematic constraints includes the following:Frame:−100≤xA0, yA0, xB0, yB0≤100, ≤100.Types: crank-rocker linkage, double-rocker linkage, or triple-rocker linkage.Transmission performance: minimum transmission angleγmin of crank-rocker mechanism ≥30°, ≥30°.Link length: link length ratioiR =  lmin/lmax≥0.1, single link length ls≤200, sum lsum of link length ≤600, ≤600.Straightness: deviation of straight lineΔh≤0.5; length L of the approximate straight line ≥ 60, as shown in Figure 8.Defect: no branch defect for double-rocker linkage and triple-rocker linkage.Coupler curve: widthW≤500, height H≤200.Optimization goal: It is desirable to have the most extended straight-line segment within the allowable deviation of the straight-line.The optimal search algorithm is as follows:(1)
Set the range of design variables and the step according to the working space and the demanded precision, such as the whole solution region (−π/2, π/2) and step Δ=1°.(2)
The dimensional and performance parameters of the mechanism are calculated by changing the design variables according to the step. This paper gives the contour sheet of approximate straight-line length when the mechanism is in various configuration conditions, as shown in Figure9.The contour sheet of the third-order osculation straight-line mechanism and the unique configuration with a coupler (AB) perpendicular to a side link (B0B) are two-dimensional graphs because there is only one design variable. The ordinate represents the approximate straight-line length, as shown in Figures 9(b) and 9(c). The other configurations are three-dimensional graphs (the values on the line are the approximate straight-line length). For the third-order mechanism, two inflection circles can be solved according to (9), so there are two solution cases, as shown in Figure 9(b). For the particular configuration 1/M = 0, two critical points P1∗, P2∗ and two direction angles θ1∗, θ2∗ of the corresponding side link A0A can be obtained according to (24), so there are two solution cases and no solution regions as shown in Figures 9(d) and 9(e).(3)
The mechanisms obtained by combining design variables in pairs are screened automatically. Check mechanism attributes item by item, as long as one does not conform to the constraints and the mechanism is unfeasible. The solution regions composed of feasible solutions that satisfy all constraints are drawn, as shown in Figure9, in thered zone, and the optimal solution position is marked by a red dot.(4)
The range of design variables and the step can be reduced to get the optimal solution for different configurations, as shown in Figure10. Points D and D′ are the junction of two branches in Figure 10, and it is easy to see that points D and D′ are not on the approximate straight-line segment, which means that these mechanisms have no branch defect. The dimensional and performance parameters are given in Tables 3 and 4, where M1–M8 correspond to the optimal solution in Figure 10.Table 3
Dimensional parameters.
MechanismθγA0B0ABM160.2125.69(40, 10)(80, 0)(98.37, 111, 96)(116.83, 103.54)M260.50—(40, 10)(80, 0)(98.66, 113.68)(117.81, 109.87)M335.05−22.41(40, 10)(−90.21, −81.34)(−19.26, −31.57)(−102.74, −48.29)M433.99−32.93(40, 10)(−100.00, −23.83)(−15.55, −27.46)(−93.35, −73.17)M559.88−11.51(40, 10)(−26.17, 68.81)(103.25, 119.02)(71.95, 146.84)M659.41−48.40(40, 10)(−22.72, −0.91)(107.1, 123.5)(84.53, 136.9)M760.11—(40, 10)(3.86, 30.77)(101.49, 116.99)(84.43, 126.79)M8−78.15−8.2(40, 73.84)(98.37, 111,96)(−23.13, −1.4)(−58.95, −77.33)Table 4
Performance parameters.
MechanismTypeWHiRLΔhM13306.34199.8570.17143.800.5M23305.13199.90.15111.40M38175.4963.330.22128.31M48222.29122.3410.37109.57M53316.74197.9040.33117.66M63325.27199.8010.14124.65M73310.98199.8920.16115.04M88160.5528.240.12121.42From the example, the following can be seen:(1)
Both the second-order and the third-order mechanism design methods can be used to solve the approximate straight-line mechanism synthesis problem.(2)
The third-order mechanism solution is a subset of the synthesized second-order mechanism set which is characterized by the curvature of the trajectory at the pointC as a stationary point. In the actual design, if the designer is very concerned about the straightness in the straight-line working segment, the location needs to be used as the link point C. Designers can consider choosing the third-order mechanism design method to converge quickly and find the mechanism with higher straight-line performance.(3)
If the third-order mechanism design method cannot get a satisfactory mechanism due to the motion space, transmission angle, branch defect, length, and so on, or the designer is more concerned about a longer straight-line working segment, the designer can consider using the second-order mechanism design method to obtain more mechanisms using the position angle of the inflection circle. Although the calculation and search time are extended, the optimal mechanism with good straight-line performance can be obtained more easily to satisfy the tolerance.


## 5.1. Mechanism Optimization Steps

The main steps of the solution region optimization method can be summarized as below:Step 1. Input initial conditions: the fixed pivotA0 (xA0, yA0), point C (xC, yC) on the expected straight line, and direction angle β of the expected straight line.Step 2. Select the range of design variables and the calculation step.Step 3. Set geometric and kinematic constraints. Choose the optimization goal according to the design purpose, such as the longest approximate straight line and the minimum deviation of straight line.Step 4. Mechanism synthesis: based on the above mathematical model, get all solutions to form solution regions and figure their attributes.Step 5. Feasible solution regions are generated by screening out solutions which are satisfying all constraints.Step 6. An optimum solution is obtained by comparison with the value of the optimization goal of the mechanism in feasible solution regions.


## 5.2. Example of Mechanism Synthesis

The expected straight-line and the fixed points are given in Table2.Figure 8
Straight-line performance parameters.Figure 9
Contour sheets of the length. (a) Second-order osculation straight-line mechanism. (b) Third-order osculation straight-line mechanism. (c) Special configuration withAB⊥B0B. (d) Special configuration with A0A and A0B0 being collinear. (e) Special configuration with A0A and AB being collinear. (f) Special configuration with AB∥A0B0. (g) Special configuration with Ball point on B0B. (h) Special configuration with A0A⊥B0B.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 10
The optimal mechanism of different configurations. (a) Two-order osculation straight-line mechanism (M1). (b) Three-order osculation straight-line mechanism (M2). (c) Special configuration with A0A and A0B0 being collinear (M3). (d) Special configuration with A0A and AB being collinear (M4). (e) Special configuration with AB∥A0B0 (M5). (f) Special configuration with Ball point on link B0B (M6). (g) Special configuration with AB∥B0B (M7). (h) Special configuration with A0A⊥B0B (M8).
(a)(b)(c)(d)(e)(f)(g)(h)Table 2
Condition parameters.
Geometric featuresxA0yA0xB0yB0xCyCβ (°)General40108002050140SpecialA0A⊥B0B———Others10The set of kinematic constraints includes the following:Frame:−100≤xA0, yA0, xB0, yB0≤100, ≤100.Types: crank-rocker linkage, double-rocker linkage, or triple-rocker linkage.Transmission performance: minimum transmission angleγmin of crank-rocker mechanism ≥30°, ≥30°.Link length: link length ratioiR =  lmin/lmax≥0.1, single link length ls≤200, sum lsum of link length ≤600, ≤600.Straightness: deviation of straight lineΔh≤0.5; length L of the approximate straight line ≥ 60, as shown in Figure 8.Defect: no branch defect for double-rocker linkage and triple-rocker linkage.Coupler curve: widthW≤500, height H≤200.Optimization goal: It is desirable to have the most extended straight-line segment within the allowable deviation of the straight-line.The optimal search algorithm is as follows:(1)
Set the range of design variables and the step according to the working space and the demanded precision, such as the whole solution region (−π/2, π/2) and step Δ=1°.(2)
The dimensional and performance parameters of the mechanism are calculated by changing the design variables according to the step. This paper gives the contour sheet of approximate straight-line length when the mechanism is in various configuration conditions, as shown in Figure9.The contour sheet of the third-order osculation straight-line mechanism and the unique configuration with a coupler (AB) perpendicular to a side link (B0B) are two-dimensional graphs because there is only one design variable. The ordinate represents the approximate straight-line length, as shown in Figures 9(b) and 9(c). The other configurations are three-dimensional graphs (the values on the line are the approximate straight-line length). For the third-order mechanism, two inflection circles can be solved according to (9), so there are two solution cases, as shown in Figure 9(b). For the particular configuration 1/M = 0, two critical points P1∗, P2∗ and two direction angles θ1∗, θ2∗ of the corresponding side link A0A can be obtained according to (24), so there are two solution cases and no solution regions as shown in Figures 9(d) and 9(e).(3)
The mechanisms obtained by combining design variables in pairs are screened automatically. Check mechanism attributes item by item, as long as one does not conform to the constraints and the mechanism is unfeasible. The solution regions composed of feasible solutions that satisfy all constraints are drawn, as shown in Figure9, in thered zone, and the optimal solution position is marked by a red dot.(4)
The range of design variables and the step can be reduced to get the optimal solution for different configurations, as shown in Figure10. Points D and D′ are the junction of two branches in Figure 10, and it is easy to see that points D and D′ are not on the approximate straight-line segment, which means that these mechanisms have no branch defect. The dimensional and performance parameters are given in Tables 3 and 4, where M1–M8 correspond to the optimal solution in Figure 10.Table 3
Dimensional parameters.
MechanismθγA0B0ABM160.2125.69(40, 10)(80, 0)(98.37, 111, 96)(116.83, 103.54)M260.50—(40, 10)(80, 0)(98.66, 113.68)(117.81, 109.87)M335.05−22.41(40, 10)(−90.21, −81.34)(−19.26, −31.57)(−102.74, −48.29)M433.99−32.93(40, 10)(−100.00, −23.83)(−15.55, −27.46)(−93.35, −73.17)M559.88−11.51(40, 10)(−26.17, 68.81)(103.25, 119.02)(71.95, 146.84)M659.41−48.40(40, 10)(−22.72, −0.91)(107.1, 123.5)(84.53, 136.9)M760.11—(40, 10)(3.86, 30.77)(101.49, 116.99)(84.43, 126.79)M8−78.15−8.2(40, 73.84)(98.37, 111,96)(−23.13, −1.4)(−58.95, −77.33)Table 4
Performance parameters.
MechanismTypeWHiRLΔhM13306.34199.8570.17143.800.5M23305.13199.90.15111.40M38175.4963.330.22128.31M48222.29122.3410.37109.57M53316.74197.9040.33117.66M63325.27199.8010.14124.65M73310.98199.8920.16115.04M88160.5528.240.12121.42From the example, the following can be seen:(1)
Both the second-order and the third-order mechanism design methods can be used to solve the approximate straight-line mechanism synthesis problem.(2)
The third-order mechanism solution is a subset of the synthesized second-order mechanism set which is characterized by the curvature of the trajectory at the pointC as a stationary point. In the actual design, if the designer is very concerned about the straightness in the straight-line working segment, the location needs to be used as the link point C. Designers can consider choosing the third-order mechanism design method to converge quickly and find the mechanism with higher straight-line performance.(3)
If the third-order mechanism design method cannot get a satisfactory mechanism due to the motion space, transmission angle, branch defect, length, and so on, or the designer is more concerned about a longer straight-line working segment, the designer can consider using the second-order mechanism design method to obtain more mechanisms using the position angle of the inflection circle. Although the calculation and search time are extended, the optimal mechanism with good straight-line performance can be obtained more easily to satisfy the tolerance.


## 6. Conclusion

This paper presents a novel method to solve the synthesis problem of the approximate straight-line mechanism, providing a general and effective synthesis method for the mechanism that realizes an approximate straight-line trajectory. The method uses the analytical geometry method combined with computer graphics technology and visual optimization design. A unified mathematical model for the synthesis of this kind of mechanism is established, and the design variables are limited to (−π/2, π/2) to generate a straight-line mechanism with at least second-order or third-order osculation. The designer can chose a general configuration or a certain specific configuration according to the design requirements. An infinite number of mechanisms will be obtained by changing the design variables. In addition to straightness, other motion conditions such as type, contour size of the coupler curve, and transmission performance must be considered for engineering purposes. To achieve efficient optimization, the kinematic properties of the mechanism that are of interest to the designer are calculated, and the mechanical properties are visualized graphically. By representing an infinite number of mechanisms in a finite solution region, the designer can get an intuitive and comprehensive understanding of the trend and distribution pattern of all mechanisms from the mechanism solution region with the design variables changing. Then, by applying kinematic constraints, the feasible mechanisms are automatically selected from the infinite number of mechanisms. The optimal high-precision straight-line mechanism satisfying linear performance and other kinematic requirements is solved according to the set optimization target, which provides a new technical means for designers to make fast and accurate decisions. The proposed method is an effective combination of traditional theory and modern design methods, which solves the synthesis problem of approximate straight-line mechanisms and the blindness of selecting from infinitely number mechanisms in the synthesis process.



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