Exhaustive Enumeration of Spatial Prime Structures

Abstract

Prime structures are link chains with 0 DoF (degrees of freedom), not including subchains with 0 or fewer DoF, which are expected to be used in systematic kinematic and dynamic analyses of link mechanisms. This paper describes the exhaustive enumeration of spatial prime structures with three–five links. There will be more types of spatial prime structures than planar prime structures due to the variety in the DoF of kinematic pairs and the existence of prime structures with idle DoF. In the enumeration, the graphs which represent the connection of links and pairs are used. The vertices of the graphs represent the links of structures, and the edges represent the kinematic pairs. To consider the pair DoF, weights are set on edges. First, the numbers of pairs are calculated using Grübler’s equation. Second, the graphs corresponding to structures are enumerated, without considering pair DoF, and the isomorphic graphs and other inappropriate graphs are eliminated. Then, the combinations of pair DoF arrangement are enumerated, and the isomorphic graphs and graphs which correspond to non-prime structures are eliminated. Finally, prime structures with idle DoF are considered. As a result, 3, 13, and 97 kinds of spatial prime structures for 3, 4, and 5 links, respectively, are obtained.

1. Introduction

The systematic kinematic analysis method using “Prime Structures [1]” has been proposed for planar mechanisms. Prime structures are the structures with zero DoF (degrees of freedom), not including subchains with 0 or fewer DoF [2], which cannot be divided into other prime structures. Prime structures are also called “Baranov Trusses [3]”, “Basic Rigid Chains [4]”, “Basic Kinematic Chains [5]”, “Assur Kinematic Chains [6]”, “Primary structures [7]”, and so on. These names represent the same thing as prime structures. Planar prime structures have been investigated by many researchers [8,9].

Planar prime structures have zero DoF and they cannot move. Therefore, if the links can be connected, the positions and postures of all links in the prime structure, namely, the “configuration” of the prime structure, can be determined and calculated. This analysis is called “configuration analysis”. Funabashi [1] established the configuration analysis method of planar prime structures with 3, 5, and 7 links and applied it to the kinematic analysis of planar link mechanisms. Rojas and Thomas [10] established the configuration analysis method of planar prime structures with up to nine links.

For example, let us consider a planar three-link kinematic chain, shown in Figure 1a. In this chain, the number of links, $N$, is three and the number of kinematic pairs, $J$, is three. Therefore, from Grübler’s equation, the DoF of the chain, $F$, can be calculated as

$$F=3\left(N-1\right)-2J=3\left(3-1\right)-2\times 3=0.$$

(1)

Also, the chain does not include rigid subchains (subchains with 0 or less DoF). Therefore, the chain in Figure 1a is a three-link planar prime structure.

If one removes the link ${\mathrm{j}}_1{\mathrm{j}}_3$ of the prime structure in Figure 1a and connects ${\mathrm{j}}_1$ and ${\mathrm{j}}_3$ to the frame and the tip of a driving link, respectively, then a planar four-bar link mechanism, shown in Figure 1c, can be obtained. On the other hand, when the motion of the driving link ${\mathrm{j}}_4{\mathrm{j}}_3$ in the mechanism in Figure 1c is given, the position of the pair ${\mathrm{j}}_3$ can be calculated; then, the remaining kinematic chain is the prime structure in Figure 1a. Therefore, the kinematic analysis of the mechanism can be achieved based on the configuration analysis of the prime structure ${\mathrm{j}}_1{\mathrm{j}}_2{\mathrm{j}}_3$.

For another example, let us consider a planar five-link kinematic chain, shown in Figure 1b. This chain has five links and six revolute pairs. Therefore, from Grübler’s equation, the DoF of the chain, $F$, can be calculated as

$$F=3\left(5-1\right)-2\times 6=0.$$

(2)

Also, the chain does not include rigid subchains, so the chain in Figure 1b is a five-link prime structure.

By removing the link ${\mathrm{j}}_1{\mathrm{j}}_5$ in Figure 1b and then connecting ${\mathrm{j}}_1$ and ${\mathrm{j}}_5$ to a frame and the tip of a moving link arm, respectively, the planar six-link mechanism shown in Figure 1d is obtained. Also, by removing the link ${\mathrm{j}}_4{\mathrm{j}}_5{\mathrm{j}}_6$, shown in Figure 1b, and then connecting ${\mathrm{j}}_4,{\mathrm{j}}_5$, and ${\mathrm{j}}_6$ to the tip of three driving links, respectively, a planar parallel mechanism with 3 DoF, shown in Figure 1e, is obtained. By applying configuration analysis of the five-link prime structure shown in Figure 1b, kinematic analyses of these two mechanisms can be achieved. As mentioned above, if we establish the configuration analyses of prime structures, we can analyze many types of link mechanisms.

In addition to kinematic analysis, planar prime structures are also applied to various types of analyses and designs of planar link mechanisms. Chung [11] calculated the motion ranges of Stephenson mechanisms using the five-link planar prime structure. Pennock and Kamthe [12] applied planar prime structures to the calculations of the dead points of planar mechanisms. Lee and Lo [13] synthesized seven-link over-constrained mechanisms, which are topologically same as seven-link planar prime structures. Kim and Cho [14] designed the static balancers using planar prime structures, and Rojas, et al. [15] applied planar prime structures to the design of underactuated grippers.

As mentioned above, prime structures are useful for the analysis and synthesis of link mechanisms. However, the research of spatial prime structures has not progressed. Liang and Takeda [7] used a specific spatial prime structure to calculate the transmission index of a spatial parallel mechanism, but systematic research of spatial prime structures has not yet been achieved.

In order to apply spatial prime structures to the systematic analysis and synthesis of spatial link mechanisms, it is strongly required to enumerate the types of spatial prime structures exhaustively. Thus, the objective of this paper is to enumerate, exhaustively, spatial prime structures. The exhaustive enumeration of planar prime structures has been researched by many researchers. Nie et al. [16] obtained planar prime structures with up to 11 links, Huang and Ding [2] obtained those with up to 13 links, and Morlin et al. [17] obtained those with up to 15 links.

The enumeration of planar kinematic chains, including planar prime structures, can be classified into three processes: an enumeration of candidate chains, an elimination of isomorphic chains, and an elimination of non-prime structures. Here, a non-prime structure is a structure including rigid subchains, which are subchains with 0 or less DoF.

For example, Huang et al. [2,18,19,20] used graphs in graph theory to enumerate planar prime structures. In the graphs, the vertices represent the links in the kinematic chains and the edges represent the kinematic pairs. First, the graphs corresponding to kinematic chains with zero DoF were exhaustively enumerated using Grübler’s equation and other conditions that prime structures must satisfy. Next, for the elimination of isomorphic graphs, the graphs were represented by adjacency matrices, and the adjacency matrices were used to determine isomorphism. For the elimination of non-prime structures, in order to detect rigid subchains, the authors proposed a method by which to create a new subchain from two subchains. Many other researchers have also studied the counting of candidate chains [16,21], the elimination of isomorphic chains [21,22,23], and the elimination of non-prime chains [17,24].

In this paper, the enumeration is extended to the spatial prime structures. In the planar prime structures, only one DoF pair is used. However, in spatial kinematic chains, there are some types of kinematic DoF pairs to be considered. Furthermore, unlike planar chains, there are spatial kinematic chains with idle DoF. The idle DoF are DoF that do not affect the positions and postures of almost all the links in the kinematic chains, and the spatial prime structures with idle DoF are also enumerated.

2. Calculation of the Numbers of Pairs and Links

For the first step of the enumeration of the spatial prime structure, the number of links is given, and the numbers of 1-, 2-, and 3-DoF pairs are calculated based on Grübler’s equation. Then, the numbers of binary, ternary, …, and $n$-pair links are calculated using the condition of the numbers of links and pairs. In the enumeration, we practically consider 1-DoF (revolute or prismatic), 2-DoF (cylindrical), and 3-DoF (spherical) pairs. In some research on the enumeration of planar kinematic chains, multiple kinematic pairs are considered [25,26], but in this research, we do not consider them because multiple kinematic pairs such as a universal pair can be realized using the combinations of some of the pairs mentioned above. For example, a two-link chain in which two links are connected by a universal pair can be considered as a special case of a three-link chain in which three links are connected by two revolute pairs. Therefore, if we exhaustively enumerate prime structures with revolute pairs, it becomes possible to consider prime structures that include universal pairs.

First, the numbers of kinematic pairs are calculated. When the number of links, $N$, is given, the numbers of pairs, $J_i$ ($i=1, 2,3$), with $i$-DoF are calculated using Grübler’s equation:

$$F=6\left(N-1\right)-5J_1-4J_2-3J_3.$$

(3)

In Equation (3), $F$ is the DoF of the chain. First, let us consider structures without idle DoF, so let us set $F=0$. The case when $F>0$, i.e., when the structure has idle DoF, is considered in Section 4.4. When the number of links, $N$, is given, all combinations of the numbers of 1-, 2-, and 3-DoF pairs $\left( J_1, J_2, J_3\right)$, which are non-negative integers, are found from Equation (3). The results for $N=3, 4, 5$ are shown in the second rows of

Table 1. The enumerated spatial prime structures with 3–4 links.

(1) Number of links $N$	3			4
(2) Numbers of 1, 2, 3-DoF pairs $\left(J_1, J_2, J_3\right)$	$\left(1, 1, 1\right)$	$\left(0, 3, 0\right)$	$\left(0, 0, 4\right)$	$\left(3, 0, 1\right)$	$\left(2, 2, 0\right)$	$\left(1, 1, 3\right)$	$\left(0, 3, 2\right)$	$\left(0, 0, 6\right)$
(3) Numbers of links with 2, 3 pairs $\left(N_2, N_3\right)$	$\left(3, 0\right)$		$\left(2, 1\right)$	$\left(4, 0\right)$		$\left(2, 2\right)$		$\left(0, 4\right)$
(4) Link connection topology			N/A
(5) Prime structures
(6) Prime structures with idle DoFs

,

Table 2. The enumerated spatial prime structures with five links and five–six pairs.

(1) Number of links $N$	5
(2) Numbers of 1, 2, 3-DoF pairs $\left(J_1, J_2, J_3\right)$	$\left(4, 1, 0\right)$	$\left(0 6, 0\right), \left(1 4, 1\right), \left(2, 2, 2\right), \left(3, 0, 3\right)$
(3) Numbers of links with 2, 3, 4 pairs $\left(N_2, N_3, N_4\right)$	$\left(5, 0, 0\right)$	$\left(3, 2, 0\right)$
(4) Link connection topology
(5) Prime structures		DOFs of $\left({\mathrm{j}}_1,{\mathrm{j}}_2,{\mathrm{j}}_3,{\mathrm{j}}_4,{\mathrm{j}}_5,{\mathrm{j}}_6\right)$ $=(2, 2, 2, 2, 2, 2),$ $(3, 2, 2, 2, 2, 1),$ $(3, 2, 2, 2, 1, 2),$ $(3, 1, 2, 2, 2, 2),$ $(3, 2, 3, 2, 1, 1),$ $(3, 2, 3, 1, 2, 1),$ $(3, 2, 3, 1, 1, 2),$ $(3, 2, 2, 3, 1, 1),$ $(3, 2, 2, 1, 1, 3),$ $(3, 2, 1, 3, 1, 2),$ $\left(3, 1, 3, 1, 2, 2\right),$ $\left(3, 1, 2, 2, 1, 3\right),$ $\left(3, 1, 3, 1, 3, 1\right),$ $\left(3, 1, 1, 3, 1, 3\right)$	$\mathrm{DOFs} \mathrm{of}$  $\left({\mathrm{j}}_1,{\mathrm{j}}_2,{\mathrm{j}}_3,{\mathrm{j}}_4,{\mathrm{j}}_5,{\mathrm{j}}_6\right)$ $=(3, 2, 2, 2, 2, 1),$ $(3, 2, 2, 2, 1, 2),$ $(3, 2, 2, 1, 2, 2),$ $(2, 2, 3, 2, 2, 1),$ $(2, 2, 3, 2, 1, 2),$ $(3, 2, 3, 2, 1, 1),$ $(3, 2, 3, 1, 2, 1),$ $(3, 2, 3, 1, 1, 2),$ $(3, 2, 2, 3, 1, 1),$ $(3, 2, 2, 1, 3, 1),$ $(3, 2, 2, 1, 1, 3),$ $(3, 1, 3, 2, 2, 1),$ $(3, 1, 3, 2, 1, 2),$ $(3, 1, 3, 1, 2, 2),$ $(2, 2, 3, 3, 1, 1),$ $(2, 2, 3, 1, 3, 1),$ $(3, 1, 3, 3, 1, 1),$ $(3, 1, 3, 1, 3, 1),$ $\left(3, 1, 3, 1, 1, 3\right)$
(6) Prime structures with idle DoFs		DOFs of $\left({\mathrm{j}}_1,{\mathrm{j}}_2,{\mathrm{j}}_3,{\mathrm{j}}_4,{\mathrm{j}}_5,{\mathrm{j}}_6\right)$ $=(3, 3, 2, 2, 2, 1),$ $(3, 3, 3, 2, 1, 1),$ $(3, 3, 3, 1, 2, 1),$ $(3, 3, 3, 1, 1, 2),$ $\left(3, 3, 3, 3, 1, 1\right)$	$\mathrm{DOFs} \mathrm{of}$ $\left({\mathrm{j}}_1,{\mathrm{j}}_2,{\mathrm{j}}_3,{\mathrm{j}}_4,{\mathrm{j}}_5,{\mathrm{j}}_6\right)$ $=(3, 3, 2, 2, 2, 1),$ $(3, 3, 2, 2, 1, 2),$ $(3, 3, 3, 2, 1, 1),$ $(3, 3, 3, 1, 2, 1),$ $(3, 3, 2, 3, 1, 1),$ $\left(3, 3, 2, 1, 3, 1\right)$

and

Table 3. The enumerated spatial prime structures with five links and seven–eight pairs.

(1) Number of links N	5
(2) Numbers of 1, 2, 3-DoF pairs $\left(J_1, J_2, J_3\right)$	$\left(0, 3, 4\right), \left(1, 1, 5\right)$			$\left(0, 0, 8\right)$
(3) Numbers of links with 2, 3, 4 pairs $\left(N_2, N_3, N_4\right)$	$\left(3, 0, 2\right)$	$\left(2, 2, 1\right)$	$\left(1, 4, 0\right)$	$\left(0, 4, 1\right)$
(4) Link connection topology
(5) Prime structures	$\mathrm{DOFs} \mathrm{of}$ $\left({\mathrm{j}}_1,{\mathrm{j}}_2,{\mathrm{j}}_3,{\mathrm{j}}_4,{\mathrm{j}}_5,{\mathrm{j}}_6,{\mathrm{j}}_7\right)$ $=\left(3, 2, 3, 3, 2, 3, 2\right),$ $\left(3, 2, 3, 2, 3, 2, 3\right)$	$\mathrm{DOFs} \mathrm{of}$ $\left({\mathrm{j}}_1,{\mathrm{j}}_2,{\mathrm{j}}_3,{\mathrm{j}}_4,{\mathrm{j}}_5,{\mathrm{j}}_6,{\mathrm{j}}_7\right)$ $=\left(3, 2, 3, 3, 3, 2, 2\right),$ $\left(3, 2, 3, 3, 2, 3, 2\right),$ $\left(3, 2, 3, 3, 2, 2, 3\right),$ $\left(3, 2, 3, 2, 3, 3, 2\right),$ $\left(3, 2, 3, 2, 3, 2, 3\right),$ $\left(3, 2, 2, 3, 3, 3, 2\right),$ $\left(2, 3, 3, 3, 3, 2, 2\right),$ $\left(2, 3, 3, 3, 2, 3, 2\right),$ $\left(2, 3, 3, 2, 3, 3, 2\right),$ $\left(3, 2, 3, 3, 3, 3, 1\right),$ $\left(3, 2, 3, 3, 3, 1, 3\right),$ $\left(3, 1, 3, 3, 3, 3, 2\right),$ $\left(2, 3, 3, 3, 3, 3, 1\right)$	$\mathrm{DOFs} \mathrm{of}$ $\left({\mathrm{j}}_1,{\mathrm{j}}_2,{\mathrm{j}}_3,{\mathrm{j}}_4,{\mathrm{j}}_5,{\mathrm{j}}_6,{\mathrm{j}}_7\right)$ $=\left(3, 3, 3, 3, 2, 2, 2\right),$ $\left(3, 3, 3, 2, 2, 3, 2\right),$ $\left(3, 3, 3, 2, 2, 2, 3\right),$ $\left(3, 3, 2, 3, 2, 3, 2\right),$ $\left(3, 3, 2, 2, 3, 3, 2\right),$ $\left(3, 2, 3, 3, 3, 2, 2\right),$ $\left(3, 2, 3, 3, 2, 3, 2\right),$ $\left(3, 2, 3, 3, 2, 2, 3\right),$ $\left(3, 3, 3, 3, 3, 2, 1\right),$ $\left(3, 3, 3, 3, 2, 3, 1\right),$ $\left(3, 3, 3, 3, 2, 1, 3\right),$ $\left(3, 3, 3, 3, 1, 3, 2\right),$ $\left(3, 3, 3, 3, 1, 2, 3\right),$ $\left(3, 2, 3, 3, 3, 3, 1\right),$ $\left(3, 1, 3, 3, 3, 3, 2\right)$
(6) Prime structures with idle DoFs	$\mathrm{DOFs} \mathrm{of}$ $\left({\mathrm{j}}_1,{\mathrm{j}}_2,{\mathrm{j}}_3,{\mathrm{j}}_4,{\mathrm{j}}_5,{\mathrm{j}}_6,{\mathrm{j}}_7\right)$ $=\left(3, 3, 3, 3, 2, 3, 2\right),$ $\left(3, 2, 3, 3, 3, 3, 2\right),$ $\left(3, 2, 3, 3, 3, 3, 3\right),$ $\left(3, 3, 3, 3, 3, 3, 3\right)$	$\mathrm{DOFs} \mathrm{of}$ $\left({\mathrm{j}}_1,{\mathrm{j}}_2,{\mathrm{j}}_3,{\mathrm{j}}_4,{\mathrm{j}}_5,{\mathrm{j}}_6,{\mathrm{j}}_7\right)$ $=\left(3, 3, 3, 3, 3, 2, 2\right),$ $\left(3, 3, 3, 3, 2, 3, 2\right),$ $\left(3, 3, 3, 3, 2, 2, 3\right),$ $\left(3, 3, 3, 2, 3, 3, 2\right),$ $\left(3, 3, 3, 2, 3, 2, 3\right),$ $\left(3, 3, 2, 3, 3, 3, 2\right),$ $\left(3, 2, 3, 2, 3, 3, 3\right),$ $\left(3, 3, 3, 3, 2, 3, 3\right),$ $\left(3, 3, 3, 2, 3, 3, 3\right),$ $\left(3, 3, 3, 3, 3, 3, 1\right),$ $\left(3, 3, 3, 3, 3, 1, 3\right)$	$\mathrm{DOFs} \mathrm{of}$ $\left({\mathrm{j}}_1,{\mathrm{j}}_2,{\mathrm{j}}_3,{\mathrm{j}}_4,{\mathrm{j}}_5,{\mathrm{j}}_6,{\mathrm{j}}_7\right)$ $=\left(3, 3, 3, 2, 2, 3, 3\right),$ $\left(3, 3, 2, 3, 2, 3, 3\right),$ $\left(3, 3, 2, 2, 3, 3, 3\right),$ $\left(3, 2, 3, 3, 2, 3, 3\right),$ $\left(3, 3, 3, 3, 1, 3, 3\right),$ $\left(3, 1, 3, 3, 3, 3, 3\right)$

.

Next, the numbers of links, $N_k \left(k=2, 3,\dots \right)$ with $k$ kinematic pairs are calculated. $N_2,N_3, \dots$ mean the numbers of binary, ternary, … links. From the stated conditions of the number of links and the number of pairs, the following equations hold.

$$\sum _{k=2}^{\infty }{N}_k=N, \sum _{k=2}^{\infty }k{N}_k=2\left(J_1+J_2+J_3\right) .$$

(4)

By exhaustively searching for all combinations, the possible combinations of non-negative integers $\left(N_2, N_3, N_4, \dots \right)$ are obtained from Equation (4). The results are shown in the third rows of Table 1, Table 2 and Table 3.

3. Link Connection Topology

In this section, the combinations of the connections between links in the prime structures are considered, without considering the DoF of kinematic pairs. We call these combinations the “Link connection topology”.

3.1. Graph Representation of Prime Structures

In the enumeration of the planar prime structures, a graph representation is used [2]. Although the graph representation may be complicated and less intuitive, we adopt the graph representation because exhaustive enumeration becomes possible by applying the concepts of adjacency matrices and dual graphs in graph theory. In this representation, the links of the chain are set as vertices of the graph and the joints of the chain are set as edges of the graph. For example, the planar five-link prime structure in Figure 2a can be represented as a graph, as shown in Figure 2d.

In planar kinematic chains, kinematic pairs are basically only 1 DoF. However, in spatial chains, 1–3 DoF pairs are used. Therefore, when extending the graph representation to spatial chains, it is necessary to add the information of the DoF of kinematic pairs. In this paper, the DoF of pairs are represented by weighting the edges of the graph. For example, let us consider the graph representation of the spatial kinematic chain, as shown in Figure 2b. In the figure, the numbers in kinematic pairs represent the DoF of the kinematic pairs. This chain is represented as a graph, as shown in Figure 2e. The information that link ${\mathrm{n}}_2$ and link ${\mathrm{n}}_5$ are connected by a 2-DoF pair in Figure 2b is represented by connecting vertex ${\mathrm{v}}_2$ and vertex ${\mathrm{v}}_5$ with an edge of weight 2 in Figure 2e.

Enumerating the graph of spatial prime structures is more complicated than that of planar prime structures since there are many kinds of pair DoF. Therefore, we first exhaustively enumerate the combinations of connections between links without considering DoF and then consider the combinations of the arrangement of the pair DoF. Since we call the connections between links “Link connection topology”, we call the corresponding graphs “Link connection graphs”. Also, we call graphs which are made by arranging pair DoF on link connection graphs as “Pair arrangement graphs”. For example, Figure 2c is the link connection topology corresponding to the spatial chain in Figure 2b, Figure 2f is the link connection graph, and Figure 2e is the pair arrangement graph.

3.2. Exhaustive Enumeration of All Possible Link Connection Graphs

First, link connection graphs can be exhaustively enumerated. From the calculation in Section 2, the number of kinematic pairs, $J=J_1+J_2+J_3$, and the numbers of links with $k$ pairs $N_k$ are obtained. For enumeration, “complete graphs” are used, and link connection graphs are obtained by deleting some edges of the complete graphs.

A complete graph is a graph in which all vertices are connected to each other by a single edge. For example, complete graphs with 3, 4, or 5 vertices are shown in Figure 3a–c, and the link connection topology corresponding to the graphs in Figure 3a,b are shown in Figure 3d,e.

Some edges are deleted from complete graphs to obtain link connection graphs. For example, by deleting edge $e_1$ from the complete graph in Figure 4a, the link connection graph in Figure 4b is obtained. This means that by deleting pair $j_1$ from the link connection topology in Figure 4c, the link connection topology in Figure 4d can be obtained.

In a complete graph with $N$ vertices, the number of edges $E_{comp, N}$ are calculated as

$$E_{comp,3}=3, E_{comp,4}=6, E_{comp,5}=10,$$

(5)

as shown in Figure 3a–c. By deleting $E_{comp,N}-J$ edges from a complete graph, a link connection graph with $N$ vertices and $J$ edges is then obtained. These link connection graphs correspond to link connection topologies with $N$ links and $J$ pairs. In this way, all possible link connection graphs can be obtained. The number of combinations of deleting edges is

$$\left(\begin{array}{c}{E}_{comp,N}\\ E_{comp,N}-J\end{array}\right)={\displaystyle \frac{E_{comp,N}!}{J!\left(E_{comp,N}-J\right)!}},$$

(6)

where $\left(\begin{array}{c}a\\ b\end{array}\right)$ is the binominal coefficient.

There are no multiple edges in the link connection graphs obtained via this method. However, we can count all possible link connection graphs since there can be no multiple edges in the link connection graphs of prime structures. This is because if there are multiple edges in a link connection graph, two links in the corresponding link connection topology are connected by two or more kinematics pairs, as shown in Figure 5, and the link connection topology cannot be a prime structure.

3.3. Elimination of the Inappropriate Link Connection Graphs

The link connection graphs which are obtained in the previous subsection include inappropriate ones. Therefore, in order to eliminate them, we set the following three conditions that the link connection graphs must satisfy.

• Condition 1: The number of vertices with $k$-degree is $N_k$.
• Condition 2: The graph must be 2-connected.
• Condition 3: There are no isomorphic graphs.

Concerning condition 1, the “degree” of a vertex is the number of edges that are connected to the vertex. If the degree of a vertex in a link connection graph is $k$, the corresponding link in the link connection topology has $k$ pairs. Thus, in order to satisfy the condition, “the number of links with 2, 3, … pairs are $N_2, N_3, \dots$”, which is calculated in Section 2, the numbers of vertices of degree 2, 3, … in link connection graphs must be $N_2, N_3, \dots$.

For the calculation of the degrees, “adjacency matrices” are used. An adjacency matrix is a matrix that indicates whether the vertices of a graph are connected by edges. If vertices ${\mathrm{v}}_i$ and ${\mathrm{v}}_j$ are connected by edges, the $(i, j)$ element and $(j, i)$ component of the adjacency matrix are 1; otherwise, the components are 0. For an example, when we consider the link connection graph in Figure 6a corresponding to the link connection topology in Figure 6b, the adjacency matrix is shown in Figure 6c. In Figure 6, for example, the vertices ${\mathrm{v}}_1$ and ${\mathrm{v}}_2$ of the graph are connected, so the $\left(1, 2\right)$ element of the adjacency matrix is 1, and ${\mathrm{v}}_3$ and ${\mathrm{v}}_4$ are not connected, so the $\left(3, 4\right)$ element is 0.

From the adjacency matrix, the degree $d\left({\mathrm{v}}_i\right)$ of a vertex ${\mathrm{v}}_i$ is calculated as

$$d\left({\mathrm{v}}_i\right)=\sum _{j=1}^N{m}_{ij},$$

(7)

where $m_{ij}$ is $\left(i, j\right)$ element of the adjacency matrix. Using the degrees, condition 1 can be considered.

Concerning condition 2, a graph is called “$k$-connected” or “$k$-vertex-connected” if the graph remains connected (not divided) when fewer than $k$ vertices are removed. For example, the graph in Figure 7a is not 1-connected because it is itself divided into two disconnected parts. The graph in Figure 7b is 1-connected because the whole graph is not divided into parts, but it is not 2-connected because it is divided into two parts when the vertex ${\mathrm{v}}_1$ is removed. The graph in Figure 7d is 1-connected because the whole graph is not divided into parts, and it is also 2-connected because it is not divided when any one vertex is removed.

If the link connection graph is not 1-connected like in Figure 7a, the corresponding kinematic chain is separated into two or more parts, so it cannot be a structure. If the graph is 1-connected but not 2-connected like in Figure 7b, the corresponding structure can be divided into two independent subchains, ${\mathrm{S}}_1$ and ${\mathrm{S}}_2$, as shown in Figure 7c. For the kinematic chain to be a structure, both independent subchains ${\mathrm{S}}_1$ and ${\mathrm{S}}_2$ must be structures. However, such a kinematic chain cannot be a prime structure because a structure containing rigid subchains cannot be a prime structure. Therefore, link connection graphs that are not 2-connected are eliminated in this paper.

To consider condition 3, adjacency matrices are used. For example, we consider whether graphs $G_1$ and $G_2$ in Figure 8a,b are isomorphic. The adjacency matrices of graphs $G_1$ and $G_2$ are as shown in Figure 8d,e. From Figure 8d,e, the adjacency matrices of $G_1$ and $G_2$ are not equal, but the matrix of $G_2$ may match that of $G_1$ by re-labeling the vertices. The number of combinations for the re-labeling is $N!$

In the graph $G_2$ in Figure 8b, the graph shown in Figure 8c can be obtained by re-labeling the graph $G_2$ as $\left({\mathrm{v}}_1^{″},{\mathrm{v}}_2^{″},{\mathrm{v}}_3^{″},{\mathrm{v}}_4^{″}\right)=\left({\mathrm{v}}_3^{\prime },{\mathrm{v}}_4^{\prime },{\mathrm{v}}_1^{\prime },{\mathrm{v}}_2^{\prime }\right)$. The adjacency matrix of the graph in Figure 8c is shown in Figure 8f, and it matches the adjacency matrix of graph $G_1$ shown in Figure 8d. Therefore, graphs $G_1$ and $G_2$ are isomorphic, and one of them is eliminated to avoid duplicates. In this way, we re-label the vertices exhaustively to consider the isomorphism.

In summary, the flowchart for the enumeration of link connection graphs is as shown in Figure 9. The results are as shown in the fourth rows of Table 1, Table 2 and Table 3.

4. Pair Arrangement

In the previous section, the DoF of the pairs are not considered. Next, the pair arrangement graphs are enumerated to complete the enumeration of the spatial prime structures.

4.1. Enumeration of the Pair Arrangement Graphs

To enumerate the pair arrangement graphs, the weights to the edges of the link connection graphs are set. The weights on the pair arrangement graphs correspond to the DoF of the pairs on the structures. The number of the combinations of setting weights on edges is

$$\left(\begin{array}{c}{J}_1+J_2+J_3\\ J_1\end{array}\right)\times \left(\begin{array}{c}{J}_2+J_3\\ J_2\end{array}\right).$$

(8)

For example, we consider enumerating the pair arrangement graphs corresponding to the link connection graph in Figure 10a. When the numbers of 1-, 2-, and 3-DoF pairs $\left(J_1, J_2, J_3\right)=\left(\mathrm{1,1},1\right)$, the number of the combination of setting weights is

$$\left(\begin{array}{c}{J}_1+J_2+J_3\\ J_1\end{array}\right)\times \left(\begin{array}{c}{J}_2+J_3\\ J_2\end{array}\right)=\left(\begin{array}{c}3\\ 1\end{array}\right)\times \left(\begin{array}{c}2\\ 1\end{array}\right)=6,$$

(9)

and when they are exhaustively enumerated, the pair arrangement graphs shown in Figure 10b–g can be obtained. For example, the graph in Figure 10b represents the prime structure in Figure 10h.

4.2. Isomorphism Test

After enumerating pair arrangement graphs, isomorphic graphs are eliminated. We call this elimination the “isomorphism test”. In the isomorphism test, the different adjacency matrices in Section 3 are used. The elements of the adjacency matrices in this section are the weights of the edges, which correspond to the DoF of pairs. If the vertices ${\mathrm{n}}_{\mathrm{i}}$ and ${\mathrm{n}}_{\mathrm{j}}$ are connected by a $k$-DoF pair, the $\left(i, j\right)$ element of the matrix is $k$. For example, the adjacency matrices of the pair arrangement graphs in Figure 10b,c are

$$\left[\begin{array}{ccc}0& 1& 2\\ 1& 0& 3\\ 2& 3& 0\end{array}\right], \left[\begin{array}{ccc}0& 1& 3\\ 1& 0& 2\\ 3& 2& 0\end{array}\right].$$

(10)

We re-label the vertices for the isomorphism test in the same way as in Section 3. From Equation (10), the adjacency matrices of Figure 10b,c are not the same, but if we re-label the vertices of Figure 10c as $\left({\mathrm{v}}_1^{\prime }, {\mathrm{v}}_2^{\prime }, {\mathrm{v}}_3^{\prime }\right)=\left({\mathrm{v}}_2,{\mathrm{v}}_1,{\mathrm{v}}_3\right)$, the adjacency matrix becomes the same as that in Figure 10b, and they are isomorphic. In this way, the vertices are exhaustively re-labeled, and if the adjacency matrix matches that of another graph, we eliminate one.

4.3. Primality Test

Among the structures corresponding to the pair arrangement graphs enumerated in the previous subsection, there can exist non-prime ones. In other words, there can exist the structures which contain subchains whose DoF are zero or less. Therefore, we conduct the detection of the rigid subchains to eliminate the non-prime structures. This procedure is called the “primality test”.

In the primality test, the method of the enumeration for the planar prime structures [17] is expanded to the spatial prime structures. First, the structure is decomposed to enumerate the subchains, and the DoF of the subchains can then be calculated.

In the decomposition of the structures to the subchains, we use “dual graphs”. When a graph $G$ is given, the dual graph of $G$ is a graph that represents the relationship of the “faces” and the edges of the original graph $G$. A face of a graph is the area that is surrounded by the edges. For example, about the graph in Figure 11a, there is a face ${\mathrm{f}}_1$ surrounded by edges ${\mathrm{e}}_1,{\mathrm{e}}_2,{\mathrm{e}}_5,{\mathrm{e}}_6$, and a face ${\mathrm{f}}_2$ surrounded by edges ${\mathrm{e}}_3,{\mathrm{e}}_4,{\mathrm{e}}_5,{\mathrm{e}}_6$, as shown in Figure 11b. Also, the infinite area ${\mathrm{f}}_3$ that exists outside the graph is also considered as a face, i.e., an “infinite face”.

In the dual graphs, the vertices correspond to the faces of the original graph. The edges in the dual graphs correspond to the edges which separate the faces of original graphs. For example, the dual graph of the graph in Figure 11a is shown in Figure 11c. In Figure 11a,b, there are two edges ${\mathrm{e}}_5$ and ${\mathrm{e}}_6$ between the faces ${\mathrm{f}}_1$ and ${\mathrm{f}}_2$. Therefore, in the dual graph shown in Figure 11c, the vertices ${\mathrm{f}}_1$ and ${\mathrm{f}}_2$ are connected by two edges ${\mathrm{e}}_5$ and ${\mathrm{e}}_6$. As a property of dual graphs, for any graph G, the dual graph of the dual graph coincides with the original graph G.

By deleting all edges between two vertices in the dual graph, all edges between the two faces are deleted, and the two faces are combined in the original graph. Therefore, we separate the vertices in the dual graphs into some groups (Morlin et al. [17] call the groups “flats”) and delete the edges in the dual graphs between the vertices in the same group. After this, the pair arrangement graphs of all subchains can be obtained by considering the dual graphs of the decomposed dual graph. Finally, the DoF of the subchain can be calculated using Grübler’s equation. If the DoF is zero or less, the original chain includes a rigid subchain; therefore, it cannot be a prime structure. Therefore, such chains can be eliminated. If the DoF of all subchains are more than zero, the original chain is identified as a prime structure.

It was not necessary to consider the DoF of pairs in the primality test of planar prime structures. However, there are some types of pair DoF for spatial prime structures, and the information of pair DoF are necessary in the calculation of the DoF of subchains. Therefore, we must keep the information of the weights on edges when we obtain the dual graph from the original graph and the dual graph.

For example, let the weights of the edges of the graph in Figure 11a be $\left({\mathrm{e}}_1,{\mathrm{e}}_2,{\mathrm{e}}_3,{\mathrm{e}}_4,{\mathrm{e}}_5,{\mathrm{e}}_6\right)=\left(1, 1, 3, 3, 2, 2\right)$, and we consider the decomposition of the dual graph shown in Figure 11c. There are three vertices, ${\mathrm{f}}_1, {\mathrm{f}}_2$, and ${\mathrm{f}}_3$, in the dual graph, so there are five combinations for the grouping, as shown in Figure 12. Edges in the dual graph are deleted in the above-mentioned process, and the dual graphs of the dual graphs should be considered to obtain the subchains. The subchains are as shown in Figure 12. Then, the DoF of each subchain can be calculated, except for the cases ($\left[{\mathrm{f}}_1\right],\left[{\mathrm{f}}_2\right],\left[{\mathrm{f}}_3\right]$) and $\left(\left[{\mathrm{f}}_1,{\mathrm{f}}_2,{\mathrm{f}}_3\right]\right)$, which are the original chain and the empty set, respectively. In this example, the DoF of the subchain $\left(\left[{\mathrm{f}}_2,{\mathrm{f}}_3\right],\left[{\mathrm{f}}_1\right]\right)$ is zero, so the original pair arrangement graph is not prime and inappropriate.

If the original pair arrangement graph is “a nonplanar graph”, i.e., if there is an edge intersection in the graph, we cannot use the dual graphs and we have to use the matroid theory [17]. But all link connection graphs of the spatial prime structures with $N\le 5$ are planar, as shown in Table 1, Table 2 and Table 3, so dual graphs can be used.

4.4. Consideration of Idle DoF

Some spatial structures and spatial mechanisms have idle DoF. For example, the RSS chain shown in Figure 13a has 1 DoF according to Grübler’s equation. This DoF corresponds to the rotation of link ${\mathrm{S}}_1{\mathrm{S}}_2$ about the segment ${\mathrm{S}}_1{\mathrm{S}}_2$, but the position and orientation of the chain are determined regardless of the DoF. Therefore, we call such DoF an “idle DoF”, and we also include such chains in prime structures. Such idle DoF do not exist in the planar prime structures or planar mechanisms, so idle DoF are the inherent characteristics for the spatial prime structures and spatial mechanisms. Note that the link ${\mathrm{S}}_1{\mathrm{S}}_2$ in Figure 13a cannot be the fixed link of the RSS structure.

As mentioned above, when there is a binary link in the kinematic chain such that both ends are spherical pairs (we call such link as “S-S link”), there is a rotational DoF about this link. Therefore, about a prime structure with 1 idle DoF, as shown in Figure 13a, if we change one of the 3-DoF pairs (spherical pairs) of the S-S link to a 2-DoF pair (cylindrical pair), we obtain a prime structure with 0 idle DoF, as shown in Figure 13b. Conversely, if there is a prime structure with 0 idle DoF, which has a binary link with 2-DoF pair and a 3-DoF pair, as shown in Figure 13d, if we replace the 2-DoF pair with a 3-DoF pair, we can obtain a new prime structure with 1 idle DoF, as shown in Figure 13c. Similarly, about a prime structure with $k$ DoF (where $k$ is a positive integer), if there is a binary link with a 2-DoF pair and 3-DoF pair, a new prime structure with $k+1$ idle DoF can be obtained by replacing the 2-DoF pair with a 3-DoF pair. In this way, the prime structures with idle DoF can be exhaustively enumerated. After the enumeration, the isomorphism test is conducted to eliminate inappropriate ones.

5. Results and Discussion

The results of the exhaustive enumeration of the spatial prime structures with 3–5 links are shown in Table 1, Table 2 and Table 3. As a result, 3, 13, and 97 types of three-, four-, and five-link spatial prime structures are obtained, respectively. There are 1, 0, and 1 types of three-, four-, and five-link planar prime structures [2]; therefore, many more types of spatial mechanisms are available compared to planar mechanisms.

In this paper, we only considered the DoF of the pairs and did not consider the type of pairs (e.g., revolute pair and prismatic pair for 1 DoF pair). Therefore, more types of spatial mechanisms are available when the types of pairs are considered. In the planar prime structures, there are none including other planar prime structures. But there exist spatial prime structures that include other spatial prime structures with idle DoF. For example, the ${\mathrm{j}}_1{\mathrm{j}}_2{\mathrm{j}}_3$ closed loop in the prime structure in Figure 14b matches the prime structure with 1 idle DoF in Figure 14a. But the rotation around two spherical pairs, ${\mathrm{j}}_1$ and ${\mathrm{j}}_2$, in the structure in Figure 14b is not the idle DoF, and it affects the motion of the pair ${\mathrm{j}}_5$.

Therefore, the structure in Figure 14b cannot be analyzed simply by using the analysis method for structure in Figure 14a, and we have to develop the analysis method for the structure Figure 14b. For this reason, structures such as Figure 14b are also treated as prime structures.

6. Kinematic Analysis of Spatial Mechanisms Using Spatial Prime Structures

Because various spatial prime structures with up to five links can be exhaustively enumerated, we will be able to analyze the kinematics of spatial closed-loop mechanisms based on the configuration analysis of each spatial prime structure in the same way as conventional planar prime structures, as mentioned in Section 1. The authors will report the configuration analysis of the enumerated spatial prime structures in detail in the next paper. Here, the authors explain how to apply the enumerated spatial prime structures to the kinematics analysis of spatial closed-loop mechanisms under the assumption that the configuration analyses of the spatial prime structures have been completed.

The first example is a kinematics analysis of spatial four-bar link mechanisms with 1 DoF [27]. After giving the motion of the input link, the position and posture of the moving pair on the input link are calculated. Then, a three-link spatial prime structure composed of two moving links and an assumed link connecting the fixed pair of the moving link and the moving pair on the input link can be found. Therefore, the motions of two moving links can be analyzed via configuration analysis of the three-link spatial prime structure. For example, Figure 15a shows an RCSR spatial four-bar link mechanism. After rotation of the crank link RC, the position and posture of the cylindrical pair are calculated. Then, the PSC three-link spatial prime structure, shown in Figure 15b,c can be found. The motions of CS and SP links can then be calculated. In this way, all kinds of spatial four-bar mechanisms with 1 DoF, even including idle DoF, can be analyzed.

The second example is an inverse kinematics analysis of a six-RSS parallel manipulator with 6 DoF, shown in Figure 16a [28]. After giving the position and posture of its moving platform, the positions of all spherical pairs, S_B,i, on the moving platform can easily be calculated. Six-RSS three-link spatial prime structures, each of which is composed of R_i-S_A,i, S_A,i-S_B,i, and S_B,i-R_i links, shown in Figure 16b,c can be analyzed in terms of their configurations. The input rotations of R_i-S_A,i can be obtained.

The third example is the kinematics analysis of the mechanism of the MacPherson strut suspension [29,30], shown in Figure 17a. When the inputs of the pairs ${\mathrm{P}}_{\mathrm{i}\mathrm{n}}$ and ${\mathrm{R}}_{\mathrm{i}\mathrm{n}}$, which are the input length of the steering and the angle of the radius rod, are decided, the positions of the spherical pairs ${\mathrm{S}}_2$ and ${\mathrm{S}}_4$ can be calculated. Then, the pairs ${\mathrm{S}}_2$ and ${\mathrm{S}}_4$ can be regarded as fixed pairs, and the remaining chain is the PSS-SS chain, as shown in Figure 17b. The PSS-SS chain is a four-link prime structure in Table 1, as shown in Figure 17c.

Therefore, we can analyze the kinematics of MacPherson strut suspension by applying configuration analysis of the PSS-SS four-link spatial prime structure.

If it becomes possible to analyze the motion of spatial mechanisms using prime structures, various applications become possible.

For example, it can be applied to the singularity analysis of spatial mechanisms. If the configurations of the prime structures are calculated algebraically, the motions of many mechanisms can be calculated algebraically, and the Jacobian matrices of the mechanisms can also be derived algebraically. Therefore, singularity analysis can be easily performed by searching mechanism configuration, which makes the determinant of the Jacobian matrix zero [31].

For another example, it can also be applied to the analysis of compliant mechanisms. In the motion analysis of compliant mechanisms, the compliant mechanisms can be simplified to pseudo-rigid link mechanisms by replacing thinner parts with kinematic pairs [32] or simplifying the large deformations of elastic beams to the rotation of two revolute pairs [33]. Therefore, if it becomes possible to analyze various spatial mechanisms using prime structures, the motion analysis of spatial compliant mechanisms becomes possible.

As described above, the results of this research are expected to have a wide range of applications.

7. Conclusions and Future Works

Aiming to obtain a strong tool for kinematics analyses of spatial closed-loop mechanisms, the exhaustive enumeration of spatial prime structures, which are minimum spatial link chains, to analyze kinematics in order is theoretically conducted. The obtained results are summarized as follows.

(1)

In order to consider the DoF of pairs, the enumeration method using two types of graphs—link connection graphs and pair arrangement graphs—are proposed.

(2)

The prime structures with idle DoF, which are given as rotation of a spherical–spherical binary link and do not exist in planar prime structure, are also enumerated.

(3)

As a result of the exhaustive enumeration, 3, 13, and 97 types of three-, four-, and five-link spatial prime structures are obtained, respectively.

(4)

Several examples of kinematics analysis based on configuration analyses of the enumerated spatial prime structures are shown.

As mentioned in Section 6, the authors will report the configuration analyses for the various spatial prime structures enumerated in this paper and will show several examples of kinematics analyses of the spatial closed-loop link mechanisms.

In this paper, we used the exhaustive method, especially in the generation of graphs and the isomorphism test. However, it is expected that there are many more types of spatial prime structures with six and more links. If we apply group theory [21] and other methods presented in the research to planar prime structures, it will be possible to achieve the enumeration of spatial prime structures with six and more links efficiently.