The classification method of spherical single-loop mechanisms

Mechanism and Machine Theory 51 (2012) 46–57

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Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmt

The classification method of spherical single-loop mechanisms Dengfeng Zhao ⁎, Guoying Zeng, Yubin Lu Ministry of Education Key Laboratory of Testing Technology for Manufacturing Process, Southwest university of science and technology, Sichuan 621010, PR China

a r t i c l e

i n f o

Article history: Received 27 August 2010 Received in revised form 15 December 2011 Accepted 2 January 2012 Available online 28 January 2012 Keywords: Singularity Complex Spherical mechanisms Classification

a b s t r a c t Based on the singularity of constraint equations and the division of dimension space of links, the classification method of spherical single-loop mechanisms was studied. Firstly, the classification conditions were derived from the singularity of constraint equations. Secondly, the complex division process of dimension space of links was investigated. Finally, the classification method was validated by using 4-, 5- and 6-link spherical mechanisms. It is shown that the classification method is comprehensive and all the classification information is included in the complex division results, and this method is appropriate for the classification of complicated mechanisms by computers. © 2012 Elsevier Ltd. All rights reserved.

1. Introduction For mechanisms with different dimension types, their topologies of motility characteristics like work-space, singularity configuration and motion decoupling are totally different. Presently, accurate numerical results can be obtained for mechanisms with fixed dimensions through numerical computation methods. However, studies on the global structure of motility characteristics and its evolution are few, and it is still hard to analyze these problems by using the computer aided tools, which heavily restrains people's understanding on the analysis and synthesis of mechanisms. The global structure analyses of mechanisms should be an important part of future computer aided software for the analysis and synthesis of mechanisms. Therefore, accurate classification and expression of complicated mechanisms is necessary. Study on the motility characteristics of spherical mechanisms, which have particular motility characteristics, has important significance for accurately ascertain the characteristics of other mechanisms’ global structure and its evolution with structural parameters. The well-known Grashof criterion used in 4-link planar mechanisms is the most classic paradigm. Since then, many scholars have expanded its application into spherical mechanisms and other complicated mechanisms. The classification of spherical 4-link mechanisms is relatively well established. Cervantes-Sanchez et al. [1] extended the Grashof criterion into 4-link spherical mechanisms and the classification scheme was given. Hang and Wang [2] studied the decoupling conditions of spherical 4-link mechanisms. Kohli et al. [3] proposed a rotatability criterion for spherical 5-bar linkages. Liu et al. [4] investigated the rotatability rule of spherical N-bar chains. The classification of planar mechanisms can provide important reference to that of spherical mechanisms. Ting et al. [5] gave the rotatability law for N-bar kinematic chains. References of [6–8] studied the workspace, solution space and singularity of planar 5-link mechanisms. Zhao [9] explored the classification method of single-loop planar mechanisms by using the convex hull division method.

⁎ Corresponding author. E-mail address: [email protected] (D. Zhao). 0094-114X/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2012.01.002

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This paper includes three main parts. First, the classification conditions of spherical single-loop mechanisms are proposed based on the singularity of constraint equations. Second, the classification method based on the division of dimension space of links is studied. Third, the classification method is validated by using 4-, 5- and 6-link spherical mechanisms. 2. The constraint equations A single-loop N-link spherical mechanism is shown in Fig. 1. The N links [l1, l2,…, lN] are connected with N revolute joints. The axis of each revolute joint intersects at the spherical center. The angles of links are φ = [φ1, φ2,…, φN], which are variables of dimension space of links and remain constant in motion. A unit vector in the Cartesian coordinate format, ri = [xi, yi, zi], is used to express the position of revolute joint ri. r = [r1,r2,…,rn] T are variables of motion space of mechanisms and vary with time. These variables satisfied the following constraint equations,


x2i þ y2i þ z2i ¼ 1 xi xiþ1 þ yi yiþ1 þ zi ziþ1 ¼ cosϕi

i∈ð1; 2; …; N Þ N þ 1→1

ð1Þ

In Eq. (1), the first formula is the module of the vector of revolute joints and the second one is the dot product of the vector of neighboring revolute joints. There are many different motion parameter sets and the formats of constraint equations are also different. But only the classification process not the classification results is influenced. The format of Eq. (1) has obvious symmetry and the classification process will be easy. The freedom of mechanisms F (including the holistic freedom) is the difference between the number of motion parameters 3N and the amount of constraint equations 2N. The solution set of constraint equations in the motion space of mechanisms is the F-dimensional manifold, which is termed as solution space. The topology of the solution space may mutate with the change of link dimensions. Accordingly, the topology of mechanism performances will also mutate. 3. The singularity of the constraint equations According to the singularity theories of multi-variables [10,11], a singular Jacobi matrix is a necessary condition for the topology of a solution space to mutate. The Jacobi matrix of the constraint equations is as follows, 2

x1 6 0 6 6 ⋮ 6 6 0 J¼6 6 x2 6 6 0 6 4 ⋮ xN

y1 0 ⋮ 0 y2 0 ⋮ yN

z1 0 ⋮ 0 z2 0 ⋮ zN

0 x2 ⋮ 0 x1 x3 ⋮ 0

0 y2 ⋮ 0 y1 y3 ⋮ 0

0 z2 ⋮ 0 z1 z3 ⋮ 0

⋯ ⋯ ⋱ ⋯ ⋯ ⋯ ⋱ ⋯

0 0 ⋮ xN 0 0 ⋮ x1

0 0 ⋮ yN 0 0 ⋮ y1

3 0 07 7 ⋮ 7 7 zN 7 7 07 7 07 7 ⋮ 5 z1

ð2Þ

In the preceding matrix, the 1 ~ N and N + 1 ~ 2N rows are the partial derivative of the first formula and second formula in Eq. (1), respectively. Since J is a 2N × 3N non-square matrix, the singular condition of the J is that the rank is less than 2N. Namely, a 2N × 2N square matrix is randomly taken from the J, its determinant is always zero. Certainly, if N independent square matrix is taken and its determinant is zero, then all the square matrix determinants are zero.

Fig. 1. A single-loop N-link spherical mechanism.

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The Jacobi matrix has an obvious feature, i.e. in the (3k-2) ~ 3k (k is an integer) columns, only three rows are nonzero elements. These nonzero elements are the coordinate components of three adjacent revolute joint vectors. If the three columns are included in the 2N × 2N square matrix and the determinant of the nonzero 3 × 3 matrix equals to zero, the determinant of the 2N × 2N matrix also equals to zero. This means that the three revolute joint vectors are coplanar. By analogy, the singular condition of the Jacobi matrix is that all the revolute joint vectors are coplanar. Thus, the angles of links, φ, satisfy the following equation, N X

 ϕi −2kπ ¼ 0

ð3Þ

k∈Z

i¼1

which is the mutational condition of topology of the solution space. When the dimensions of links exceed the mutational condition, the mechanism type will transfer and the topology characteristics of the solution space will mutate. Accordingly, the global structure of all mechanism performances will mutate. As an essential basis of the classification of spherical mechanisms, the mutational condition, Eq. (3), can be proved to do not relate with the specific forms of the constraint equations. In Eq. (3), many “±” symbols and various k values can form a lot of mutational conditions. Because all the mutational conditions are linear equations, the dimension space of links is divided into many complexes or simplexes. The topology of mechanism performances in different areas will be different. 4. Additive restrictions for the classification conditions A link with two revolute joints is shown in Fig. 2a. The link's angle is φ. But the revolute joint rA is equal to r′A and the revolute joint rB is also equal to r′B. Hence φ, π–φ or 2π–φ expresses the same link. The relative position of the two revolute joints on the link will not change. Accordingly, the range of every link's angle should be restricted in the following range, 0 b ϕi ≤

π 2

ð4Þ

i ∈ ð1; 2; ⋯; NÞ

Under the constraint of Eq. (4), the overlap of mechanism types caused by the range of angle can be avoided. In a mechanism, if any link's angle φ is replaced by 2π–φ, the constraint equations, Eq. (1), will not change. As shown in Fig. 2b, if a revolute joint rk is replaced by −rk and the corresponding angles φk, φk − 1 are replaced by π–φk, π–φk − 1, respectively, the constraint equations, Eq. (1), will also not change. Certainly, the mechanism type will be same. Because of this limit condition, spherical mechanisms can be divided into two classes, i.e., one is the general closed mechanism, as shown in Fig. 1, which is designated as 0-close mechanism; another is designated as π-close mechanism, as shown in Fig. 2c. In a general mechanism, if the number of links whose angles exceed π/2 is an even or odd, the mechanism is an equivalent 0-close mechanism or π-close mechanism, respectively. The mutational condition of π-close mechanisms is N X

 ϕi −ð2k þ 1Þπ ¼ 0

k∈Z

ð5Þ

i¼1

Eqs. (3) and (5) have a very important feature, viz., when the order of links is arbitrarily changed, Eqs. (3) and (5) remain unchanged. But Eqs. (1) and (2) do not have this feature. Namely, when the order of links is changed, the metric characters of the solution space will be changed, but the topology of the solution space will remain unchanged, the mechanism type and its evolvement will also not change. Accordingly, all angles of links should satisfy the following inequation, ϕ1 ≤ ϕ2 ≤ ⋯ ≤ ϕN

ð6Þ

Eq. (6) is the sort condition of links. Under the constraint of Eq. (6), the overlap of mechanism types induced by the order of links will be avoided. In the mutational conditions, Eqs. (3) and (5), have N “±” symbols, but the number of independent equations should be 2 N − 1, not 2 N. The reason is that Eqs. (3) and (5) multiplied by −1 makes a half of the total equations becoming dependent. The independent equations could be divided into several groups according to the total number of “+” symbol, N +, or the total number of “−” symbol, N − (N − + N + = N). Within the group N +, the range of k value should satisfy the following condition,


−

þ

−N b4kbN −N− b4k−2bNþ

0− close mechanism π− close mechanism

Under the constraint of Eq. (6), the number of independent equations will be reduced.

ð7Þ

b

c

D. Zhao et al. / Mechanism and Machine Theory 51 (2012) 46–57

a

Fig. 2. The restriction of link angles and the π-close spherical mechanism.

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5. The expression of classification factors The preceding analyses show that there are many classification conditions and the division situation of dimension space of links is also very complicated. The primary problem is to accurately express the division conditions, the geometry objects and the correlations between them. (1). The division conditions All division conditions in Eqs. (3)–(6) are linear equations. The matrix format is the simplest one and is named as condition matrix, designated as T. A row of T represents a division condition. The 1 ~ N columns correspond to the coefficients of the angles φ and the elements are 0, 1 or − 1. The N + 1 column is the constant term in the division condition and the elements are 2kπ or (2k + 1)π for 0-close mechanisms or π-close mechanisms, respectively. (2). The geometry objects For N-link spherical mechanism, the division results are many complexes in the N-dimensional dimension space of links. The universal set is expressed as S. The set of m-dimensional geometric objects is expressed as s(m), and any geometric object in s(m) is expressed as s(m, n), where n is the serial number. The boundary geometric objects of s(m, n) are expressed as ∂s(m, n). The relational matrixes express the topology structure of geometric objects. If there are I elements in s(k − 1) and J elements in s(k), then relational matrix L k is a I × J matrix and the elements are

k Li;j

8 <1 ¼ −1 : 0

sðk−1; iÞ∈ þ ∂sðk; jÞ sðk−1; iÞ∈−∂sðk; jÞ sðk−1; iÞ∉∂sðk; jÞ

ð8Þ

There are N relational matrixes which provide completed description of the topology of S. Additionally, a relational matrixes L 0 is appended to represent the position coordinates of each vertex (0-dimensional geometric objects). The coordinate components are the link's angles. (3). The relation between division conditions and geometry objects For any interior point p in the geometric object s(k, j) and any division condition ti, there is a condition value vi(p, tj). The relation between division conditions and geometry objects is named as object–condition matrix. If there is I division conditions and J geometric objects in s(k), the object–condition matrix K k is a I × J matrix and the elements are   8 > 1 vi p; t j > 0 ∀p ∈ sðk; jÞ > > <   k K ij ¼ −1 vi p; t j b 0 ∀p ∈ sðk; jÞ >   > > :0 vi p; t j ¼ 0 ∀p ∈ sðk; jÞ

ð9Þ

Eq. (9) does not require condition values of all the interior points but those of vertexes of s(k, j) to get the elements k of Kij. A simple example in 3-dimensional space is shown in Fig. 3. There are 4 vertexes (0-simplex), 5 lines (1-simplex), 2 faces (2-simplex) and 4 division conditions. The condition matrix T is 2

1 0 60 1 6 T¼4 0 0 1 −1

0 0 1 0

3 0 07 7 05 0

ð10Þ

The relational matrixes L 0, L 1, L 2 are 2

1 0 6 0 1 0 6 L ¼4 0 0 1=2 1=2

3 0 07 7 15 0

2

−1 6 0 6 1 L ¼6 6 0 4 1 0

3 0 1 0 1 −1 07 7 0 1 −1 7 7 0 0 −1 5 −1 0 1

2

L ¼



1 0

0 −1 1 0 1 1 0 1

 ð11Þ

D. Zhao et al. / Mechanism and Machine Theory 51 (2012) 46–57

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P3(0,0,1) L2

L3

L1

S2

S1 L4 P1(1,0,0)

P2(0,1,0)

L5 P4(1/2,1/2,0)

Fig. 3. A simple example of complex.

The elements in L 0 are the coordinate components of 4 vertexes. The object–condition matrixes K 0, K 1, K 2 are 2

1 60 0 6 K ¼4 0 1

0 1 0 1

3 0 1 0 −1 7 7 1 0 5 0 0

2

1 60 6 1 K ¼6 61 41 1

0 1 1 1 1

1 1 1 0 0

3 1 −1 7 7 07 7 15 −1

2

K ¼



1 1

1 1

1 1

1 −1

 ð12Þ

All information of mechanism classification is included in the aforementioned three types of matrixes. Those are the bases of numerical computation programs. In the division process, the division state of dimension space of links is always described by the three types of matrixes. 6. The division of dimension space of links The division procedure of the dimension space of links consists of three steps. (1) An N-dimensional cube is created according to the range of every link's angle [Eq. (4)], named as a basic cube. It has 2 N vertexes and all the coordinate components can only be 0 or π/2. There are many complexes (or simplex) in the basic cube. The number of K-dimensional complexes C(N, K) is

C ðN; K Þ ¼

N! N−K 2 K!ðN−K Þ!

K∈ð0; 1; ⋯; N Þ

ð13Þ

In the basic cube, any K-dimensional complex is a K-dimensional cube and has 2 K vertexes. In the coordinate components of vertexes, N − K numbers of coordinate components are always same. (2) The sort condition of Eq. (6) is used to divide the basic cube and an N-dimensional simplex is obtained, named as a basic simplex. It has N + 1 vertexes and all the coordinate components also can only be 0 or π/2. If the right order of vertexes is used, the L 0 can form a triangular matrix. There are many simplexes in the basic simplex. The number of K-dimensional simplexes S(N, K) is

SðN; K Þ ¼

ðN þ 1Þ! ðK þ 1Þ!ðN−K Þ!

K∈ð0; 1; ⋯; NÞ

ð14Þ

(3) The mutational conditions of Eqs. (3) and (5) are used to divide the basic simplex and then many simplex or complex regions are produced. Firstly, regions which do not satisfy the assembly condition, which is the condition of links to form a closed loop, are eliminated. Subsequently, the type of mechanism and the evolutionary relations among mechanism types are studied. If all interior points in a simplex or a complex do not satisfy the assembly condition, the mechanism constraint equations do not have real solutions and the topology of the solution space is a null set. For planar mechanisms, the length of the longest link should be longer than the sum of the length of all the other links. For spherical mechanisms, if one vertex in a simplex or a complex does not satisfy the assembly condition, all interior points in this simplex also do not satisfy the assembly condition.

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Although the dimension space of links is a high dimensional space, the division operations of complexes and the topology relation among complexes are similar to those of a common 3-dimensional space. The division operation consists of two steps. The flow chart of the division operation is shown in Fig. 4. (1) First, obtaining the condition matrix T, the relational matrixes L 0, L 1,…, L N and the object–condition matrixes K 0, K 1,…, K N of divided geometric objects and the current division condition t. Subsequently, calculating the condition value v(L 0, t) of all vertexes (0-dimensional geometric objects) and updating the condition matrix T and the object–condition matrix K 0 as h

T

T ;t

i

T T

update

→T h   i update K 0 ; sign v L0 ; t →K 0

ð15Þ

where sign(x) is the signum function. The vertexes are divided into three groups as per the condition value being greater than or less than or equal to 0. (2) From 1-dimensional to N-dimensional geometric objects, the grouping and division operation is conducted on every geometric object. Simultaneously, the relational matrixes and the object–condition matrixes must be updated. There exist two situations, i.e., (1) in a geometric object, if the condition values of all boundary geometric objects are 0, not less than 0, or not larger than 0, the division operation is not required and the condition values are correspondingly updated as 0, 1 or − 1; (2) in a geometric object, if the condition values of some boundary geometric objects are larger than 0 and those of others are less than 0, then the division operation must be performed, and the geometric object is divided into two and a mutual boundary is added. Simultaneously, the relational matrixes and the object–condition matrixes related with the geometric objects are updated.

Fig. 4. The flow chart of the division operation.

D. Zhao et al. / Mechanism and Machine Theory 51 (2012) 46–57

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7. Verification examples Based on the foregoing analyses, a computer program which can classify various spherical single-loop mechanisms is written. To save the space, only the classification results of the 3-, 4-, 5- and 6-link spherical mechanisms are presented here. Although three links can only form a spherical triangle not a mechanism, but its dimension space of links is a 3-dimensional space and the division situation can be intuitively revealed, which is important to understand the classification method of this study. The division results of 4-, 5- and 6-link 0-close and π-close spherical mechanisms are summarized in Table 1 and Table 2, respectively. The transition among the types of π-close spherical mechanisms is the same as that of 0-close spherical mechanisms, as shown in Fig. 6. Over the years, the word “crank” normally restricts to a link which couples on the fixed link, which perhaps results in controversies during the classification of mechanisms. Thus, an extended definition of “crank” is introduced, i.e., a link which can fully rotate relatively to its two adjacent links is defined as “crank”.

Table 1 The division results of 4-, 5- and 6-link 0-close spherical mechanisms.

4-link

5-link

6-link

Vertex 2 1 6 0 6 6 0 6 6 0 6 6 0 6 4 1=3 0 2 1 6 0 6 6 0 6 6 0 6 6 0 6 6 0 6 6 1=2 6 6 2=3 6 6 3=4 6 6 4=5 6 6 0 6 6 0 6 4 1=4 1=2 2 1 6 0 6 6 0 6 6 0 6 6 0 6 6 0 6 6 0 6 6 1=3 6 6 6 1=2 6 3=5 6 6 2=3 6 6 0 6 6 0 6 6 0 6 6 0 6 6 0 6 6 0 6 6 1=5 6 6 0 6 6 1=3 6 6 0 6 6 1=3 6 6 1=2 6 6 1=2 6 4 1=4 1=3

coordinates 1 1 0 0 0 1=3 1=2

1 1 1 0 0 1=3 1=2

3 1 17 7 17 7π 17 72 07 7 15 1

1 1 0 0 0 0 1=2 2=3 3=4 4=5 1=3 0 1=4 1=2

1 1 1 0 0 0 1 2=3 3=4 4=5 1=3 1=2 1=4 1=2

1 1 1 1 0 0 1 1 3=4 4=5 1=3 1=2 1=4 1=2

3 1 1 7 7 1 7 7 1 7 7 1 7 7 0 7 7 1 7 7π 2 1 7 7 1 7 7 4=5 7 7 1 7 7 1 7 7 1 5 1

1 1 0 0 0 0 0 1=3 1=2 3=5 2=3 1=2 2=3 3=4 4=5 0 0 1=5 1=4 1=3 1=2 2=3 1=2 1=2 1=4 1=3

1 1 1 0 0 0 0 1=3 1=2 3=5 2=3 1=2 2=3 3=4 4=5 1=3 0 1=5 1=4 1=3 1=2 2=3 2=3 1=2 1=2 1=3

1 1 1 1 0 0 0 1 1=2 3=5 2=3 1 2=3 3=4 4=5 1=3 1=2 1=5 1=4 1=3 1=2 2=3 2=3 3=4 1=2 2=3

1 1 1 1 1 0 0 1 1 3=5 2=3 1 1 3=4 4=5 1=3 1=2 1=5 1=4 1=3 1=2 2=3 2=3 3=4 1=2 2=3

3 1 1 7 7 1 7 7 1 7 7 1 7 7 1 7 7 0 7 7 1 7 7 1 7 7 1 7 7 2=3 7 7 1 7 7 1 7 7π 2 1 7 7 4=5 7 7 1 7 7 1 7 7 1 7 7 1 7 7 1 7 7 1 7 7 1 7 7 1 7 7 1 7 7 1 5 1

Simplexes (or complexes)

Mechanism characteristics

v1b3,4,5,6,7> v2b1,3,5,6,7> v3b1,2,3,5,7> s1b3,5,6,7> s2b1,3,5,7>

Cannot be assembled Crank: none Crank: l1 v1 ↔ v2, φ1 + φ2 + φ3 = φ4 v2 ↔ v3 , φ 1 + φ 4 = φ 2 + φ 3

v1b4,5,6,11,12,13> v2b4,6,11,12,13,14> v3b2,4,6,11,12,14> v4b2,4,6,7,12,14> v5b2,4,6,7,8,14> v6b2,6,7,8,9,10,14> V7b2,3,4,6,7,12> v8b1,2,7,8,9,10>

Cannot be assembled Crank: none Crank: l1 Crank: l1,l2 Crank: (l1,l2) Crank: l1,l2,l3 Crank: l1,l2,l3,l4 Crank: (li,lj)

v1b5,6,7,16,17,18,19> v2b5,7,16,17,18,19,20> v3b5,7,16,17,19,20,21> v4b5,7,16,17,20,21,25> v5b5,7,17,20,21,25,26> v6b5,7,9,20,21,25,26> v7b7,9,10,11,20,21,22,23,24,25,26> v8b3,5,7,16,17,21,25> v9b3,5,7,17,21,25,26> v10b3,5,7,9,21,25,26> v11b3,7,9,11,21,22,23,24,25,26> v12b3,5,7,12,17,21,26> v13b3,5,7,9,12,21,26> v14b3,7,9,11,12,21,22,24,26> v15b3,5,7,8,12,17,26> v16b3,5,7,8,9,12,26> v17b3,7,8,9,11,12,24,26> v18b3,5,7,9,12,13,21> v19b3,7,9,11,12,13,21,22> v20b3,7,11,12,13,14,15,21,22> v21b3,4,5,7,8,12,17> v22b1,9,10,11,22,23,24> v23b1,3,9,11,22,23,24> v24b1,3,9,11,12,22,24> v25b1,3,8,9,11,12,24> v26b1,3,9,11,12,13,22> v27b1,3,11,12,13,14,15,22> v28b1,2,3,12,13,14,15>

Cannot be assembled Crank: none Crank: l1 Crank: l1,l2 Crank: l1,l2,l3 Crank: l1,l2,l3,l4 Crank: l1,l2,l3,l4,l5 Crank: (l1,l2) Crank: (l1,l2), l3 Crank: (l1,l2),l3,l4 Crank: (l1,l2),l3,l4,l5 Crank: (l1,l2),(l1,l3) Crank: (l1,l2),(l1,l3),l4 Crank: (l1,l2),(l1,l3),l4,l5 Crank: (l1,l2),(l1,l3),(l2,l3) Crank: (l1,l2),(l1,l3),(l2,l3), l4 Crank: (l1,l2),(l1,l3),(l2,l3), l4,l5 Crank: (l1,l2),(l1,l3),(l1,l4) Crank: (l1,l2),(l1,l3),(l1,l4),l5 Crank: (l1,l2),(l1,l3),(l1,l4),(l1, l5) Crank: (l1,l2,l3) Crank: (li,lj), j = 1 Crank: (li,lj), j = 1,2 Crank: (li,lj), j = 1,2,3 Crank: (li,lj), j = 1,2,3,4 Crank: (li,lj), j = 1,2,3,4,5 Crank: (li,lj) Crank: (l1,li,lj)

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Table 2 The division results of 4-, 5- and 6-link π-close spherical mechanisms.

4-link

5-link

6-link

Vertex 2 1 6 0 6 6 0 6 6 0 6 6 0 6 6 1=3 6 6 1=2 6 4 0 0 2 1 6 0 6 6 6 0 6 0 6 6 0 6 6 0 6 6 1=4 6 6 2=5 6 6 0 6 6 0 6 6 0 6 6 6 0 6 1=2 6 6 2=3 6 6 3=4 6 4 2=3 1=2 2 1 6 0 6 6 0 6 6 0 6 6 0 6 6 0 6 6 0 6 6 1=5 6 6 1=3 6 6 0 6 6 0 6 6 0 6 6 0 6 6 0 6 6 0 6 6 1=3 6 6 1=2 6 6 3=5 6 6 1=2 6 6 0 6 6 0 6 6 0 6 6 0 6 6 1=3 6 6 0 6 6 1=3 6 6 1=2 6 6 1=3 6 6 1=2 6 6 2=5 6 4 1=4 1=3

coordinates 1 1 0 0 0 1=3 1=2 1=2 2=3

1 1 1 0 0 1=3 1=2 1=2 2=3

3 1 1 7 7 1 7 7 1 7 7π 0 7 72 1 7 7 1=2 7 7 1 5 2=3

1 1 0 0 0 0 1=4 2=5 1=3 1=2 0 0 1=2 2=3 3=4 2=3 1=2 1 1 0 0 0 0 0 1=5 1=3 1=4 2=5 0 0 0 0 1=3 1=2 3=5 1=2 1=2 2=3 3=4 2=3 1=3 1=2 2=3 1=2 1=3 1=2 2=5 1=4 1=3

1 1 1 0 0 0 1=4 2=5 1=3 1=2 1=2 2=3 1 2=3 3=4 2=3 1=2 1 1 1 0 0 0 0 1=5 1=3 1=4 2=5 1=3 1=2 0 0 1=3 1=2 3=5 1=2 1=2 2=3 3=4 2=3 1=3 1=2 2=3 2=3 2=3 1=2 2=5 1=2 1=3

1 1 1 1 0 0 1=4 2=5 1=3 1=2 1=2 2=3 1 1 3=4 2=3 1=2 1 1 1 1 0 0 0 1=5 1=3 1=4 2=5 1=3 1=2 1=2 2=3 1 1=2 3=5 1=2 1 2=3 3=4 2=3 1=3 1=2 2=3 2=3 2=3 3=4 4=5 1=2 2=3

3 1 1 7 7 1 7 7 1 7 7 1 7 7 0 7 7 1 7 7 2=5 7 7π 1 7 72 1=2 7 7 1 7 7 2=3 7 7 1 7 7 1 7 7 1 7 7 2=3 5 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 1=5 1 1=3 1=3 1=4 1 2=5 2=5 1=3 1 1=2 1=2 1=2 1 2=3 2=3 1 1 1 1 1 3=5 1=2 1=2 1 1 1 1 3=4 1 2=3 2=3 1=3 1 1=2 1 2=3 1 2=3 1 2=2 2=3 3=4 1 4=5 4=5 1=2 1 2=3 1

3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7π 7 72 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5

Simplexes (or complexes)

Mechanism characteristics

v1b3,4,5,6,7,8,9> v2b1,3,6,7,8,9> v3b1,2,3,8,9> s1b3,6,7,8,9> s2b1,3,8,9>

Cannot be assembled Crank: none Crank: l1 v1↔v2, φ1 + φ2 + φ3 = φ4 v2↔v3, φ1 + φ4 = φ2 + φ3

v1b4,5,6,7,8,9,10,11,12> v2b4,7,8,9,10,11,12,16,17> v3b2,4,9,10,11,12,16,17> v4b2,4,11,12,13,16,17> v5b2,4,13,14,16,17> v6b2,13,14,15,16,17> v7b2,3,4,11,12,13> v8b1,2,13,14,15,16>

Cannot be assembled Crank: none Crank: l1 Crank: l1,l2 Crank: (l1,l2) Crank: l1,l2,l3 Crank: l1,l2,l3,l4 Crank: (li,lj)

v1b5,6,7,8,9,10,11,12,13,14,15> v2b5,8,9,10,11,12,13,14,15,19,24> v3b5,10,11,12,13,14,15,19,23,24,25> v4b5,12,13,14,15,19,23,24,25,28,31> v5b5,14,15,19,23,24,25,28,30,31,32> v6b5,17,19,23,24,25,28,30,31,32> v7b17,18,19,23,24,25,26,27,28,29,30,31, 32> v8b3,5,12,13,14,15,23,25,28,31> v9b3,5,14,15,23,25,28,30,31,32> v10b3,5,17,23,25,28,30,31,32> v11b3,17,23,25,26,27,28,29,30,31,32> v12b3,5,14,15,20,23,25,30,32> v13b3,5,17,20,23,25,30,32> v14b3,17,20,23,25,26,29,30,32> v15b3,5,14,15,16,20,30,32> v16b3,5,16,17,20,30,32> v17b3,16,17,20,29,30,32> v18b3,5,17,20,21,23,25> v19b3,17,20,21,23,25,26> v20b3,20,21,22,23,25,26> v21b3,4,5,14,15,16,20> v22b1,17,18,19,23,26,27,28,29,30> v23b1,3,17,23,26,27,28,29,30> v24b1,3,17,20,23,26,29,30> v25b1,3,16,17,20,29,30> v26b1,3,17,20,21,23,26> v27b1,3,20,21,22,23,26> v28b1,2,3,20,21,22,23>

Cannot be assembled Crank: none Crank: l1 Crank: l1,l2 Crank: l1,l2,l3 Crank: l1,l2,l3,l4 Crank: l1,l2,l3,l4,l5 Crank: (l1,l2) Crank: (l1,l2), l3 Crank: (l1,l2),l3,l4 Crank: (l1,l2),l3,l4,l5 Crank: (l1,l2),(l1,l3) Crank: (l1,l2),(l1,l3),l4 Crank: (l1,l2),(l1,l3),l4,l5 Crank: (l1,l2),(l1,l3),(l2,l3) Crank: (l1,l2),(l1,l3),(l2,l3), l4 Crank: (l1,l2),(l1,l3),(l2,l3), l4,l5 Crank: (l1,l2),(l1,l3),(l1,l4) Crank: (l1,l2),(l1,l3),(l1,l4),l5 Crank: (l1,l2),(l1,l3),(l1,l4),(l1, l5) Crank: (l1,l2,l3) Crank: (li,lj), j = 1 Crank: (li,lj), j = 1,2 Crank: (li,lj), j = 1,2,3 Crank: (li,lj), j = 1,2,3,4 Crank: (li,lj), j = 1,2,3,4,5 Crank: (li,lj) Crank: (l1,li,lj)

7.1. The classification of spherical triangles The classification of spherical triangles is shown in Fig. 5. There are 10 vertexes in the division results. The basic cube which satisfies Eq. (4) is a complex ba, b, c, d, e, f, g, h>. The basic simplex which satisfies Eq. (6) is a simplex ba, e, f, g>. The classification condition and the condition matrix are πclosemechanism :

ϕ1 þ ϕ2 þ ϕ3 ¼ π

0closemechanism :

−ϕ1 þ ϕ2 þ ϕ3 ¼ 0 þϕ1 −ϕ2 þ ϕ3 ¼ 0 −ϕ1 −ϕ2 þ ϕ3 ¼ 0

T ¼ ½1

1

1 −π 

3 −1 1 1 0 T ¼ 4 1 −1 1 0 5 −1 −1 1 0

ð16aÞ

2

ð16bÞ

D. Zhao et al. / Mechanism and Machine Theory 51 (2012) 46–57

a

b

55

c

Fig. 5. The classification of spherical triangles.

For 0-close mechanisms, only the third condition in Eq. (16b) intersects with the basic simplex and divides the basic simplex into two 3-simplexes (Fig. 5b). A vertex e in the 3-simplex ba, e, i, f> does not satisfy the assembly condition, so all the interior points of the simplexes do not satisfy the assembly condition and the solution space is a null set. The existence domain of spherical triangles is the 3-simplex ba, i, f, g> and the solution space is two isolated points, thus, spherical triangles only have one type. The 2-simplex ba, i, f> satisfies the third condition in Eq. (16b), i.e., the critical assembly condition. The 2-simplexes ba, i, g> and ba, g, f> satisfy the sort condition of links, Eq. (6), and all the interior points represent spherical isosceles triangles. In the 1-simplex ba, g>, all the interior points represent spherical equilateral triangles. For π-close spherical triangles, the classification condition divides the basic simplex into two parts, as shown in Fig. 5c. Vertexes a and e in the 3-complex ba, e, i, f, j> do not satisfy the assembly condition, so all the interior points of the complex do not satisfy the assembly condition and the solution space is a null set. The existence domain of π-close spherical triangles is the 3-simplex bi, j, f, g> and the solution space also is two isolated points. In the existence domain, every coordinate component φi of interior points is between 0 and π/2. If replacing any φi with π–φi or 2π–φi, the relatively position of links does not change. Spherical mechanisms with infinite radius and infinitesimal link's angles correspond to planar mechanisms. Accordingly, the division situation in the neighborhood of 0-simplex ba> corresponds to the planar mechanism classification. For N-link mechanisms, the N-dimensional geometry objects correspond to the basic types of mechanisms and non N-dimensional geometry objects correspond to transition types among the basic types. Those geometry objects whose dimensions are less than N have many special transmission characteristics and are also very important for variable topology mechanisms. According to the Carathéodory theory [13], for a simplex consisting of n vertexes, p k represents the coordinate of the vertex k and any interior point p can be expressed as   n n X X  k αk p α k ∈Rα k > 0; αk ¼ 1 ð17Þ p¼  k¼1

k¼1

where [α1, α2,…, αn] are called barycentric coordinates. Eq. (17) is significant for both the division results and the expression of mechanism dimensions. Every vertex corresponds to a special mechanism which has special transmission characteristics. The components of barycentric coordinates correspond to the weight of different transmission characteristics. Eq. (17) is useful to rapidly get an expected mechanism. 7.2. The classification of 4-link spherical mechanisms The classification of 4-link 0-close spherical mechanisms is shown in Fig. 6a and Table 1. There are 7 vertexes, 18 lines, 22 planes, 13 volumes and 3 4-simplexes in the division results. Each 4-simplex corresponds to a basic type of mechanisms. Main results are summarized as follows, (1) As shown in Table 1, in the v1, because the fourth vertex (0, 0, 0, 1) does not satisfy the assembly condition, the interior points of v1 cannot constitute a 4-link mechanism and the solution space is a null set. (2) v1 enters into v2 when passing through the 3-simplex s1. The solution space is homeomorphic to a circle. All revolute joints are not full revolution and there is no crank. (3) v2 enters into v3 when passing through the 3-simplex s2. The solution space is homeomorphic to two circles and the two revolute joints on the shortest link l1 are always full revolution and the link l1 is the crank. (4) In v1, v2 and v3, there is a common vertex (0, 0, 0, 0). Accordingly, these simplexes correspond to the classification of planar mechanisms and the division is equivalent to the Glashof criterion. The closeness of sphere does not add new mechanism types.

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D. Zhao et al. / Mechanism and Machine Theory 51 (2012) 46–57

v8

v18

v6

v21

v5 v3 p2

v7

v2 p5 v1

null set

v1

v20

v27

v19

v26

v28

v15

v16

v17

v25

v12

v13

v14

v24

v4

v8

v9

v10

v11

v23

v3

v4

v5

v6

v7

v22

v3

v2

v1

v2

a

null set

b

c

Fig. 6. The evolvement process of mechanism type and solution space.

The classification of 4-link π-close spherical mechanisms is listed in Table 2. In some complexes, the number of vertexes is different, but the transition among basic types is the same as 0-close spherical mechanisms, as shown in Fig. 6a. 7.3. The classification of 5-link spherical mechanisms The classification of 5-link 0-close spherical mechanisms is shown in Fig. 6b and Table 1. There are 14 vertexes, 54 lines, 96 planes, 89 volumes, 42 4-dimensional geometry objects and 8 5-dimensional geometry objects in the division results. Each 5-dimensional geometry objects corresponds to a basic mechanism type. Main results are summarized as follows, (1) As shown in Table 1, in v1, the vertex (0, 0, 0, 0, 1) does not satisfy the assembly condition. Accordingly, the interior points of v1 cannot constitute a 5-link mechanism and the topology of solution space always is a null set. (2) In v2–v6, links which are cranks are none → l1 → l1l2 → l1l2l3 → l1l2l3l4, respectively, and the topology of solution space is in the order null set → a sphere → a torus → a double-torus → a triple-torus → a quadruple-torus. Namely, the genus of solution space increases one at each evolvement. (3) When v4 enters into v7, the topology of solution space splits into two tori from a double-torus. Links l1 and l2 are cranks at the same time, which is expressed as (l1l2). When v6 enters into v8, the topology of solution space evolves into a quintupletorus and any two links can be cranks at the same time. (4) v1–v7 have a common vertex (0, 0, 0, 0). Accordingly, these geometry objects correspond to the classification of planar 5-link mechanisms. Only v8 is a new type of spherical mechanism because of the closeness of sphere. In Ref. [12], the evolvement of the work-space of 5-link planar mechanisms was reported. Because the work-space is the projection of the solution space, the evolvement of solution space can be indirectly observed. (5) Only v6 is a complex and the rest are simplexes. The classification of 5-link π-close spherical mechanisms is listed in Table 2. The transition among basic types of mechanisms is the same as 0-close spherical mechanisms, as shown in Fig. 6b. 7.4. The classification of 6-link spherical mechanisms The classification of 6-link 0-close spherical mechanisms is shown in Fig. 6c, Tables 1 and 2. There are 26 vertexes, 145 lines, 368 planes, 507 volumes, 394 4-dimensional geometry objects, 163 5-dimensional geometry objects and 28 6-dimensional geometry objects in the division results. Each 6-dimensional geometry objects corresponds to a basic mechanism type. Main results are summarized as follows, (1) In v1, vertex (0, 0, 0, 0, 0, 1) does not satisfy the assembly condition. Accordingly, the interior points of v1 can not constitute a 6-link mechanism and the topology of solution space always is a null set. (2) The evolvement of v2–v7 is similar to that of the 5-link mechanism. The genus of solution space also increases 1 at each evolvement. Links which are cranks are in the order of none → l1 → l1l2 → l1l2l3 → l1l2l3l4 → l1l2l3l4l5.

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(3) As shown in Fig. 6c, along a complicated route, v4–v7 evolve into v8–v21, and the topology of solution space becomes more complicated. In v21, links l1, l2 and l3 can be cranks at the same time, which is expressed as (l1l2l3). The topology of solution space is homeomorphic to two 3-dimensional closed geometry objects. (4) v7, v11, v14, v17, v19 and v20 evolve into v22–v27. There are at least two links which can be cranks at the same time. In v28, the link l1 together with any two links can be cranks at the same time. (5) v1–v21 have a common vertex (0, 0, 0, 0, 0, 0). Accordingly, those geometry objects correspond to the classification of 6-link planar mechanisms. v22–v28 are new spherical mechanisms because of the closeness of sphere. (6) v7, v11, v14, v19, v20 and v27 are complexes and the rest are simplexes. 8. Conclusion The classification method discussed in this paper is mainly consisted of two parts, i.e., the classification condition of all the spherical mechanisms obtained from the singularity of constraint equations, and the mechanism types and the transformation among these mechanism types based on the complex division of dimension space of linkages and its expression. Comparing with the research method of other investigators, the research methodology in this study has many advantages, 1. The topology of the solution space is the only criterion of classification. This criterion is the most fundamental condition to distinguish mechanisms, as all the characteristics of mechanisms will mutate with the change of the topology of solution space. Therefore, confusion caused by the use of multiple classification criteria can be avoided. 2. The classification condition obtained from the singularity of constraint equations is completed and can be used to classify all the spherical mechanisms. 3. In the division results, the mechanism types and the transformation among these mechanism types are comprehensive, and the geometry topology theories can be used to analyze these results. 4. The dimension range of each mechanism type is expressed by the vertexes of the complex, which is appropriate for the synthesis of mechanisms. 5. The classification method is suitable to implement in computer programs to classify complicated mechanisms. However, the current study is still inadequate. Further research is conducting in the following aspects, i.e., proposing a more proper representation of topology of the solution space; studying the mechanism characteristic features in all divided geometry objects; and developing a analysis software concerning the global characteristics of the mechanisms. References [1] J.J. Cervantes-Sanchez, H.I. Medellin-Castillo, A robust classification scheme for spherical 4R linkages, Mechanism and Machine Theory 37 (10) (2002) 1145–1163. [2] L.B. Hang, Y. Wang, Study on the decoupling conditions of the spherical parallel mechanism based on the criterion for topologically decoupled parallel mechanisms, Chinese Journal of Mechanical Engineering 41 (9) (2005) 28–32. [3] D. Kohli, A. Khonji, Grashof-type rotatability criteria of spherical five-bar linkages, Transactions of the ASME Journal of Mechanical Design 116 (99) (1994) 99–104. [4] Y.W. Liu, K.L. Ting, On the rotatability of spherical N-bar chains, AMSE Journal of Mechanical Design 116 (3) (1994) 920–923. [5] K.L. Ting, Y.W. Liu, Rotatability law for N-bar kinematic chains and their proof, ASME Journal of Mechanisms Transmissions and Automation in Design 108 (1) (1991) 32–39. [6] B. Fallahi, H.Y. Lai, R. Naghibi, A study of workspace of five-bar closed loops manipulator, Mechanism and Machine Theory 29 (5) (1994) 759–765. [7] F. Gao, X.Q. Zhang, Y.S. Zhao, A physical model of the solution space and the atlas of reachable workspace for 2-DOF parallel planar manipulators, Mechanism and Machine Theory 31 (2) (1996) 173–184. [8] G. Alici, Determination of singularity contours for five-bar planar parallel manipulators, Robotica 18 (5) (2000) 569–575. [9] D.F. Zhao, Dimension classification method of planar mechanism based on convex hull division, Chinese Journal of Mechanical Engineering 45 (4) (2009) 100–104. [10] R. Gilmore, Catastrophe Theory for Scientists and Engineers, Dover, UK, 1993. [11] V.I. Arnold, Catastrophe Theory, third ed. Springer, Berlin, 1992. [12] D.F. Zhao, Study on the workspace and its change rule of the plane five-bar, Machine Design and Research 23 (5) (2007) 22–25. [13] C. Carathéodory, Über den Variabilitätsbereich der Fourierschen Konstanten von positiven harmonischen Funktionen, Rendiconti del Circolo Matematico di Palermo 32 (1911) 193–217.