Reconfiguration Analysis and Characteristics of a Novel 8-Link Variable-DOF Planar Mechanism with Five Motion Modes

Abstract

Variable-DOF (or kinematotropic) mechanisms are a class of reconfigurable mechanisms that have varying degrees of freedom (DOF) in different motion modes and can be reconfigured without disassembly. However, the number of proposed variable-DOF multi-loop planar mechanisms is currently limited. This paper introduces a new 8-link variable-DOF planar mechanism that has five motion modes. Firstly, the 8-link variable-DOF planar mechanism is described. Then, reconfiguration analysis of the mechanism is performed using a hybrid approach that combines elimination and computer algebraic geometry methods. The analysis reveals that the 8-link mechanism has one 2-DOF motion mode and four 1-DOF motion modes. It can switch among three motion modes at four transition configurations and between two motion modes at the remaining four transition configurations. The paper also highlights the geometric characteristics of the mechanism in different motion modes. In contrast to variable-DOF planar mechanisms presented in the literature, the proposed 8-link mechanism has two inactive joints in one of its 1-DOF motion modes. Moreover, both closed-loop 4R kinematic sub-chains of the mechanism must appear as either a pair of parallelograms or a pair of anti-parallelograms in the same motion mode. As a by-product of this research, a method for factoring trigonometric functions in two angles is also proposed.

1. Introduction

One of the current research focuses in mechanisms and robotics is reconfigurable mechanisms and robots [1,2], which could help meet the needs of robots and manufacturing systems that can rapidly adapt to changes in environment and production.

Variable-DOF (or kinematotropic) mechanisms [3,4,5,6,7,8,9,10,11,12,13,14,15] are a class of reconfigurable mechanisms that have varying degrees of freedom (DOF) in different motion modes and can be reconfigured without disassembly. Considerable progress has been made in the type synthesis and reconfiguration analysis of variable-DOF mechanisms, including single-loop spatial mechanisms [5,7,12], parallel mechanisms [11,13,14,15,16,17,18,19,20], multi-mode mobile parallel mechanisms [21], and multi-loop mechanisms [4,5,22,23,24,25,26,27,28,29]. It should be noted that variable-DOF mechanisms are composed of conventional kinematic joints and do not involve variable kinematic joints [30], reconfigurable kinematic joints [31] or metamorphic kinematic joints [32].

Apart from the construction methods [7,23,27], most approaches for the type synthesis of variable-DOF mechanisms are based on different mathematical methods ranging from displacement group theory [5,11], intersection of surfaces [10,20], factorization of polynomials [12,33], and primary decomposition of ideals [14,17,18,19] to the comprehensive Gröbner basis of parametric polynomial equations [19,34]. Through the construction methods in [7,23,27], a number of variable-DOF mechanisms have been constructed from existing overconstrained mechanisms. No overconstrained mechanisms are required in advance if using methods in [5,10,11,12,33]; however, only several variable-DOF mechanisms have been obtained by using these methods. Using the methods in [14,17,18], one can detect whether a multi-DOF overconstrained parallel mechanism is a variable-DOF parallel mechanism. Using the methods in [19,34], one can investigate the impact of link parameters of multi-DOF overconstrained parallel mechanism on the number and types of motion modes and identify different variable-DOF parallel mechanisms of the same topological structure. Variable-DOF mechanisms were obtained by using multi-mode single-loop kinematic chains as building blocks in [16,23]. With further development and application of the above methods, more and more variable-DOF mechanisms are expected to be revealed.

Methods for the reconfiguration analysis of variable-DOF mechanisms mainly include the elimination approaches [22], algebraic geometry methods [24,35,36], numerical algebraic geometry methods [37], branch-and-prune methods [38], singular value decomposition approaches [39,40,41,42], and the higher-order kinematics based approaches [43,44,45,46]. The first five methods can be used to identify all the motion modes of a variable-DOF mechanism as long as the link parameters of the mechanism are given, whereas a singular or transition configuration of the variable-DOF mechanism must be given in advance if the last method, which is more computationally efficient, is used for the reconfiguration analysis.

It is noted that there are no variable-DOF planar single-loop mechanisms composed of R (revolute) and P (prismatic) joints, and the number of variable-DOF multi-loop planar mechanisms is still very limited. The only four variable-DOF multi-loop planar mechanisms proposed so far are the 12-link Wunderlich mechanism in [3], the 10-link Kovalev mechanism in [4], the 8-link variable-DOF planar mechanism in [5], and the 8-link variable-DOF planar mechanism in [47].

One difference between these four variable-DOF multi-loop planar mechanisms lies in the number of their inactive joints. An inactive joint in a variable-DOF mechanism under a specified motion mode is a kinematic joint that loses its DOF due to intrinsic constraints within the mechanism. The 8-link variable-DOF planar mechanism proposed in [5] has four inactive joints in its 3-DOF motion mode and no inactive joint in its 1-DOF motion mode. In the 3-DOF motion mode, this 8-link mechanism degenerates to a planar serial 3R mechanism. The 8-link variable-DOF planar mechanism presented in [47] has four inactive joints in four of its 1-DOF motion modes and no inactive joint in its 2-DOF motion mode. In these four 1-DOF motion modes, a closed-loop 4R kinematic sub-chain of the 8-link mechanism degenerates (or loses its DOF).

One question arising from the above observations is the following: Are there variable-DOF 8-link planar mechanisms which have neither a serial mechanism motion mode nor a motion mode with a degenerated closed-loop 4R sub-kinematic chain? This paper will answer the above question by presenting a novel 8-link variable-DOF planar mechanism. As will be shown later, this 8-link mechanism has two inactive joints in one of its 1-DOF motion modes and no inactive joints in the other motion modes.

This paper is organized as follows. In Section 2, a geometric description of a novel variable-DOF 8-link planar mechanism is given. A set of kinematic equations is set up in Section 3 with the variables selected to better reflect the geometric characteristics of the mechanism in different motion modes. The motion modes and transition configurations of the variable-DOF 8-link planar mechanism are identified using a hybrid approach that combines elimination and computer algebraic geometry methods in Section 4 and Section 5. The reconfiguration of the variable-DOF 8-link planar mechanism is detailed in Section 6. Finally, conclusions are drawn.

2. Geometric Description of a Novel 8-Link Variable-Dof Planar Mechanism

The 8-link variable-DOF planar mechanism [47] constructed using two parallelograms is composed of four binary links and four triangular ternary links and has four inactive joints in four of its 1-DOF motion modes and no inactive joint in its 2-DOF motion mode. Recently, it was revealed in [48] that in addition to the 1-DOF motion mode, a 3-RR planar parallelogram may have up to two structure modes if the two ternary links are triangular or no structure mode if the two ternary links are collinear. The four inactive joints in a motion mode of the 8-link variable-DOF planar mechanism correspond to the structure mode of a 3-RR planar parallelogram. Using two parallelograms with no structure mode, we can construct an 8-link variable-DOF planar mechanism that does not have 1-DOF motion mode with four inactive joints. Alternatively, by simply replacing each triangular ternary link in the 8-link variable-DOF planar mechanism in [47] with a collinear ternary link, the 1-DOF motion modes with four inactive joints of the original 8-link mechanism will be eliminated. This would lead to a novel 8-link variable-DOF planar mechanism (Figure 1).

The novel 8-link variable-DOF planar mechanism is composed of four identical binary links, A${}_{11}$B${}_{11}$, A${}_{12}$B${}_{12}$, A${}_{21}$B${}_{21}$, and A${}_{22}$B${}_{22}$, and four identical collinear ternary links, AA${}_{11}$A${}_{12}$, BB${}_{11}$B${}_{12}$, AA${}_{21}$A${}_{22}$, and BB${}_{21}$B${}_{22}$, connected by 10 R joints. Link A${}_{22}$B${}_{22}$ is the frame. The link parameters of the 8-link variable-DOF planar mechanism are

$A{A}_{11}=B{B}_{11}=A{A}_{21}=B{B}_{21}=a_1$, $A{A}_{12}=B{B}_{12}=A{A}_{22}=B{B}_{22}=a_2$, and $A_{11}{B}_{11}=A_{12}{B}_{12}=A_{21}{B}_{21}=A_{22}{B}_{22}=L_1$.

The link parameters of an example 8-link variable-DOF planar mechanism are $a_1=45$, $a_2=75$, and $L_1=25$. Here, link lengths are represented by dimensionless numbers that indicate their relative length, subject to the condition that $L_1<(a_2-a_1)$ in order to avoid link interference.

3. Kinematic Equations

To facilitate the identification of the geometric characteristics of the 8-link mechanism, the coordinate system O-XY is set up such that O coincides with R joint center A, and R joint center B is located on the positive X-axis. Let ${\varphi }_a$ (${\psi }_a$) denote the angle between the negative direction of the X-axis and the link AA${}_{11}$ (AA${}_{21}$) measured clockwise and ${\varphi }_b$ (${\psi }_b$) the angle between the positive direction of the X-axis and the link BB${}_{11}$ (BB${}_{21}$) measured anti-clockwise. An auxiliary variable, $L=AB$ ($L>0$), is introduced to simplify the reconfiguration analysis.

The loop closure equations of loops ABB${}_{11}$A${}_{11}$A, ABB${}_{12}$A${}_{12}$A, ABB${}_{21}$A${}_{21}$A, and ABB${}_{22}$A${}_{22}$A written in vector form are

$$\begin{array}{c}\hfill \left\{\begin{array}{c}(\overrightarrow{AB}+\overrightarrow{B{B}_{11}}-\overrightarrow{A{A}_{11}})·(\overrightarrow{AB}+\overrightarrow{B{B}_{11}}-\overrightarrow{A{A}_{11}})=L_1^2\hfill \\ (\overrightarrow{AB}+\overrightarrow{B{B}_{12}}-\overrightarrow{A{A}_{12}})·(\overrightarrow{AB}+\overrightarrow{B{B}_{12}}-\overrightarrow{A{A}_{12}})=L_1^2\hfill \\ (\overrightarrow{AB}+\overrightarrow{B{B}_{21}}-\overrightarrow{A{A}_{21}})·(\overrightarrow{AB}+\overrightarrow{B{B}_{21}}-\overrightarrow{A{A}_{21}})=L_1^2\hfill \\ (\overrightarrow{AB}+\overrightarrow{B{B}_{22}}-\overrightarrow{A{A}_{22}})·(\overrightarrow{AB}+\overrightarrow{B{B}_{22}}-\overrightarrow{A{A}_{22}})=L_1^2\hfill \end{array}\right.\end{array}$$

Rewriting the above equation in complex number form, we have

$$\begin{array}{c}\hfill \left\{\begin{array}{c}(L+a_1{e}^{i{\varphi }_b}-a_1{e}^{i(\pi -{\varphi }_a})(L+a_1{e}^{-i{\varphi }_b}-a_1{e}^{-i(\pi -{\varphi }_a})-L_1^2=0\hfill \\ (L+a_2{e}^{i{\varphi }_b}-a_2{e}^{i(\pi -{\varphi }_a})(L+a_2{e}^{-i{\varphi }_b}-a_2{e}^{-i(\pi -{\varphi }_a})-L_1^2=0\hfill \\ (L+a_1{e}^{i{\psi }_b}-a_1{e}^{i(\pi -{\psi }_a})(L+a_1{e}^{-i{\psi }_b}-a_1{e}^{-i(\pi -{\psi }_a})-L_1^2=0\hfill \\ (L+a_2{e}^{i{\psi }_b}-a_2{e}^{i(\pi -{\psi }_a})(L+a_2{e}^{-i{\psi }_b}-a_2{e}^{-i(\pi -{\psi }_a})-L_1^2=0\hfill \end{array}\right.\end{array}$$

Simplifying the above equation, we obtain

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{a}_{1}C({\varphi }_b+{\varphi }_a)+L(C{\varphi }_b+C{\varphi }_a)+(L^2-L_1^2)/\left(2a_1\right)+a_1=0\hfill \\ a_{2}C({\varphi }_b+{\varphi }_a)+L(C{\varphi }_b+C{\varphi }_a)+(L^2-L_1^2)/\left(2a_2\right)+a_2=0\hfill \\ a_{1}C({\psi }_b+{\psi }_a)+L(C{\psi }_b+C{\psi }_a)+(L^2-L_1^2)/\left(2a_1\right)+a_1=0\hfill \\ a_{2}C({\psi }_b+{\psi }_a)+L(C{\psi }_b+C{\psi }_a)+(L^2-L_1^2)/\left(2a_2\right)+a_2=0\hfill \end{array}\right.\end{array}$$

(1)

where $S\ast$ and $C\ast$ denote $sin\ast$ and $cos\ast$, respectively.

4. Motion Mode Analysis of an 8-Link Variable-Dof Planar Mechanism Using a Hybrid Approach

In this section, we will reveal all the motion modes of the novel 8-link variable-DOF mechanism (Figure 1) by using resultant elimination, which has been extensively used in the kinematic analysis of mechanisms [49,50], and the primary decomposition of ideals from computer algebraic geometry [35], which has been used in the reconfiguration analysis of multi-mode mechanisms [17,24,36].

Eliminating ${\varphi }_b$ from the first and second equations of Equation (1) and ${\psi }_b$ from the third and fourth equations of Equation (1), Equation (1) is reduced to the following set of two equations in three variables ${\varphi }_a$, ${\psi }_a$, and L (see Appendix A for details)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}(1-C^2{\varphi }_a)[1-{(k_2-C{\varphi }_a)}^2]-{[k_1-C{\varphi }_a(k_2-C{\varphi }_a)]}^2=0\hfill \\ (1-C^2{\psi }_a)[1-{(k_2-C{\psi }_a)}^2]-{[k_1-C{\psi }_a(k_2-C{\psi }_a)]}^2=0\hfill \end{array}\right.\end{array}$$

(2)

where $k_1=(L^2-{L_1}^2)/\left(2a_1{a}_2\right)-1$ and $k_2=-(a_1+a_2)(L^2-{L_1}^2)/\left(2L{a}_1{a}_2\right)$.

For simplicity reasons and without loss of generality, we will investigate the reconfiguration analysis of the 8-link variable-DOF mechanism via the example mechanism given in Section 2. Substituting the link parameters of the example 8-link mechanism into Equation (2), we obtain

$$\begin{array}{c}\hfill \left\{\begin{array}{c}(L-25)(L+25)f_1=0\hfill \\ (L-25)(L+25)f_2=0\hfill \end{array}\right.\end{array}$$

(3)

where $f_1=L^4+240L^{3}C{\varphi }_a+6750C\left(2{\varphi }_a\right)L^2+7025L^2-150,000C{\varphi }_{a}L-9,000,000$ and $f_2=L^4+240L^{3}C{\psi }_a+6750C\left(2{\psi }_a\right)L^2+7025L^2-150,000C{\psi }_{a}L-9,000,000$.

Since $L>0$, Equation (3) leads to two cases:

Case A

$$L-25=0$$

(4)

Equation (4) represents a 2-DOF motion mode, motion mode 1 (Figure 2a), of the 8-link mechanism. In motion mode 1, both closed-loop 4R kinematic sub-chains, A${}_{11}$B${}_{11}$B${}_{12}$A${}_{12}$ and A${}_{21}$B${}_{21}$B${}_{22}$A${}_{22}$, are parallelograms. Throughout the remainder of this paper, including Figure 2, line AB will be kept in a horizontal position by releasing the frame. This is to ensure that the geometric characteristics of the 8-link mechanism are clearly illustrated.

Case B

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{L}^4+240L^{3}C{\varphi }_a+6750C\left(2{\varphi }_a\right)L^2+7025L^2-150,000C{\varphi }_{a}L-9,000,000=0\hfill \\ L^4+240L^{3}C{\psi }_a+6750C\left(2{\psi }_a\right)L^2+7025L^2-150,000C{\psi }_{a}L-9,000,000=0\hfill \end{array}\right.\end{array}$$

(5)

In the following, we will identify the motion modes associated with Equation (5) by eliminating L using resultant first and then using the primary decomposition of ideals from the algebraic geometry [35] to find the positive-dimensional solutions to the resulted equations.

Eliminating L using resultants from Equation (5), we obtain

$$\begin{array}{c}\hfill 36,905,625,000,000,000,000,000,000g{(C{\varphi }_a-C{\psi }_a)}^4=0\end{array}$$

(6)

where $g=72,900C{\varphi }_a^4-5940C{\varphi }_a^{3}C{\psi }_a-215,280C{\varphi }_a^{2}C{\psi }_a^2-5940C{\varphi }_{a}C{\psi }_a^3+72,900C{\psi }_a^4+63,661C{\varphi }_a^2+12,122C{\varphi }_{a}C{\psi }_a+63,661C{\psi }_a^2-57,600$.

From Equation (6), we obtain the following two sub-cases

$$\begin{array}{c}\hfill C{\varphi }_a-C{\psi }_a=0\end{array}$$

(7)

and

$$\begin{array}{c}\hfill g=0\end{array}$$

(8)

Equation (7) has two solutions:

$$\begin{array}{c}\hfill {\psi }_a=-{\varphi }_a\end{array}$$

(9)

and

$$\begin{array}{c}\hfill {\psi }_a={\varphi }_a\end{array}$$

(10)

Equation (9) represents a 1-DOF motion mode, motion mode 2 (Figure 2b), of the 8-link mechanism. In motion mode 2, both closed-loop 4R sub-kinematic chains, A${}_{11}$B${}_{11}$B${}_{12}$A${}_{12}$ and A${}_{21}$B${}_{21}$B${}_{22}$A${}_{22}$, are anti-parallelograms, and the 8-link mechanism is symmetric about line AB. Motion mode 2 is called the kite motion mode because, in this motion mode, the 8-link mechanism takes the shape of a kite.

Equation (10) represents a 1-DOF motion mode, motion mode 3 (Figure 2c), of the 8-link mechanism. In motion mode 3, both closed-loop 4R kinematic sub-chains, A${}_{11}$B${}_{11}$B${}_{12}$A${}_{12}$ and A${}_{21}$B${}_{21}$B${}_{22}$A${}_{22}$, coincide, and the 8-link mechanism has two inactive joints A and B. Motion mode 3 is called the planar 4R mechanism mode because, in this motion mode, the 8-link mechanism degenerates to a planar 4R mechanism.

Using the primary decomposition of ideals from computer algebraic geometry, Equation (8) can be rewritten as (See Appendix B for details)

$$g_1{g}_2=0$$

(11)

where $g_1=-281+135C\left(2{\varphi }_a\right)-306C({\varphi }_a-{\psi }_a)+295C({\varphi }_a+{\psi }_a)+135C\left(2{\psi }_a\right)$ and $g_2=-281+135C\left(2{\varphi }_a\right)+295C({\varphi }_a-{\psi }_a)-306C({\varphi }_a+{\psi }_a)+135C\left(2{\psi }_a\right)$.

Equation (11) has two solutions:

$$g_1=0$$

(12)

and

$$g_2=0$$

(13)

Equation (12) represents a 1-DOF motion mode, motion mode 4 (Figure 2d), of the 8-link mechanism. In motion mode 4, both closed-loop 4R sub-kinematic chains, A${}_{11}$B${}_{11}$B${}_{12}$A${}_{12}$ and A${}_{21}$B${}_{21}$B${}_{22}$A${}_{22}$, are anti-parallelograms, and the 8-link mechanism is rotational symmetric. Motion mode 4 is called the parallelogram motion mode because, in this motion mode, the 8-link mechanism is in the shape of a parallelogram.

Equation (13) represents a 1-DOF motion mode, motion mode 5 (Figure 2e), of the 8-link mechanism. In motion mode 5, both closed-loop 4R kinematic sub-chains, A${}_{11}$B${}_{11}$B${}_{12}$A${}_{12}$ and A${}_{21}$B${}_{21}$B${}_{22}$A${}_{22}$, are anti-parallelograms, and the 8-link mechanism is symmetric about the perpendicular bisector of $AB$. Motion mode 5 is called the isosceles trapezium motion mode because, in this motion mode, the 8-link mechanism is in the shape of an isosceles trapezium.

It can be observed that if (${\varphi }_a^{*}$, ${\psi }_a^{*}$, $L^{*}$) is a set of solution to Equation (5), then ($\pi \pm {\varphi }_a^{*}$, $\pi \pm {\psi }_a^{*}$, $-L^{*}$) are also solutions to Equation (5). Since $L>0$, the ${\varphi }_a$ − ${\psi }_a$ curve for motion mode 4 (or 5) (see Figure 3) is only one half of the curve obtained using Equation (12) (or Equation (13)) that lies outside of the region enclosed by lines ${\psi }_a-{\varphi }_a=\pm \pi$ (or ${\varphi }_a+{\psi }_a=\pm \pi$). In other words, in motion mode 4, we have ${\psi }_a-{\varphi }_a>\pi$ or ${\psi }_a-{\varphi }_a<-\pi$. In motion mode 5, we have ${\psi }_a+{\varphi }_a>\pi$ or ${\psi }_a+{\varphi }_a<-\pi$.

In summary, the 8-link variable-DOF mechanism has one 2-DOF motion mode and four 1-DOF motion modes (Figure 2).

Table 1. Five motion modes of the 8-link variable-DOF mechanism.

No	DOF	Constraint Equations	Description
1	2	L = 25	Both closed-loop 4R sub-kinematic chains are parallelograms (Figure 2a). ${\varphi }_a$ and ${\psi }_a$ are independent.
2	1	${\psi }_a=-{\varphi }_a$	Both closed-loop 4R kinematic sub-chains are anti-parallelograms. The 8-link mechanism is symmetric about line $AB$ (Figure 2b).
3	1	${\psi }_a={\varphi }_a$	Both closed-loop 4R sub-kinematic chains are anti-parallelograms that coincide with each other (Figure 2c), and the 8-link mechanism has two inactive joints A and B.
4	1	$$\begin{array}{c}\hfill -281+135C\left(2{\varphi }_a\right)\\ \hfill -306C({\varphi }_a-{\psi }_a)\\ \hfill +295C({\varphi }_a+{\psi }_a)\\ \hfill +135C\left(2{\psi }_a\right)=0\end{array}$$	Both closed-loop 4R kinematic sub-chains are anti-parallelograms. The 8-link mechanism is rotational symmetric (Figure 2d).
5	1	$$\begin{array}{c}\hfill -281+135C\left(2{\varphi }_a\right)\\ \hfill +295C({\varphi }_a-{\psi }_a)\\ \hfill -306C({\varphi }_a+{\psi }_a)\\ \hfill +135C\left(2{\psi }_a\right)=0\end{array}$$	Two closed-loop 4R sub-kinematic chains are anti-parallelograms. The 8-link mechanism is symmetric about the perpendicular bisector of $AB$ (Figure 2e).

summarizes all the five motion modes of the 8-link mechanism and their geometric characteristics. Figure 4 shows the ${\varphi }_a$ − ${\psi }_a$ curves for the four 1-DOF motion modes 2 (Equation (9)), 3 (Equation (10)), 4 (Equation (12)) and 5 (Equation (13)).

Unlike the two 8-link variable-DOF planar mechanisms in [5,47] which have four inactive joints in some of their motion modes, this 8-link variable-DOF planar mechanism has two inactive joints A and B in one of the 1-DOF motion modes (motion mode 3).

Figure 2 shows that both closed-loop 4R kinematic sub-chains of this novel 8-link variable-DOF planar mechanism must appear as parallelograms in its 2-DOF motion mode or anti-parallelograms in any of its 1-DOF motion mode, whereas both the 8-link variable-DOF planar mechanism [47] and the 12-link Wunderlich mechanism have a motion mode with an odd number of 4R sub-kinematic chains appearing as anti-parallelograms.

Although one can solve Equation (5) directly using the primary decomposition of ideals to identify all the motion modes of the 8-link variable-DOF mechanism, it was found hard to obtain the concise equations (Equations (12) and (13)) for motion modes 4 and 5 that were obtained by eliminating L before calculating the primary decomposition of ideals.

It is important to note that the reconfiguration analysis approach described in [47] is not applicable to the 8-link variable-DOF planar mechanism shown in Figure 1. This is because the method in [47] cannot distinguish between motion mode 2 (Figure 2b) and motion mode 4 (Figure 2d), or between motion mode 3 (Figure 2c) and motion mode 5 (Figure 2e), for this mechanism.

5. Transition Configuration Analysis of the 8-Link Variable-Dof Planar Mechanism

The transition configurations between two or more motion modes can be obtained by solving the kinematic equations composed of equations of these motion modes [36].

Let us take the transition configurations between motion modes 2 and 4, T(2 ⋀ 4), of the 8-link mechanism as an example.

The set of equations composed of Equations (9) (motion mode 2) and (12) (motion mode 4) is

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\psi }_a=-{\varphi }_a\hfill \\ -281+135C\left(2{\varphi }_a\right)-306C({\varphi }_a-{\psi }_a)+295C({\varphi }_a+{\psi }_a)+135C\left(2{\psi }_a\right)=0\hfill \end{array}\right.\end{array}$$

(14)

Section 4 shows that in motion mode 4, we have ${\psi }_a-{\varphi }_a>\pi$ or ${\psi }_a-{\varphi }_a<-\pi$. Solving Equation (14) under these conditions, we obtain two solutions:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\varphi }_a=2.5559\left(rad\right)\hfill \\ {\psi }_a=-{\varphi }_a\hfill \end{array}\right.\end{array}$$

(15)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\varphi }_a=-2.5559\left(rad\right)\hfill \\ {\psi }_a=-{\varphi }_a\hfill \end{array}\right.\end{array}$$

(16)

Equations (15) and (16) show that there are two transition configurations between motion modes 2 and 4, T(2 ⋀ 4)${}_I$ (Figure 5a) and T(2 ⋀ 4)${}_{II}$ (Figure 5b). In these two transition configurations, links A${}_{i1}$B${}_{i1}$ and B${}_{i2}$A${}_{i2}$ are parallel to AB. One can readily obtain that the instantaneous DOF of the 8-link variable-DOF mechanism is two in these transition configurations. The details are omitted here since the calculation of instantaneous DOF of a mechanism has been well-documented in the literature.

Following the above steps, we can identify six more transition configurations of the 8-link mechanism, including two transition configurations, T(3 ⋀ 5)${}_I$ (Figure 5c) and T(3 ⋀ 5)${}_{II}$) (Figure 5d), in which the mechanism can switch between two motion modes and four transition configurations, (T(1 ⋀ 2 ⋀ 3)${}_I$ (Figure 5e), T(1 ⋀ 2 ⋀ 3)${}_{II}$ (Figure 5f), T(1 ⋀ 4 ⋀ 5)${}_I$ (Figure 5g), and T(1 ⋀ 4 ⋀ 5)${}_{II}$) (Figure 5h), in which the mechanism can switch among three motion modes. It is noted that there are no transition configurations between motion modes 2 and 5 or between motion modes 3 and 4. The geometric characteristics of the 8-link mechanism in all the eight transition configurations are summarized in

Table 2. Transition configurations of the 8-link mechanism.

No	${\mathit{\varphi }}_{\mathit{a}}$ and ${\mathit{\psi }}_{\mathit{a}}$ (rad)	Description	Instantaneous DOF
T(2 ⋀ 4)${}_I$	$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\varphi }_a=2.5559\hfill \\ {\psi }_a=-{\varphi }_a\hfill \end{array}\right.\end{array}$$	Links A${}_{i1}$B${}_{i1}$ and B${}_{i2}$A${}_{i2}$ are parallel to AB (Figure 5a).	2
T(2 ⋀ 4)${}_{II}$	$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\varphi }_a=-2.5559\hfill \\ {\psi }_a=-{\varphi }_a\hfill \end{array}\right.\end{array}$$	Links A${}_{i1}$B${}_{i1}$ and B${}_{i2}$A${}_{i2}$ are parallel to AB (Figure 5b).
T(3 ⋀ 5)${}_I$	$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\varphi }_a=2.5559\hfill \\ {\psi }_a={\varphi }_a\hfill \end{array}\right.\end{array}$$	Links A${}_{i1}$B${}_{i1}$ and B${}_{i2}$A${}_{i2}$ (i = 1 and 2) are parallel to AB (Figure 5c).
T(3 ⋀ 5)${}_{II}$	$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\varphi }_a=-2.5559\hfill \\ {\psi }_a={\varphi }_a\hfill \end{array}\right.\end{array}$$	Links A${}_{i1}$B${}_{i1}$ and B${}_{i2}$A${}_{i2}$ (i = 1 and 2) are parallel to AB (Figure 5d).
T(1 ⋀ 2 ⋀ 3)${}_I$	$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\varphi }_a=0\hfill \\ {\psi }_a=0\hfill \end{array}\right.\end{array}$$	All the R joint centers are collinear (Figure 5e).	4
T(1 ⋀ 2 ⋀ 3)${}_{II}$	$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\varphi }_a=\pi \hfill \\ {\psi }_a=\pi \hfill \end{array}\right.\end{array}$$	All the R joint centers are collinear (Figure 5f).
T(1 ⋀ 4 ⋀ 5)${}_I$	$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\varphi }_a=\pi \hfill \\ {\psi }_a=0\hfill \end{array}\right.\end{array}$$	All the R joint centers are collinear (Figure 5g).
T(1 ⋀ 4 ⋀ 5)${}_{II}$	$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\varphi }_a=0\hfill \\ {\psi }_a=\pi \hfill \end{array}\right.\end{array}$$	All the R joint centers are collinear (Figure 5h).

. All the transition configurations are singular configurations. The instantaneous DOF of the 8-link variable-DOF in transition configurations can be readily obtained as two in transition configurations T(2 ⋀ 4)${}_I$, T(2 ⋀ 4)${}_{II}$, T(3 ⋀ 5)${}_I$, and T(3 ⋀ 5)${}_{II}$ and four in transition configurations T(1 ⋀ 2 ⋀ 3)${}_I$, T(1 ⋀ 2 ⋀ 3)${}_{II}$, T(1 ⋀ 4 ⋀ 5)${}_I$, and T(1 ⋀ 4 ⋀ 5)${}_{II}$.

6. Reconfiguration of the Variable-Dof 8-Link Planar Mechanism

Figure 6 illustrates the reconfiguration of the 8-link planar mechanism among the five motion modes via the eight transition configurations in the ${\varphi }_a-{\psi }_a$ plane. The curves in the ${\varphi }_a-{\psi }_a$ plane of the four 1-DOF motion modes, motion modes 2, 3, 4, and 5, are shown in green, pink, red, and blue, respectively. The 2-DOF motion mode, motion mode 1 in which $L=25$, covers the whole ${\varphi }_a-{\psi }_a$ plane. However, the 8-link mechanism can only transit among motion modes 1, 2, and 3 at two transition configurations T(1 ⋀ 2 ⋀ 3)${}_I$ and T(1 ⋀ 2 ⋀ 3)${}_{II}$ and among motion modes 1, 4, and 5 at two transition configurations T(1 ⋀ 4 ⋀ 5)${}_I$ and T(1 ⋀ 4 ⋀ 5)${}_{II}$ since we have $L>25$ in the other configurations in motion modes 2, 3, 4, and 5.

Photos of the LEGO model of this 8-link mechanism at all the transition configurations, configurations with ${\varphi }_a=\pm \pi /2$ and/or ${\psi }_a=\pm \pi /2$ in 1-DOF motion modes 2, 3, 4, and 5, and a configuration with ${\varphi }_a=\pi /2$ and ${\psi }_a=-\pi /2$ in 2-DOF motion mode 1 are given. To distinguish the only 2-DOF motion mode from the 1-DOF motion modes, the photos of the sample configuration and the four transition configurations associated with motion mode 1 are framed. In the LEGO model, the links are allocated in six layers, and the axis of R joint A is in a curved shape to allow the mechanism to switch among all the five motion modes through the eight transition configurations without link interference. An animation of the reconfiguration of the 8-link mechanism among the five motion modes can be found in the Supplementary Materials. In the animation, link AA${}_{21}$A${}_{22}$ is selected as the frame of the mechanism, and all “⋀” have been omitted in the notations for transition configurations for simplicity reasons.

The mechanism could be more compact if it is only required to switch among some but not all of its motion modes. For example, if one needs the 8-link mechanism to switch among four motion modes 1, 2, 4 and 5 only (Figure 7), all the links can be located in four layers without encountering link interference as shown in the CAD model of the mechanism in transition configuration T(1 ⋀ 4 ⋀ 5)${}_I$. This four-layer 8-link planar mechanism could be used as a construction unit of new variable-DOF multi-loop mechanisms, which would enrich the types of reconfigurable/multi-mode deployable mechanisms [25,26,27,42,51,52].

7. Conclusions

A novel 8-link variable-DOF planar mechanism with five motion modes has been proposed. Reconfiguration analysis has shown that the mechanism has one 2-DOF double parallelogram motion mode and four 1-DOF motion modes. In addition, the mechanism can switch among three motion modes at four transition configurations and between two motion modes at four other transition configurations.

In contrast to the two 8-link variable-DOF planar mechanisms in [5,47], which have four inactive joints in some of their motion modes, this novel 8-link variable-DOF planar mechanism has two inactive joints in one of its 1-DOF motion modes. The two closed-loop 4R kinematic sub-chains of the novel mechanism must appear either as a pair of parallelograms in the 2-DOF motion mode or a pair of anti-parallelograms in a 1-DOF motion mode.

The hybrid approach that combines elimination and computer algebraic geometry methods has been found to be more efficient than the algebraic geometry approach without elimination. As a by-product, a method for factoring trigonometric functions in two angles has been proposed.

This work, together with reference [47], provides a starting point for the design and analysis of variable-DOF multi-loop mechanisms constructed using more than two parallelograms, which could be used as reconfigurable/multi-mode deployable mechanisms.