Structural–Parametric Synthesis of the Planar Four-Bar and Six-Bar Function Generators With Revolute Joints

Baigunchekov, Zhumadil; Laribi, Med Amine; Carbone, Giuseppe; Kaiyrov, Rustem; Tolenov, Serik; Dosmagambet, Nurdaulet

doi:10.1115/1.4064253

Abstract

This paper studies a structural–parametric synthesis of the four-bar and Stephenson II, Stephenson III A, and Stephenson III B six-bar function generators. A four-bar function generator is formed by connecting two coordinate systems with given angles of rotation using a negative closing kinematic chain (CKC) of the RR type. Six-bar function generators are formed by connecting two coordinate systems using two CKCs: a passive CKC of the RRR type and a negative CKC of the RR type. The negative CKC of the RR type imposes one geometrical constraint to the relative motion of the links, and its geometric parameters are defined by least-squares approximation. Passive CKC of the RRR type does not impose a geometrical constraint, and the geometric parameters of its links are varied to satisfy the geometrical constraint of the negative CKC. Numerical results of the four-bar and six-bar function generators parametric synthesis are presented.

1 Introduction

The first studies on the design of function-generating linkages are due to Svoboda [1,2], who designed a Watt II function generator for the generation of the logarithmic function. Kinematic synthesis (dimensional or parametric synthesis) of mechanisms, including function-generating linkages, is carried out on the basis of the kinematic geometry of finite positions of a rigid body, approximation methods (polynomials), and computers [3]. The kinematic geometry of finite positions of a rigid body, which in the case of plane motion is known as the Burmester theory [4], is used for the synthesis of function generators in the works of Hunt [5], Bottema and Roth [6], Angeles and Bai [7,8], Pira and Cunaku [9], McCarthy and Soh [10], and others. The synthesis of mechanisms by kinematic geometry is clear and simple. However, these methods are applicable for a limited number of positions. For the kinematic synthesis of mechanisms with unlimited positions of the output links, the approximation methods are used, the foundations of which were laid by Chebyshev [11]. Approximation (algebraic, optimization) methods for the kinematic synthesis of four-bar and six-bar function generators Watt II, Stephenson II, and Stephenson III were used in the works of Freudenstein [12], Hartenberg and Denavit [13], McLarnan [14], Subbian and Flugrad [15,16], Kiper et al. [17], Hwang and Chen [18], Bulatovic et al. [19], Plecnik and McCarthy [20–22], and others. In Refs. [18,19,20], the polynomial homotopic software bertini [23] was used. At the intersection of kinematic geometry and approximation synthesis, a new direction in the kinematic synthesis of mechanisms—approximation kinematic geometry—has been created by Sarkissyan et al. [24–26]. Approximation kinematic geometry combines the advantages of methods of kinematic geometry and approximate synthesis of mechanisms, such as simplicity and unlimited positions of the output links. Based on approximation kinematic geometry by Baigunchekov et al. [27–31], the parallel mechanism and manipulator are synthesized.

In this work, a structural–parametric synthesis of four-bar and six-bar function generators is carried out, where the structures and geometric parameters of the links of the synthesized mechanisms are determined in arbitrary given discrete values of the input and output link angles. In structural–parametric synthesis, according to the given laws of motion of the input and output links, structural schemes and geometric parameters of the links of the synthesized mechanisms are simultaneously determined. At the same time, the structural–parametric synthesis of the designed mechanisms begins with the smallest number of links and becomes more complicated depending on the implementation of the specified laws of motion of the output links within the required accuracy. Depending on the complexity of the given laws of motion of the output links, it is possible to form mechanisms with complex structural schemes. Structural and kinematic analysis of complex mechanisms containing Assur groups of higher classes [32] is the subject of work [33–41]. In this paper, to analyze the positions of complex mechanisms, which is necessary to evaluate the results of parametric synthesis, a simple method of conditional generalized coordinates is proposed.

2 Structural Synthesis of the Planar Four-Bar and Six-Bar Function Generators With Revolute Joints

According to the developed principle of mechanism formation, they are formed by connecting the output object to the base using active, passive, and negative CKCs [27,28]. Active CKCs have active joints, passive CKCs have zero DOF, and negative CKCs have negative DOF. The active and passive CKCs impose geometrical constraints on the motion of the output object. The passive CKCs do not impose geometrical constraints.

The output object of the function generators is a link that performs a given rotary or translational motion relative to the base at a given motion of the input link. Let the input link and the output object perform rotational movements. We choose as the input and output links the coordinate systems Ax₁y₁ and Bx₂y₂ rotating relative to the base with rotation angles φ_i and ψ_i (Fig. 1(a)).

Fig. 1

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Structural synthesis of planar function generators

If we connect the planes of two moving coordinate systems Ax₁y₁ and Bx₂y₂ by a negative CKC CD of the RR type, then we obtain a structural scheme of a four-bar function generator ACDB. Connection of the planes of two moving coordinate systems Ax₁y₁ and Bx₂y₂ by binary link CD of the type RR is possible when the plane of the moving coordinate system Bx₂y₂ has a circular point (a point moving along a circle) D in relative motion to the coordinate system Ax₁y₁, or vice versa, i.e., when there is a circular point C in the plane of the coordinate system Ax₁y₁ in relative motion with respect to the coordinate system Bx₂y₂.

If none of the planes of the moving coordinate systems Ax₁y₁ and Bx₂y₂ do not have circular points in relative motion, then the planes of the two coordinate systems are connected by passive CKC CED of the RRR type. As a result, we obtain a structural scheme of the five-bar linkage ACEDB with two DOF (Fig. 1(b)). To form six-bar function generators from this five-bar linkage, we connect its non-adjacent links by binary link FG of the type RR, having one negative DOF (or a constraint that reduces the DOF of the system by one), defined by the Chebyshev formula

F = 3 n - 2 p_{5}

(1)

where n = 1 is the number of moving links, p₅ = 2 is the number of kinematic pairs of the fifth class, then F = −1.

Consequently, the negative CKC FG, imposing one geometrical constraint on the five-bar linkage with two DOF, forms six-bar function generators with one DOF. Figures 1(c)–1(f) show the structural schemes of the formed six-bar function generators. If links 1 and 2 of the five-bar linkage are connected by binary link FG, we obtain a Stephenson I function generator (Fig. 1(c)). If links 3 and 2 of the five-bar linkage are connected by binary link FG, we obtain a Stephenson II function generator (Fig. 1(d)). When link 3 of the five-bar linkage is connected to the base by binary link FG, we obtain a Stephenson III A function generator (Fig. 1(e)). When link 4 of the five-bar linkage is connected to the base by binary link FG, we obtain a Stephenson III B function generator (Fig. 1(f)).

3 Parametric Synthesis of Four-Bar Function Generator

Let given the function

ψ_{i} = f (φ_{i}), i = 1, 2, \dots, N

(2)

of the four-bar function generator (Fig. 2), where N is the number of finite positions of the moving planes Ax₁y₁ and Bx₂y₂.

Fig. 2

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Four-bar function generator

It is necessary to determine the synthesis parameters (geometric parameters) of the four-bar function generator that implements function . The synthesis parameters are

x_{C}^{(1)}, y_{C}^{(1)}, x_{D}^{(2)}, y_{D}^{(2)}

and l_CD, where

x_{C}^{(1)}, y_{C}^{(1)}

and

x_{D}^{(2)}, y_{D}^{(2)}

are the coordinates of the joints C and D in coordinate systems Ax₁y₁ and Bx₂y₂, respectively, and l_CD is the length of the CD link. Consider the movement of the coordinate system Bx₂y₂ relative to the coordinate system Ax₁y₁. In this case, point D moves along a circle centered at point C and with radius l_CD. Derive an equation

(x_{D_{i}}^{(1)} - x_{C}^{(1)})^{2} + (y_{D_{i}}^{(1)} - x_{C}^{(1)})^{2} - l_{C D}^{2} = 0

(3)

where

[\begin{matrix} x_{D i}^{(1)} \\ y_{D i}^{(1)} \end{matrix}] = [\begin{matrix} \cos φ_{i} & \sin φ_{i} \\ - \sin φ_{i} & \cos φ_{i} \end{matrix}] \cdot [\begin{matrix} X_{D i} - X_{A} \\ Y_{D i} - Y_{A} \end{matrix}]

(4)

[\begin{matrix} X_{D i} \\ Y_{D i} \end{matrix}] = [\begin{matrix} X_{B} \\ Y_{B} \end{matrix}] + [\begin{matrix} \cos ψ_{i} & - \sin ψ_{i} \\ \sin ψ & \cos ψ_{i} \end{matrix}] \cdot [\begin{matrix} x_{D}^{(2)} \\ y_{D}^{(2)} \end{matrix}]

(5)

Equation is an equation of geometrical constraint imposed by binary link CD of the type RR on the movements of two moving coordinate systems Ax₁y₁ and Bx₂y₂. The left side of Eq. will be denoted by Δq_i and called the weighted difference function

Δ q_{i} = (x_{D i}^{(1)} - x_{C}^{(1)})^{2} + (y_{D i}^{(1)} - x_{C}^{(1)})^{2} - l_{C D}^{2}

(6)

Parametric synthesis of a four-bar function generator is to determine five geometric parameters

x_{C}^{(1)}, y_{C}^{(1)}, x_{D}^{(2)}, y_{D}^{(2)}, and l_{C D}

from the minimum of function . Substituting expressions and into Eq. , we obtain

\begin{aligned} Δ q_{i} & = 2 {[- (X_{-} X_{A}) \cos φ_{i} - (Y_{B} - Y_{A}) \sin φ_{i}] \cdot x_{C}^{(1)} + \\ + [(X_{B} - X_{A}) \sin φ_{i} - (Y_{B} - Y_{A}) \cos φ_{i}] \cdot y_{C}^{(1)} + \\ + [\frac{1}{2} (x_{C}^{{(1)}^{2}} + y_{C}^{{(1)}^{2}} + x_{D}^{{(2)}^{2}} + y_{D}^{{(2)}^{2}} - l_{C D}^{2})] + \\ + [(X_{B} - X_{A}) \cos ψ_{i} + [(Y_{B} - Y_{A}) \sin ψ_{i}] \cdot x_{D}^{(2)} + \\ + [- (X_{B} - X_{A}) \sin ψ_{i} + (Y_{D} - Y_{A}) \cos ψ_{i}] \cdot y_{D}^{(2)} + \\ + [- \cos (ψ_{i} - φ_{i}) \cdot (x_{C}^{(1)} \cdot x_{D}^{(2)} + y_{C}^{(1)} \cdot y_{D}^{(2)})] + \\ + [\sin (ψ_{i} - φ_{i}) \cdot (x_{C}^{(1)} \cdot y_{D}^{(2)} - y_{C}^{(1)} \cdot x_{D}^{(2)})] + \\ + \frac{1}{2} [{(X_{B} - X_{A})}^{2} + (X_{B} - Y_{A})^{2}]} \end{aligned}

(7)

If we introduce the notations

\begin{aligned} p_{1} = x_{C}^{(1)}, p_{2} = y_{C}^{(1)}, p_{3} = \frac{1}{2} (x_{C}^{{(1)}^{2}} + y_{C}^{{(1)}^{2}} + x_{D}^{{(2)}^{2}} + y_{D}^{{(2)}^{2}} - l_{C D}^{2}), \\ p_{4} = x_{D}^{(2)}, p_{5} = y_{D}^{(2)}, \\ f_{1 i} = - (X_{B} - X_{A}) \cos φ_{i} - (Y_{B} - Y_{A}) \sin φ_{i}, \\ f_{2 i} = (X_{B} - X_{A}) \sin φ_{i} - (Y_{B} - Y_{A}) \cos φ_{i}, \\ f_{3} = 1, f_{4 i} = (X_{B} - X_{A}) \cos ψ_{i} + (Y_{B} - Y_{A}) \sin ψ_{i}, \\ f_{5 i} = - (X_{B} - X_{A}) \sin ψ_{i} + (Y_{B} - Y_{A}) \cos ψ_{i}, f_{6 i} = - \cos (ψ_{i} - φ_{i}), \\ f_{7 i} = \sin (ψ_{i} - φ_{i}), \\ f_{0 i} = \frac{1}{2} [{(X_{B} - X_{A})}^{2} + {(X_{B} - Y_{A})}^{2}] \end{aligned}

then Eq. takes the form

\begin{aligned} Δ q_{i} & = 2 [p_{1} f_{1 i} + p_{2} f_{2 i} + p_{3} f_{3} + p_{4} f_{4 i} + p_{5} f_{5 i} + (p_{1} p_{4} + p_{2} p_{5}) f_{6 i} + \\ + (p_{1} p_{5} - p_{2} p_{4}) f_{7 i} - f_{0 i}] \end{aligned}

(8)

Equation is linear in the following two groups of synthesis parameters:

p^{(1)} = [p_{1}, p_{2}, p_{3}], p^{(2)} = [p_{4}, p_{5}, p_{3}]

(9)

and is represented in two linear forms

\begin{aligned} Δ q_{i}^{(1)} & = 2 [p_{1} (f_{1 i} + p_{4} f_{6 i} + p_{5} f_{7 i}) + p_{2} (f_{2 i} + p_{5} f_{6 i} + p_{4} f_{7 i}) + p_{3} f_{3} + \\ + (p_{4} f_{4 i} + p_{5} f_{5 i} - f_{0 i})] \end{aligned}

(10)

and

\begin{aligned} Δ q_{i}^{(2)} & = 2 [p_{4} (f_{4 i} + p_{1} f_{6 i} - p_{2} f_{7 i}) + p_{5} (f_{5 i} + p_{2} f_{6 i} + p_{1} f_{7 i}) + p_{3} f_{3} + \\ + (p_{1} f_{1 i} + p_{2} f_{2 i -} f_{0 i})] \end{aligned}

(11)

If the following notations are introduced

g_{i}^{(1)} = [g_{1 i}^{(1)}, g_{2 i}^{(1)}, g_{3 i}^{(1)}], g_{i}^{(2)} = [g_{1 i}^{(2)}, g_{2 i}^{(2)}, g_{3 i}^{(2)}]

where

\begin{aligned} g_{1 i}^{(1)} = f_{1 i} + p_{4} f_{6 i} + p_{5} f_{7 i}, g_{2 i}^{(1)} = f_{2 i} + p_{5} f_{6 i} + p_{4} f_{7 i}, g_{3 i}^{(1)} = f_{3}, \\ g_{1 i}^{(2)} = f_{4 i} + p_{1} f_{6 i} + p_{2} f_{7 i}, g_{2 i}^{(2)} = f_{5 i} + p_{2} f_{6 i} + p_{1} f_{7 i}, g_{3 i}^{(2)} = f_{3}, \\ g_{0 i}^{(1)} = p_{4} f_{4 i} + p_{5} f_{5 i} - f_{0 i}, g_{0 i}^{(2)} = p_{1} f_{1 i} + p_{2} f_{2 i} - f_{0 i} \end{aligned}

then Eqs. and take the form

Δ q_{i}^{(k)} = 2 (g_{i}^{{(k)}^{T}} p^{(k)} - g_{0 i}^{(k)}), k = 1, 2

(12)

To determine the vectors p^(k) of synthesis parameters, we minimize function by least-square optimization, i.e., derive the sums

S^{(k)} = \sum_{i = 1}^{N} (Δ q_{i}^{(k)})^{2}

(13)

and consider the necessary conditions for the minimum of function over two groups of synthesis parameters p^(k)

\frac{\partial S^{(1)}}{\partial p_{1}} = 0, \frac{\partial S^{(1)}}{\partial p_{2}} = 0, \frac{\partial S^{(1)}}{\partial p_{3}} = 0

(14)

and

\frac{\partial S^{(2)}}{\partial p_{4}} = 0, \frac{\partial S^{(2)}}{\partial p_{5}} = 0, \frac{\partial S^{(2)}}{\partial p_{3}} = 0

(15)

Conditions and lead to the following two systems of linear equations for two groups of synthesis parameters:

D^{(1)} \cdot p^{{(1)}^{T}} = d^{(1)}

(16)

and

D^{(2)} \cdot p^{{(2)}^{T}} = d^{(2)}

(17)

where

\begin{aligned} D^{(1)} & = \sum_{i = 1}^{N} [\begin{matrix} g_{1 i}^{{(1)}^{2}} & g_{1 i}^{(1)} \cdot g_{2 i}^{(1)} & g_{1 i}^{(1)} \\ g_{1 i}^{(1)} \cdot g_{2 i}^{(1)} & g_{2 i}^{{(1)}^{2}} & g_{2 i}^{(1)} \\ g_{1 i}^{(1)} & g_{2 i}^{(1)} & 1 \end{matrix}], \\ d^{(1)} & = \sum_{i = 1}^{N} [\begin{matrix} g_{1 i}^{(1)} \cdot g_{0 i}^{(1)} \\ g_{2 i}^{(1)} \cdot g_{0 i}^{(1)} \\ g_{0 i}^{(1)} \end{matrix}] \end{aligned}

(18)

\begin{aligned} D^{(2)} & = \sum_{i = 1}^{N} [\begin{matrix} g_{1 i}^{{(2)}^{2}} & g_{1 i}^{(2)} \cdot g_{2 i}^{(22)} & g_{1 i}^{(2)} \\ g_{1 i}^{(2)} \cdot g_{2 i}^{(2)} & g_{2 i}^{{(2)}^{2}} & g_{2 i}^{(2)} \\ g_{1 i}^{(2)} & g_{2 i}^{(2)} & 1 \end{matrix}], \\ d^{(1)} & = \sum_{i = 1}^{N} [\begin{matrix} g_{1 i}^{(2)} \cdot g_{0 i}^{(2)} \\ g_{2 i}^{(2)} \cdot g_{0 i}^{(2)} \\ g_{0 i}^{(2)} \end{matrix}] \end{aligned}

(19)

It is easy to show that the Hessian of matrices D⁽¹⁾ and D⁽²⁾ are positive definite together with the principal minors [23], and then the solutions of the systems of linear Eqs. (16) and (17) correspond to the minimum of function (13). Therefore, for given values of the vector parameters p⁽²⁾ = [p₄, p₅, p₃], the vector parameters p⁽¹⁾ = [p₁, p₂, p₃] are determined by solving the system of linear Eq. (16). Based on the obtained values of the vector parameters p⁽¹⁾, the vector parameters p⁽²⁾ are determined by solving the system of linear Eq. (17). In this case, the sequence of function S^(k) values will be decreasing and have a limit as a sequence bounded from below. This allows using the linear iteration method based on kinematic inversion to solve the least-square approximation.

4 Parametric Synthesis of Six-Bar Function Generators

Parametric synthesis of six-bar function generators (Figs. 1(c)–1(f)) consists of the parametric synthesis of the passive CKC CED and the negative CKC FG. Synthesis parameters of the passive CKC CED of all six-bar function generators are

x_{C}^{(1)}, y_{C}^{(1)}, x_{D}^{(2)}, y_{D}^{(2)}, l_{C E}, and l_{E D}

⁠, where

x_{C}^{(1)}, y_{C}^{(1)}

and

x_{D}^{(2)}, y_{D}^{(2)}

are the coordinates of the joints C and D in the coordinate systems Ax₁y₁ and Bx₂y₂ of the links 1 and 2, respectively, l_CE and l_ED are the lengths of the CE and ED links. Since the passive CKC CED of the type RRR has zero DOF, it does not impose a geometrical constraint on the motion of the coordinate systems Ax₁y₁ and Bx₂y₂. Geometric parameters of the passive CKC CED links are varied, and the synthesis parameters of the negative CKC FG are approximated. For the parametric synthesis of the Stephenson I (Fig. 1(c)), Stephenson II (Fig. 1(d)), Stephenson III A (Fig. 1(e)), and Stephenson III B (Fig. 1(f)) function generators, we determine the positions of the links CE and ED of the passive CKC CED by the equations

φ_{3 i} = φ_{{(C D)}_{i}} + \cos^{- 1} \frac{l_{C E}^{2} + l_{{(C D)}_{i}}^{2} - l_{E D}^{2}}{2 l_{C E}^{} \cdot l_{{(C D)}_{i}}^{}}

(20)

φ_{4 i} = t g^{- 1} \frac{Y_{E_{i}} - Y_{D_{i}}}{X_{E_{i}} - X_{D_{i}}}

(21)

where

l_{{(C D)}_{i}} = [{(X_{D_{i}} - X_{C_{i}})}^{2} + {(Y_{D_{i}} - Y_{C_{i}})}^{2}]^{\frac{1}{2}}

(22)

φ_{{(C D)}_{i}} = t g^{- 1} \frac{Y_{D_{i}} - Y_{C_{i}}}{X_{D_{i}} - X_{C_{i}}}

(23)

[\begin{matrix} X_{C_{i}} \\ Y_{C_{i}} \end{matrix}] = [\begin{matrix} X_{A} \\ Y_{A} \end{matrix}] + [\begin{matrix} \cos φ_{i} & - \sin φ_{i} \\ \sin φ_{i} & \cos φ_{i} \end{matrix}] \cdot [\begin{matrix} x_{C}^{(1)} \\ y_{C}^{(1)} \end{matrix}]

(24)

[\begin{matrix} X_{D_{i}} \\ Y_{D_{i}} \end{matrix}] = [\begin{matrix} X_{B} \\ Y_{B} \end{matrix}] + [\begin{matrix} \cos ψ_{i} & - \sin ψ_{i} \\ \sin ψ_{i} & \cos ψ_{i} \end{matrix}] \cdot [\begin{matrix} x_{D}^{(2)} \\ y_{D}^{(2)} \end{matrix}]

(25)

Synthesis parameters for the negative CKC FG of the Stephenson I mechanism (Fig. 1(c)) are $x_{F}^{(1)}, y_{F}^{(1)}, x_{G}^{(2)}, y_{G}^{(2)}, and l_{F G}$ ⁠, which are determined similar to the parametric synthesis of the four-bar function generator (Fig. 2). Therefore, the functionality of the Stephenson I mechanism is the same as the functionality of the four-bar function generator.

Synthesis parameters for the negative CKC FG of the Stephenson II function generator (Fig. 3) are $x_{F}^{(3)}, y_{F}^{(3)}, x_{G}^{(2)}, y_{G}^{(2)}, and l_{F G}$ ⁠, where $x_{F}^{(3)}, y_{F}^{(3)}$ and $x_{G}^{(2)}, y_{G}^{(2)}$ are the coordinates of the joints F and G in the coordinate systems Cx₃y₃ and Bx₂y₂ of the links 3 and 2, respectively.

Fig. 3

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Stephenson II function generator

For parametric synthesis of the link FG, we consider the movement of the coordinate system Bx₂y₂ relative to the coordinate system Cx₃y₃ and derive the equation of geometrical constraint

(x_{G_{i}}^{(3)} - x_{F}^{(3)})^{2} + (y_{G_{i}}^{(3)} - y_{F}^{(3)})^{2} - l_{F G}^{2} = 0

(26)

where

[\begin{matrix} X_{G_{i}}^{(3)} \\ Y_{G_{i}}^{(3)} \end{matrix}] = [\begin{matrix} \cos φ_{3 i} & \sin φ_{3 i} \\ - \sin φ_{3 i} & \cos φ_{3 i} \end{matrix}] \cdot [\begin{matrix} X_{G_{i}} - X_{C_{i}} \\ Y_{G_{i}} - Y_{C_{i}} \end{matrix}]

(27)

[\begin{matrix} X_{G_{i}} \\ Y G_{i} \end{matrix}] = [\begin{matrix} X_{B} \\ Y_{B} \end{matrix}] + [\begin{matrix} \cos ψ_{i} & - \sin ψ_{i} \\ \sin ψ_{i} & \cos ψ_{i} \end{matrix}] \cdot [\begin{matrix} x_{G}^{(2)} \\ y_{G}^{(2)} \end{matrix}]

(28)

Further, the synthesis parameters of the link FG are determined similarly to the determination of synthesis parameters of the link CD of the four-bar function generator.

Synthesis parameters of the binary link FG of the Stephenson III A function generator (Fig. 4) and the Stephenson III B function generator (Fig. 5) are $x_{F}^{(3)}, y_{F}^{(3)}$ and $x_{F}^{(4)}, y_{F}^{(4)}$ ⁠, respectively, and $X_{G}^{}, Y_{G}^{}, and l_{F G}$ are for both function generators, where $x_{F}^{(3)}, y_{F}^{(3)}$ and $x_{F}^{(4)}, y_{F}^{(4)}$ are the coordinates of the joint F in the coordinate systems Cx₃y₃ and Dx₄y₄, respectively, $X_{G}^{}$ and $Y_{G}^{}$ are the coordinates of the joint G in the absolute coordinate system OXY.

Fig. 4

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Stephenson III A function generator

Fig. 5

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Stephenson III B function generator

For parametric synthesis of the link FG of the Stephenson III A and Stephenson III B function generators, we derive the following geometrical constraint equation:

(X_{F_{i}} - X_{G})^{2} + (Y_{F_{i}} - Y_{G})^{2} - l_{F G}^{2} = 0

(29)

where for the Stephenson III A function generator

[\begin{matrix} X_{F_{i}} \\ Y_{F_{i}} \end{matrix}] = [\begin{matrix} X_{C_{i}} \\ Y_{C_{i}} \end{matrix}] + [\begin{matrix} \cos φ_{3 i} & - \sin φ_{3 i} \\ \sin φ_{3 i} & \cos φ_{3 i} \end{matrix}] \cdot [\begin{matrix} x_{F}^{(3)} \\ y_{F}^{(3)} \end{matrix}]

(30)

for the Stephenson III B function generator

[\begin{matrix} X_{F_{i}} \\ Y_{F_{i}} \end{matrix}] = [\begin{matrix} X_{D_{i}} \\ Y_{D_{i}} \end{matrix}] + [\begin{matrix} \cos φ_{4 i} & - \sin φ_{4 i} \\ \sin φ_{4 i} & \cos φ_{4 i} \end{matrix}] \cdot [\begin{matrix} x_{F}^{(4)} \\ y_{F}^{(4)} \end{matrix}]

(31)

Further, synthesis parameters of the binary link FG are determined similar to the parametric synthesis of the four-bar function generator.

5 Numerical Results

Let us consider the parametric synthesis of the four-bar, Stephenson II, Stephenson III A, and Stephenson III B function generators. The following coordinates X_A = 10.0, Y_A = 20.0, X_B = 70.0, and Y_B = 20.0 of the four-bar Stephenson II, Stephenson III A, and Stephenson III B function generators pivots A and B are given in the absolute coordinate system OXY.

5.1 Four-Bar Function Generator.

Table 1 gives the values of the angles φ_i and ψ_i of the input and output links for N = 12 of the four-bar function generator (Fig. 6).

Fig. 6

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Four-bar function generator with scale size

Table 1

The values of the angles φ_i and ψ_i of the four-bar function generator

N	1	2	3	4	5	6	7	8	9	10	11	12
φ_i	0 deg	30 deg	60 deg	90 deg	120 deg	150 deg	180 deg	210 deg	240 deg	270 deg	300 deg	330 deg
ψ_i	36 deg	38 deg	51 deg	69 deg	88 deg	102 deg	108 deg	108 deg	104 deg	98 deg	84 deg	57 deg

Table 2 presents the obtained values of the synthesis parameters of the four-bar function generator.

Table 2

The values of the synthesis parameters of the four-bar function generator

$x_{C}^{(1)}$	$y_{C}^{(1)}$	$x_{D}^{(2)}$	$y_{D}^{(2)}$	l_CD
25.980	15.008	25.007	43.300	45.012

To check the parametric synthesis results of the four-bar function generator, values of the angle ψ_i corresponding to the values of the angle φ_i are determined by the equation

ψ_{i} = φ_{{(B C)}_{i}} - \cos^{- 1} \frac{l_{B D}^{2} + l_{{(B C)}_{i}}^{2} - l_{C D}^{2}}{2 l_{B D}^{} \cdot l_{{(B C)}_{i}}^{}}

(32)

where

φ_{{(B C)}_{i}} = t g^{- 1} \frac{Y_{C_{i}}^{} - Y_{B}^{}}{X_{C_{i}}^{} - X_{B}^{}}

(33)

l_{{(B C)}_{i}} = [{(X_{C_{i}} - X_{B})}^{2} + {(Y_{C_{i}} - Y_{B})}^{2}]^{\frac{1}{2}}

(34)

l_{B D} = (x_{D}^{{(2)}^{2}} + y_{D}^{{(2)}^{2}})^{\frac{1}{2}}

(35)

In the expressions and , coordinates of the joint С in the absolute coordinate system OXY are determined by the equation

[\begin{matrix} X_{C_{i}} \\ Y_{C_{i}} \end{matrix}] = [\begin{matrix} X_{A} \\ Y_{A} \end{matrix}] + [\begin{matrix} \cos φ_{i} & - \sin φ_{i} \\ \sin φ_{i} & \cos φ_{i} \end{matrix}] \cdot [\begin{matrix} x_{C}^{(1)} \\ y_{C}^{(1)} \end{matrix}]

(36)

Table 3 presents the obtained values of the angle ψ_i, and Fig. 7 shows a graph of its change.

Fig. 7

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Graph of the angle ψ_i change of the four-bar function generator

Table 3

The obtained values of the angle ψ_i

N	1	2	3	4	5	6
φ_i	0 deg	30 deg	60 deg	90 deg	120 deg	150 deg
ψ_i	36.17 deg	37.60 deg	51.30 deg	69.09 deg	88.10 deg	102.32 deg
N	7	8	9	10	11	12
φ_i	180 deg	2100	2400	2700	3000	3300
ψ_i	107.91 deg	107.71 deg	104.44 deg	97.67 deg	83.77 deg	57.27 deg

N	1	2	3	4	5	6
φ_i	0 deg	30 deg	60 deg	90 deg	120 deg	150 deg
ψ_i	36.17 deg	37.60 deg	51.30 deg	69.09 deg	88.10 deg	102.32 deg
N	7	8	9	10	11	12
φ_i	180 deg	2100	2400	2700	3000	3300
ψ_i	107.91 deg	107.71 deg	104.44 deg	97.67 deg	83.77 deg	57.27 deg

5.2 Stephenson II Function Generator.

Table 4 gives the values of the angles φ_i and ψ_i of the input and output links for N = 12 of the Stephenson II function generator (Fig. 8).

Fig. 8

View large Download slide

Stephenson II function generator with scale size

Table 4

The values of the angles φ_i and ψ_i of the Stephenson II function generator

N	1	2	3	4	5	6	7	8	9	10	11	12
φ_i	0 deg	30 deg	60 deg	90 deg	120 deg	150 deg	180 deg	210 deg	240 deg	270 deg	300 deg	330 deg
ψ_i	104 deg	98 deg	93 deg	90 deg	90 deg	92 deg	98 deg	105 deg	112 deg	116 deg	116 deg	111 deg

Table 5 presents the obtained values of the synthesis parameters of the Stephenson II function generator.

Table 5

The values of the synthesis parameters of the Stephenson II function generator

$x_{C}^{(1)}$	$y_{C}^{(1)}$	$l_{C E}^{}$	$l_{E D}^{}$	$x_{D}^{(2)}$	$y_{D}^{(2)}$	$x_{F}^{(3)}$	$y_{F}^{(3)}$	$x_{G}^{(2)}$	$y_{G}^{(2)}$	$l_{G F}^{}$
8.658	5.003	45.983	50.017	12.496	21.645	23.775	−7.725	20.715	13.975	40.211

To check the parametric synthesis results of the Stephenson II function generator, we determine the values of the angle ψ_i corresponding to the values of the angle φ_i. To do this, the positions of the Stephenson II function generator are analyzed.

Stephenson II function generator contains an input link 1 and an Assur group of the fourth class. It has the following structural formula:

I (1) \to IV (2, 3, 4, 5)

(37)

For position analysis of the Stephenson II function generator or the mechanism of the fourth class, a simple method of conditional generalized coordinate is proposed, which is the opposite of the method of structural synthesis. If we remove a negative CKC—a binary link FG of the type of RR by disconnecting the elements of the joints F and G—then the considered mechanism acquires one additional DOF. If we take the link 2 as a conditional input link and the angle ψ_i as a conditional generalized coordinate, then this mechanism of the fourth class is transformed into a mechanism of the second class with the structural formula

I (1) \to II (3, 4) \leftarrow I (2)

(38)

For a given value of the angle φ_i and in changing the value of the conditional generalized coordinate ψ_i, the distance between the centers of the disconnected joints F and G is changed. A residual function is derived below:

Δ_{i} = l_{F G} - l_{{(F G)}_{i}}^{*}

(39)

where

l_{{(F G)}_{i}}^{*}

is a variable distance between the centers of the joints F and G, which is determined by the equation

l_{{(F G)}_{i}}^{*} = [{(X_{G_{i}} - X_{F_{i}})}^{2} + {(Y_{G_{i}} - Y_{F_{i}})}^{2}]^{\frac{1}{2}}

(40)

Coordinates of the joints F and G in Eq. are determined by the expressions

[\begin{matrix} X_{F_{i}} \\ Y_{F_{i}} \end{matrix}] = [\begin{matrix} X_{C_{i}} \\ Y_{C_{i}} \end{matrix}] + [\begin{matrix} \cos φ_{3 i} & - \sin φ_{3 i} \\ \sin φ_{3 i} & \cos φ_{3 i} \end{matrix}] \cdot [\begin{matrix} x_{F}^{(3)} \\ y_{F}^{(3)} \end{matrix}]

(41)

[\begin{matrix} X_{G_{i}} \\ Y_{G_{i}} \end{matrix}] = [\begin{matrix} X_{B} \\ Y_{B} \end{matrix}] + [\begin{matrix} \cos ψ_{i} & - \sin ψ_{i} \\ \sin ψ_{i} & \cos ψ_{i} \end{matrix}] \cdot [\begin{matrix} x_{G}^{(2)} \\ y_{G}^{(2)} \end{matrix}]

(42)

where

[\begin{matrix} X_{C_{i}} \\ Y_{C_{i}} \end{matrix}] = [\begin{matrix} X_{A} \\ Y_{A} \end{matrix}] + [\begin{matrix} \cos φ_{i} & - \sin φ_{i} \\ \sin φ_{i} & \cos φ_{i} \end{matrix}] \cdot [\begin{matrix} x_{C}^{(1)} \\ y_{C}^{(1)} \end{matrix}]

(43)

To determine the angle φ_3i in Eq. , we analyze the positions of the dyad II (3,4) and obtain

φ_{3 i} = φ_{{(C D)}_{i}} + \cos^{- 1} \frac{l_{C E}^{2} + l_{{(C D)}_{i}}^{2} - l_{E D}^{2}}{2 l_{C E}^{} \cdot l_{{(C D)}_{i}}^{}}

(44)

where

φ_{{(C D)}_{i}} = t g^{- 1} \frac{Y_{D_{i}}^{} - Y_{C_{i}}^{}}{X_{D_{i}}^{} - X_{C_{i}}^{}}

(45)

l_{{(C D)}_{i}} = [{(X_{D_{i}} - X_{C_{i}})}^{2} + {(Y_{D_{i}} - Y_{C_{i}})}^{2}]^{\frac{1}{2}}

(46)

[\begin{matrix} X_{D_{i}} \\ Y D_{i} \end{matrix}] = [\begin{matrix} X_{B} \\ Y_{B} \end{matrix}] + [\begin{matrix} \cos ψ_{i} & - \sin ψ_{i} \\ \sin ψ_{i} & \cos ψ_{i} \end{matrix}] \cdot [\begin{matrix} x_{D}^{(2)} \\ y_{D}^{(2)} \end{matrix}]

(47)

Consequently, the residual (39) is a function of the conditional generalized coordinate ψ_i. Minimizing this function by the bisection method, we obtain the values of the angle ψ_i for a given value of the angle φ_i. By changing the values of the angle φ_i, we similarly find the corresponding values of the angle ψ_i.

Table 6 presents the obtained values of the angle ψ_i, and Fig. 9 shows a graph of its change.

Fig. 9

View large Download slide

Graph of the angle ψ_i change of the Stephenson II function generator

Table 6

The obtained values of the angle ψ_i

N	1	2	3	4	5	6
φ_i	0 deg	30 deg	60 deg	90 deg	120 deg	150 deg
ψ_i	104.12 deg	98.35 deg	93.18 deg	90.24 deg	89.54 deg	92.07 deg
N	7	8	9	10	11	12
φ_i	180 deg	210 deg	240 deg	270 deg	300 deg	330 deg
ψ_i	97.57 deg	105.21 deg	111.87 deg	116.16 deg	115.75 deg	110.83 deg

N	1	2	3	4	5	6
φ_i	0 deg	30 deg	60 deg	90 deg	120 deg	150 deg
ψ_i	104.12 deg	98.35 deg	93.18 deg	90.24 deg	89.54 deg	92.07 deg
N	7	8	9	10	11	12
φ_i	180 deg	210 deg	240 deg	270 deg	300 deg	330 deg
ψ_i	97.57 deg	105.21 deg	111.87 deg	116.16 deg	115.75 deg	110.83 deg

5.3 Stephenson III A Function Generator.

Table 7 gives values of the angles φ_i and ψ_i of the input and output links of the Stephenson III A function generator (Fig. 10).

Fig. 10

View large Download slide

Stephenson III A function generator with scale size

Table 7

The values of the angles φ_i and ψ_i of the Stephenson III A function generator

N	1	2	3	4	5	6	7	8	9	10	11	12
φ_i	0 deg	30 deg	60 deg	90 deg	120 deg	150 deg	180 deg	210 deg	240 deg	270 deg	300 deg	330 deg
ψ_i	11 deg	03 deg	07 deg	15 deg	24 deg	33 deg	44 deg	57 deg	71 deg	83 deg	77 deg	43 deg

Table 8 presents the obtained values of the synthesis parameters of the Stephenson III A function generator.

Table 8

The values of the synthesis parameters of the Stephenson III A function generator

$x_{C}^{(1)}$	$y_{C}^{(1)}$	$l_{C E}^{}$	$l_{E D}^{}$	$x_{D}^{(2)}$	$y_{D}^{(2)}$	$x_{F}^{(3)}$	$y_{F}^{(3)}$	$X_{G}^{}$	$Y_{G}^{}$	$l_{G F}^{}$
12.979	7.503	45.947	40.014	16.971	29.438	23.775	−7.725	39.983	21.087	24.897

To check the parametric synthesis results of the Stephenson III A function generator, let us determine the values of the angle ψ_i corresponding to the values of the angle φ_i. To do this, we analyze the positions of the Stephenson III A function generator. This function generator has a structural formula

I (1) \to II (3, 5) \to II (4, 2)

(48)

i.e., it belongs to the second class mechanism. Therefore, it is necessary to consistently solve the position problems of the dyads CFG and EDB.

From the position analysis of the dyad CFG, we obtain

φ_{5_{i}} = φ_{{(G C)}_{i}} - \cos^{- 1} \frac{l_{G F}^{2} + l_{{(G C)}_{i}}^{2} - l_{C F}^{2}}{2 l_{G F}^{} \cdot l_{{(G C)}_{i}}^{}}

(49)

φ_{{(C F)}_{i}} = t g^{- 1} \frac{Y_{F_{i}} - Y_{C_{i}}}{X_{F_{i}} - X_{C_{i}}}

(50)

where

l_{{(G C)}_{i}} = [{(X_{C_{i}} - X_{G})}^{2} + {(Y_{C_{i}} - Y_{G})}^{2}]^{\frac{1}{2}}

(51)

φ_{{(G C)}_{i}} = t g^{- 1} \frac{Y_{C_{i}} - Y_{G}}{X_{C_{i}} - X_{G}}

(52)

Based on the obtained values of the angle

φ_{{(C F)}_{i}}

⁠, we determine the value of the angle

φ_{3_{i}}

φ_{3_{_{i}}} = φ_{{(C F)}_{i}} + α_{3}

(53)

and the coordinates of the joint E

[\begin{matrix} X_{E_{i}} \\ Y_{E_{i}} \end{matrix}] = [\begin{matrix} X_{C_{i}} \\ Y_{C_{i}} \end{matrix}] + l_{C E} [\begin{matrix} \cos φ_{3_{i}} \\ \sin φ_{3_{i}} \end{matrix}]

(54)

where

α_{3} = t g^{- 1} \frac{y_{F}^{(3)}}{x_{F}^{(3)}}

(55)

Then, we solve the position problem of the dyad EDB and determine the value of the angle ψ_i

ψ_{i} = φ_{{(B E)}_{i}} - \cos^{- 1} \frac{l_{B D}^{2} + l_{{(B E)}_{i}}^{2} - l_{E D}^{2}}{2 l_{B D} \cdot l_{{(B E)}_{i}}} - ψ^{0}

(56)

where

l_{{(B E)}_{i}} = [{(X_{E_{i}} - X_{B})}^{2} + {(Y_{E_{i}} - Y_{B})}^{2}]^{\frac{1}{2}}

(57)

φ_{{(B E)}_{i}} = t g^{- 1} \frac{Y_{E_{i}} - Y_{B}}{X_{E_{i}} - X_{B}}

(58)

ψ^{0} = t g^{- 1} \frac{y_{D}^{(2)}}{x_{D}^{(2)}}

(59)

Table 9 presents the obtained values of the angle ψ_i, and Fig. 11 shows a graph of its change.

Fig. 11

View large Download slide

Graph of the angle ψ_i change of the Stephenson III A function generator

Table 9

The obtained values of the angle ψ_i

N	1	2	3	4	5	6
φ_i	0 deg	30 deg	60 deg	90 deg	120 deg	150 deg
ψ_i	11.21 deg	02.98 deg	06.87 deg	14.91 deg	23.93 deg	33.16 deg
N	7	8	9	10	11	12
φ_i	180 deg	210 deg	240 deg	270 deg	300 deg	330 deg
ψ_i	43.70 deg	57.01 deg	70.96 deg	82.91 deg	77.04 deg	42.83 deg

N	1	2	3	4	5	6
φ_i	0 deg	30 deg	60 deg	90 deg	120 deg	150 deg
ψ_i	11.21 deg	02.98 deg	06.87 deg	14.91 deg	23.93 deg	33.16 deg
N	7	8	9	10	11	12
φ_i	180 deg	210 deg	240 deg	270 deg	300 deg	330 deg
ψ_i	43.70 deg	57.01 deg	70.96 deg	82.91 deg	77.04 deg	42.83 deg

5.4 Stephenson III B Function Generator.

Table 10 gives the values of the angles φ_i and ψ_i of the input and output links of the Stephenson III B function generator (Fig. 12).

Fig. 12

View large Download slide

Stephenson III B function generator with scale size

Table 10

The values of the angles φ_i and ψ_i of the Stephenson III B function generator

N	1	2	3	4	5	6	7	8	9	10	11	12
φ_i	0 deg	30 deg	60 deg	90 deg	120 deg	150 deg	180 deg	210 deg	240 deg	270 deg	300 deg	330 deg
ψ_i	223 deg	244 deg	266 deg	264 deg	245 deg	240 deg	245 deg	256 deg	268 deg	267 deg	245 deg	221 deg

Table 11 presents the obtained values of the synthesis parameters of the Stephenson III B function generator.

Table 11

The values of the synthesis parameters of the Stephenson III B function generator

$x_{C}^{(1)}$	$y_{C}^{(1)}$	$l_{C E}^{}$	$l_{E D}^{}$	$x_{D}^{(2)}$	$y_{D}^{(2)}$	$x_{F}^{(3)}$	$y_{F}^{(3)}$	$X_{G}^{}$	$Y_{G}^{}$	$l_{G F}^{}$
12.978	7.493	55.017	29.962	7.507	13.004	15.029	10.134	39.995	19.917	25.545

Let us determine the values of the angle ψ_i corresponding to the values of the angle φ_i. To do this, we analyze the positions of the Stephenson III B function generator. This function generator has a structural formula

I (1) \to III (2, 3, 4, 5)

(60)

i.e., it belongs to the mechanism of the third class. For position analysis of this mechanism, we remove the negative CKC FG by disconnecting the elements of the joints F and G. Then, the considered mechanism acquires one additional DOF. If we take link 2 as a conditional input link, due to the appeared DOF, then the third class mechanism is transformed into the second class mechanism with structural formula .

Let us derive a residual function of the form , where the coordinates of the joint F are determined by the equation

[\begin{matrix} X_{F i} \\ Y_{F i} \end{matrix}] = [\begin{matrix} X_{D i} \\ Y_{D i} \end{matrix}] + [\begin{matrix} \cos φ_{4 i} & - \sin φ_{4 i} \\ \sin φ_{4 i} & \cos φ_{4 i} \end{matrix}] \cdot [\begin{matrix} x_{F}^{(4)} \\ y_{F}^{(4)} \end{matrix}]

(61)

To determine the angle φ_4i in Eq. , we solve a position analysis problem of the dyad CED and obtain

φ_{4 i} = φ_{(D C) i} - \cos^{- 1} \frac{l_{E D}^{2} + l_{(D C) i}^{2} - l_{C E}^{2}}{2 l_{E D}^{} \cdot l_{(D C) i}^{}}

(62)

where

φ_{(D C) i} = t g^{- 1} \frac{Y_{C i} - Y_{D i}}{X_{C i} - X_{D i}}

(63)

l_{(D C) i} = [{(X_{C i} - X_{D i})}^{2} + {(Y_{C i} - Y_{D i})}^{2}]^{\frac{1}{2}}

(64)

Consequently, the residual (39) is a function of the conditional generalized coordinate ψ_i. Minimizing this function by the bisection method, we obtain the values of the angle ψ_i. Table 12 presents the obtained values of the angle ψ_i, and Fig. 13 shows a graph of its change.

Fig. 13

View large Download slide

Graph of the angle ψ_i change of the Stephenson III B function generator

Table 12

The obtained values of the angle ψ_i

N	1	2	3	4	5	6
φ_i	0 deg	30 deg	60 deg	90 deg	120 deg	150 deg
ψ_i	223.29 deg	244.38 deg	265.81 deg	264.01 deg	245.44 deg	240.18 deg
N	7	8	9	10	11	12
φ_i	180 deg	210 deg	240 deg	270 deg	300 deg	330 deg
ψ_i	244.96 deg	255.87 deg	267.45 deg	267.04 deg	244.83 deg	220.71 deg

N	1	2	3	4	5	6
φ_i	0 deg	30 deg	60 deg	90 deg	120 deg	150 deg
ψ_i	223.29 deg	244.38 deg	265.81 deg	264.01 deg	245.44 deg	240.18 deg
N	7	8	9	10	11	12
φ_i	180 deg	210 deg	240 deg	270 deg	300 deg	330 deg
ψ_i	244.96 deg	255.87 deg	267.45 deg	267.04 deg	244.83 deg	220.71 deg

6 Conclusion

Structural synthesis of four-bar and six-bar function generators with revolute joints has been carried out. A four-bar function generator is formed by connecting two rotating coordinate systems with given rotation angles using a binary link of the type RR, which is a negative CKC that imposes one geometrical constraint. Six-bar function generators are formed by connecting these two rotationally moving coordinate systems using a passive CKC of the type RRR and by connecting non-adjacent links of the resulting five-bar linkage by binary link of the type RR. As a result, Stephenson I, Stephenson II, Stephenson III A, and Stephenson III B function generators have been formed. Passive CKC of the type RRR does not impose a geometrical constraint on the movement of two moving coordinate systems, and its geometric parameters are varied to satisfy the constraint of the negative CKC. Geometric parameters of the negative CKC of the type RR are determined by least-square approximation. In this case, the least-square approximation problem is reduced to a simple kinematic inversion problem based on linear iteration. Structural and parametric synthesis of the four-bar and six-bar function generators are carried out simultaneously, starting with the smallest number of links of CKCs. Numerical results of parametric synthesis of four-bar and six-bar function generators are presented.

Acknowledgment

This work was funded by the Science Committee of the Ministry of Science and High Education of Kazakhstan (Grant No. AP14872115 “Development and Research of the Novel Tripod Type Parallel Manipulators With Six Degrees of Freedom”).

Conflict of Interest

There are no conflicts of interest.

Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

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Structural–Parametric Synthesis of the Planar Four-Bar and Six-Bar Function Generators With Revolute Joints

Abstract

1 Introduction

2 Structural Synthesis of the Planar Four-Bar and Six-Bar Function Generators With Revolute Joints

3 Parametric Synthesis of Four-Bar Function Generator

4 Parametric Synthesis of Six-Bar Function Generators

5 Numerical Results

5.1 Four-Bar Function Generator.

5.2 Stephenson II Function Generator.

5.3 Stephenson III A Function Generator.

5.4 Stephenson III B Function Generator.

6 Conclusion

Acknowledgment

Conflict of Interest

Data Availability Statement

References

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Structural–Parametric Synthesis of the Planar Four-Bar and Six-Bar Function Generators With Revolute Joints

Abstract

1 Introduction

2 Structural Synthesis of the Planar Four-Bar and Six-Bar Function Generators With Revolute Joints

3 Parametric Synthesis of Four-Bar Function Generator

4 Parametric Synthesis of Six-Bar Function Generators

5 Numerical Results

5.1 Four-Bar Function Generator.

5.2 Stephenson II Function Generator.

5.3 Stephenson III A Function Generator.

5.4 Stephenson III B Function Generator.

6 Conclusion

Acknowledgment

Conflict of Interest

Data Availability Statement

References

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