Schematic Diagrams Design of Displacement Suppression Mechanism with One Degree-of-Freedom in a Rope-Guided Hoisting System

: Since it is di ﬃ cult for lateral sti ﬀ ness of rope-guided rails to meet industry criteria in deep construction shaft, schematic diagrams of displacement suppression mechanisms (DSMs) are designed with a systematic approach demonstrated to reduce the lateral displacement of rope-guided rails in this paper. DSMs are simpliﬁed as planar four-bar and six-bar topological graphs based on topological theory. Each corresponding mechanical chain of these four-bar and six-bar mechanisms is divided into a rack, mechanical parts, prismatic, and revolute joints. An extended adjacency matrix is deﬁned to represent the rack position, speciﬁc types of kinematic joints, and adjacency relationships between kinematic parts. Then, a symmetric vertex identiﬁcation method is proposed with regard to planar 1-DOF (one degree of freedom) four-bar and six-bar topological graphs to get the sequences of prismatic joints for kinematic chains of DSMs. Finally, the alternative schematic diagrams of DSMs are obtained. The results show four-bar mechanisms with simple structure; few kinematical parts but less resident force are suitable for a mine shaft with small space and small swing. Six-bar mechanisms with two prismatic joints and three mechanical rack degree are applicable for wide shaft space in deep shaft, due to their stable structure and double resistant force. This development is helpful for DSM dimension synthesis design in future.


Introduction
With the increasing shortage of global resources and energy, many countries in the world are developing and utilizing mineral resources deeper in the earth. The excavation depth of newly-built mines in China has increased at an average speed of 10 m per year and mine hoisting in deep shaft requires higher speed, heavier load, and more stability. Wire ropes are widely used for the guidance of hosting conveyances in the process of shaft construction or in a mine hoisting system. Increasing depth of mine shaft leads to more insufficient stiffness of guiding ropes, thereby causing the substantial swinging of hoisting conveyances and increasing insecurity. Researchers made many studies about the stiffness [1,2] of guiding ropes and the vibration [3,4] in a rope-guided mine hoisting system. Therefore, the lateral deflection of conveyance and guiding ropes catch more attention in rope-guided hoisting Hamming number isomorphism determination based on a link-link adjacency matrix. This matrix contains the topological structure information of the kinematic chain. In order to obtain the required schematic diagrams of DSM, a new method is proposed to identity the schematic diagram isomorphism based on the unique nature of each kinematic chain. According to the definition of adjacency matrix, extended adjacency matrix is defined in this paper, which describes the certain type of kinematic joints, adjacency relations between two kinematic parts, and two mechanical rack positions. Then, the isomorphism identification procedure for schematic diagrams of DSM are derived based on four-bar and six-bar mechanisms. Sequences of prismatic joints for DSMs based on canonical perimeter are determined, and the alternative schemes for DSMs are obtained. This provides a new method for type synthesis of schematic diagram and also offers alternative models for the design of DSM.
The remainder of this paper is organized as follows. In Section 2, possible topological graphs for DSMs is presented and corresponding types of kinematic joints are enumerated. In Section 3, connected relations of graphs are defined by two kinds of matrices to identify different topological graphs in Section 3.1 and corresponding schematic diagrams in Section 3.2, respectively. In addition, a symmetry discrimination method by two steps is proposed to identify whether any two vertices (or any two schematic diagrams) are symmetric in Section 3.3. In Section 4, detailed computational isomorphic discrimination steps are presented, and all the non-isomorphic vertexes, racks, as well as schematic diagrams of DSMs, are obtained by special types of sequence numbers. In Section 5, alternative schematic diagrams of DSM are screened and analyzed for further use in a rope-guided mine shaft. In Section 6, concluding remarks are presented.    However, different from topological graph, the schematic diagram of mechanism includes different joint types and mechanism rack types. Since the generation from a topological graph or a kinematic chain to a schematic diagram does not make a one-to-one correspondence, due to symmetrical vertices and edges in topological graphs, i.e., several schematic diagrams may be obtained by one topological graph or kinematic chain. Therefore, Rao [28] used values of 2 and 1 in the link-link adjacency matrix to represent prismatic and revolute joints, respectively. Thus, schematic diagram isomorphism identification is less defined and concerned. Eleashy [29] obtained all solutions of planar 8-bar kinematic chains with up to three prismatic pairs by a systematic method. Dharanipragada [30] thoroughly considered the type difference between prismatic and revolute joints and studied the isomorphism for 1-DOF six-bar and an eight-bar mechanism by splitting Hamming String. However, it is worth noting that the existence of a mechanism rack may also lead to different schematic diagrams. Thus, it is necessary to identify the isomorphic schematic diagrams and obtain all these schematic diagrams of DSM. Finally, these mechanisms can be designed according to their working condition requirements, which are: (1) One of the mechanical parts of DSM is capable of swinging within a certain angle. (2) The mechanism can be locked at the limited position.
(3) It is a 1-DOF planar structure in order to implement easily.
Vertex symmetry in schematic diagram is different from that in a topology graph due to the distinct type among mechanical parts, such as rack and binary bar. Sun [31] proposed to improve Hamming number isomorphism determination based on a link-link adjacency matrix. This matrix contains the topological structure information of the kinematic chain. In order to obtain the required schematic diagrams of DSM, a new method is proposed to identity the schematic diagram isomorphism based on the unique nature of each kinematic chain. According to the definition of adjacency matrix, extended adjacency matrix is defined in this paper, which describes the certain type of kinematic joints, adjacency relations between two kinematic parts, and two mechanical rack positions. Then, the isomorphism identification procedure for schematic diagrams of DSM are derived based on four-bar and six-bar mechanisms. Sequences of prismatic joints for DSMs based on canonical perimeter are determined, and the alternative schemes for DSMs are obtained. This provides a new method for type synthesis of schematic diagram and also offers alternative models for the design of DSM.
The remainder of this paper is organized as follows. In Section 2, possible topological graphs for DSMs is presented and corresponding types of kinematic joints are enumerated. In Section 3, connected relations of graphs are defined by two kinds of matrices to identify different topological graphs in Section 3.1 and corresponding schematic diagrams in Section 3.2, respectively. In addition, a symmetry discrimination method by two steps is proposed to identify whether any two vertices (or any two schematic diagrams) are symmetric in Section 3.3. In Section 4, detailed computational Symmetry 2020, 12, 474 4 of 16 isomorphic discrimination steps are presented, and all the non-isomorphic vertexes, racks, as well as schematic diagrams of DSMs, are obtained by special types of sequence numbers. In Section 5, alternative schematic diagrams of DSM are screened and analyzed for further use in a rope-guided mine shaft. In Section 6, concluding remarks are presented.

Topology Design for DSM
According to the three working condition requirements of DSM, the number of its kinematic joints P h can be derived as where n is the total number of topological vertexes (mechanical bars). F denotes the number of DOFs and F = 1. It is obvious that n is an even number and n ≥ 4 for the reason that P h and n must be positive integers. Thus, n = 4, 6, 8, 10, etc. In this case, let n = 4 or n = 6 to simplify the mechanical structure, that is, four-bar and six-bar linkage mechanisms are chosen for schematic diagram design. Topological graph and the corresponding kinematic chain of planar 1-DOF four-bar and six-bar mechanisms are shown in Figure 3a,b, Figure 4a,b and Figure 5a,b, respectively. According to canonical perimeter loop theory in Ref. [21], the loop number "r" of four-bar and six-bar mechanism are r = 4 and r = 5, 6, respectively. Moreover, the way of labelling for topological graphs is determined based on this theory as shown in  Symmetry 2020, 12, x FOR PEER REVIEW 4 of 16

Topology Design for DSM
According to the three working condition requirements of DSM, the number of its kinematic joints Ph can be derived as where n is the total number of topological vertexes (mechanical bars). F denotes the number of DOFs and F = 1. It is obvious that n is an even number and n ≥ 4 for the reason that Ph and n must be positive integers. Thus, n = 4, 6, 8, 10, etc. In this case, let n = 4 or n = 6 to simplify the mechanical structure, that is, four-bar and six-bar linkage mechanisms are chosen for schematic diagram design.
Topological graph and the corresponding kinematic chain of planar 1-DOF four-bar and six-bar mechanisms are shown in Figures 3a,b, 4a,b and 5a,b, respectively. According to canonical perimeter loop theory in Ref. [21], the loop number "r" of four-bar and six-bar mechanism are r = 4 and r = 5, 6, respectively. Moreover, the way of labelling for topological graphs is determined based on this theory as shown in    Kinematic joints of DSM include prismatic joint P and revolute joint R in order to meet the condition that the mechanism can swing within a certain angle and be locked at a limited position. As shown in Figure 6, prismatic joint P consists of a piston and a cylinder, and revolute joint R is formed with any two of binary bars, pistons, and cylinders. A spring is used between the prismatic joint to ensure the DSMs locking in a limited position.

Topology Design for DSM
According to the three working condition requirements of DSM, the number of its kinematic joints Ph can be derived as where n is the total number of topological vertexes (mechanical bars). F denotes the number of DOFs and F = 1. It is obvious that n is an even number and n ≥ 4 for the reason that Ph and n must be positive integers. Thus, n = 4, 6, 8, 10, etc. In this case, let n = 4 or n = 6 to simplify the mechanical structure, that is, four-bar and six-bar linkage mechanisms are chosen for schematic diagram design.
Topological graph and the corresponding kinematic chain of planar 1-DOF four-bar and six-bar mechanisms are shown in Figures 3a,b, 4a,b and 5a,b, respectively. According to canonical perimeter loop theory in Ref. [21], the loop number "r" of four-bar and six-bar mechanism are r = 4 and r = 5, 6, respectively. Moreover, the way of labelling for topological graphs is determined based on this theory as shown in    Kinematic joints of DSM include prismatic joint P and revolute joint R in order to meet the condition that the mechanism can swing within a certain angle and be locked at a limited position. As shown in Figure 6, prismatic joint P consists of a piston and a cylinder, and revolute joint R is formed with any two of binary bars, pistons, and cylinders. A spring is used between the prismatic joint to ensure the DSMs locking in a limited position.

Topology Design for DSM
According to the three working condition requirements of DSM, the number of its kinematic joints Ph can be derived as where n is the total number of topological vertexes (mechanical bars). F denotes the number of DOFs and F = 1. It is obvious that n is an even number and n ≥ 4 for the reason that Ph and n must be positive integers. Thus, n = 4, 6, 8, 10, etc. In this case, let n = 4 or n = 6 to simplify the mechanical structure, that is, four-bar and six-bar linkage mechanisms are chosen for schematic diagram design.
Topological graph and the corresponding kinematic chain of planar 1-DOF four-bar and six-bar mechanisms are shown in Figures 3a,b, 4a,b and 5a,b, respectively. According to canonical perimeter loop theory in Ref. [21], the loop number "r" of four-bar and six-bar mechanism are r = 4 and r = 5, 6, respectively. Moreover, the way of labelling for topological graphs is determined based on this theory as shown in    Kinematic joints of DSM include prismatic joint P and revolute joint R in order to meet the condition that the mechanism can swing within a certain angle and be locked at a limited position. As shown in Figure 6, prismatic joint P consists of a piston and a cylinder, and revolute joint R is formed with any two of binary bars, pistons, and cylinders. A spring is used between the prismatic joint to ensure the DSMs locking in a limited position.  Kinematic joints of DSM include prismatic joint P and revolute joint R in order to meet the condition that the mechanism can swing within a certain angle and be locked at a limited position. As shown in Figure 6, prismatic joint P consists of a piston and a cylinder, and revolute joint R is formed with any two of binary bars, pistons, and cylinders. A spring is used between the prismatic joint to ensure the DSMs locking in a limited position.
Kinematic joints of DSM include prismatic joint P and revolute joint R in order to meet the condition that the mechanism can swing within a certain angle and be locked at a limited position. As shown in Figure 6, prismatic joint P consists of a piston and a cylinder, and revolute joint R is formed with any two of binary bars, pistons, and cylinders. A spring is used between the prismatic joint to ensure the DSMs locking in a limited position.  In order to describe the adjacent relations between kinematic parts, each kinematic part is presented with detailed kinematic joints and "RR, RP" and "RRR, RRP, RPP, PPP" are respectively used to represent different types of kinematic joints. "RR" denotes the kinematic part, which is connected to another two parts by two revolute joints. "RP" means that the kinematic part connected to another two bars is respectively by a revolute joint R and a prismatic joint P. Similarly, "RRR", "RRP", "RPP", and "PPP" are the kinematic parts connected to another three parts by three R joints, two R joints, a P joint, an R joint, and two P joints, as well as three P joints, respectively. Since the prismatic joint is constituted by a cylinder block and a piston, there are several types of kinematic parts corresponding to "RP", "RRP", "RPP", and "PPP". All detailed kinematic parts in schematic diagrams of DSMs are shown in Figure 7. Mechanism racks are connected with these kinematic parts; hence, their structures include two types as shown in Figure 8a,b, respectively. In order to describe the adjacent relations between kinematic parts, each kinematic part is presented with detailed kinematic joints and "RR, RP" and "RRR, RRP, RPP, PPP" are respectively used to represent different types of kinematic joints. "RR" denotes the kinematic part, which is connected to another two parts by two revolute joints. "RP" means that the kinematic part connected to another two bars is respectively by a revolute joint R and a prismatic joint P. Similarly, "RRR", "RRP", "RPP", and "PPP" are the kinematic parts connected to another three parts by three R joints, two R joints, a P joint, an R joint, and two P joints, as well as three P joints, respectively. Since the prismatic joint is constituted by a cylinder block and a piston, there are several types of kinematic parts corresponding to "RP", "RRP", "RPP", and "PPP". All detailed kinematic parts in schematic diagrams of DSMs are shown in Figure 7. Mechanism racks are connected with these kinematic parts; hence, their structures include two types as shown in Figure 8a,b, respectively.

Adjacency Relation of Topological Graph
is a topological graph where V and E are respectively defined as mathematical aggregations of vertexes and edges in Ref. [21]. According to the adjacent relationship between vertexes, adjacency matrix AM (G) of topological graph G is defined as: where n is the vertex number of topological graph G and 1, if vertexes and are connected by one edge

Adjacency Relation of Mechanism Diagram
Adjacency matrix represents the adjacent relationships of vertexes in a topological graph; however, in the corresponding schematic diagram of mechanism, it can neither express the position of mechanical racks nor describe the type of kinematic joints. As with the definition of G (V, E), M (R, C, J) is defined as a schematic diagram of mechanism where R, C, and J denote the mathematical aggregations of mechanical racks, kinematic parts and kinematic joints, respectively. Therefore, In order to describe the adjacent relations between kinematic parts, each kinematic part is presented with detailed kinematic joints and "RR, RP" and "RRR, RRP, RPP, PPP" are respectively used to represent different types of kinematic joints. "RR" denotes the kinematic part, which is connected to another two parts by two revolute joints. "RP" means that the kinematic part connected to another two bars is respectively by a revolute joint R and a prismatic joint P. Similarly, "RRR", "RRP", "RPP", and "PPP" are the kinematic parts connected to another three parts by three R joints, two R joints, a P joint, an R joint, and two P joints, as well as three P joints, respectively. Since the prismatic joint is constituted by a cylinder block and a piston, there are several types of kinematic parts corresponding to "RP", "RRP", "RPP", and "PPP". All detailed kinematic parts in schematic diagrams of DSMs are shown in Figure 7. Mechanism racks are connected with these kinematic parts; hence, their structures include two types as shown in Figure 8a,b, respectively.

Adjacency Relation of Topological Graph
is a topological graph where V and E are respectively defined as mathematical aggregations of vertexes and edges in Ref. [21]. According to the adjacent relationship between vertexes, adjacency matrix AM (G) of topological graph G is defined as: where n is the vertex number of topological graph G and 1, if vertexes and are connected by one edge

Adjacency Relation of Mechanism Diagram
Adjacency matrix represents the adjacent relationships of vertexes in a topological graph; however, in the corresponding schematic diagram of mechanism, it can neither express the position of mechanical racks nor describe the type of kinematic joints. As with the definition of G (V, E), M (R, C, J) is defined as a schematic diagram of mechanism where R, C, and J denote the mathematical aggregations of mechanical racks, kinematic parts and kinematic joints, respectively. Therefore, N (V) = N (R) + N (C) and N (E) = N (J) when N (X) is defined to represent the element number of mathematical aggregation X. Since one adjacency matrix of a topological graph corresponds to

Adjacency Relation of Topological Graph
is a topological graph where V and E are respectively defined as mathematical aggregations of vertexes and edges in Ref. [21]. According to the adjacent relationship between vertexes, adjacency matrix AM (G) of topological graph G is defined as: where n is the vertex number of topological graph G and am ij = 1, if vertexes i and j are connected by one edge 0, else

Adjacency Relation of Mechanism Diagram
Adjacency matrix represents the adjacent relationships of vertexes in a topological graph; however, in the corresponding schematic diagram of mechanism, it can neither express the position of mechanical Symmetry 2020, 12, 474 6 of 16 racks nor describe the type of kinematic joints. As with the definition of G (V, E), M (R, C, J) is defined as a schematic diagram of mechanism where R, C, and J denote the mathematical aggregations of mechanical racks, kinematic parts and kinematic joints, respectively. Therefore, N (V) = N (R) + N (C) and N (E) = N (J) when N (X) is defined to represent the element number of mathematical aggregation X. Since one adjacency matrix of a topological graph corresponds to several schematic diagrams of mechanism, an extended adjacency matrix EAM (M) is defined as follows to accurately describe the position of mechanical rack and the joint type and f is defined as any real number other than 0 and ±1 to represent mechanical rack: where n = N (V) and where n = N (V) and For example, both four-bar mechanisms, with a mechanical rack of label 1 in Figure 9, are developed from the topological graph in Figure 3a and include a prismatic joint and three revolute joints. Their adjacency matrix AM (F3) of ( )

Vertexes Symmetry Identification
Two symmetrical vertexes are the same in vertex degree and incident characteristics according to Kuo's symmetry identification [16]. Therefore, Kuo's method can identify vertexes with equal or equivalent adjacency symmetry. However, Kuo's method is complicated, due to generation matrix and cross matrix required to obtain fitness value and vertex family string. In addition, Kuo's method cannot be used for schematic diagram symmetry identification. Since a topological graph with its vertexes numbered corresponds to a certain adjacency matrix, adjacency matrixes of a topological graph with vertexes numbered by different methods can be applied to identify whether two vertexes are symmetric or equivalent. Liu [32] pointed out that, if there is a matrix T making TAT T = B, where A and B are two adjacent matrixes of topological graphs, then two topological graphs are isomorphic. This method can be also used for schematic diagram symmetry identification.
Two types of symmetric vertexes exist in Figures 3a, 4a and 5a, that is, equal adjacency symmetry and equivalent adjacency symmetry. Taking Figure 4a for example, vertexes 2 and 6 are equal adjacency symmetry because they are both connected with vertexes 1 and 3, while vertexes 1 and 3 are equivalent adjacency symmetry, and they are symmetrical in topological structure; even if vertex 1 is connected with vertexes 2 and 5, vertex 3 is connected with vertexes 2 and 4. Therefore, two steps of symmetry discrimination method are proposed to identify whether any two vertices are

Vertexes Symmetry Identification
Two symmetrical vertexes are the same in vertex degree and incident characteristics according to Kuo's symmetry identification [16]. Therefore, Kuo's method can identify vertexes with equal or equivalent adjacency symmetry. However, Kuo's method is complicated, due to generation matrix and cross matrix required to obtain fitness value and vertex family string. In addition, Kuo's method cannot be used for schematic diagram symmetry identification. Since a topological graph with its vertexes numbered corresponds to a certain adjacency matrix, adjacency matrixes of a topological graph with vertexes numbered by different methods can be applied to identify whether two vertexes are symmetric or equivalent. Liu [32] pointed out that, if there is a matrix T making TAT T = B, where A and B are two adjacent matrixes of topological graphs, then two topological graphs are isomorphic. This method can be also used for schematic diagram symmetry identification.
Two types of symmetric vertexes exist in Figures 3a, 4a and 5a, that is, equal adjacency symmetry and equivalent adjacency symmetry. Taking Figure 4a for example, vertexes 2 and 6 are equal adjacency symmetry because they are both connected with vertexes 1 and 3, while vertexes 1 and 3 are equivalent adjacency symmetry, and they are symmetrical in topological structure; even if vertex 1 is connected with vertexes 2 and 5, vertex 3 is connected with vertexes 2 and 4. Therefore, two steps of symmetry discrimination method are proposed to identify whether any two vertices are symmetric. Detailed explanations are as Step (I) and Step (II): Step (I)-Equal adjacency symmetry: Two vertexes with equal adjacency symmetry exactly as vertexes 2 and 6 in Figure 4a have the same adjacent relationship with other vertexes, i.e., their adjacency matrixes are the same. Therefore, if two vertexes are equal adjacency symmetry, the corresponding row and column of a vertex in adjacent matrix are respectively equal to those of the other. Thus, transformation matrix TM (p, q) is defined as Equation (4) to identify whether two vertexes in a topological graph have equal adjacency symmetry, where p and q are the label number of two vertexes: where In addition, the relationship between the new adjacency matrix AM (j) and the original adjacency matrix AM (G) is derived as: where superscript T denotes a transpose of the matrix, and j represents the graph number in each identification step. If AM(j) = AM(G), vertexes p and q have equal adjacency symmetry. Taking transformation matrixes TM (2, 6) and TM (1, 3), for example, their expressions are: According to Equation ( Since AM(9) = AM(F4) and AM(2) AM(F4), it can be concluded that vertexes 2 and 6 have equal adjacency symmetry, while vertexes 1 and 3 are not. Therefore, this identification method is to detect whether the vertexes connected with two objective vertexes are the same.
Step (II)-Equivalent adjacency symmetry: If the adjacency matrix of a topological graph with vertexes numbered by a method is the same as that of the graph with vertexes numbered by another way, it can be concluded that two vertexes, with the same number respectively obtained through the above two methods, are symmetrical. This symmetry identification includes equivalent and equal adjacency symmetry. New adjacency matrixes will be obtained by reversely numbering and then sequentially renumbering the vertices of a graph, which are originally numbered by canonical perimeter numbering defined in Ref. [21]. Figure 4a is a canonical perimeter topological graph for a six bar mechanism. In order to find the vertexes symmetrical to vertex 1, the process in Figure 10 is presented as an example to describe the detailed steps. Figure 10a is the process of sequential numbering, and Figure 10b includes two processes of reversely and sequentially numbering. The adjacent matrix of each topological graph after every sequential numbering process is different from that of the original one labeled with red number in Figure 10a, which indicates that there is no vertex symmetrical with vertex 1. However, the adjacent matrix of a topological graph highlighted in red after step (4) in Figure 10b is equal to that of the original, which shows that vertexes 1 and 3 are symmetrical. It is easy to detect that the result of this process is sufficient but not the necessary condition to prove that two vertexes are symmetrical. This will be a necessary condition when the following preconditions are satisfied simultaneously:

1)
Topological graphs are numbered on the basis of the perimeter loop theory and two vertexes to be identified are both in the canonical perimeter loop. 2) There is only one branch path outside the canonical perimeter loop.
3) Topological vertexes in sub-chain are no more than one. In order to perform a traversal search on the perimeter loop with loop number r, the vertex numbered with i in perimeter loop will be renumbered with i − 1 (i = 2, 3, …, r) or r (i = 1) in the sequential numbering process. The vertex numbered with i in perimeter loop will be renumbered with r + 2 − i (i = 2, 3, …, r) or 1 (i = 1) in a reverse numbering process. The corresponding adjacency matrix generated from each step in Figure 10 can be described with transformation matrixes STM and RTM as Equations (6) and (7) to respectively express the steps of sequential numbering and reverse numbering: The given adjacency matrix of a topological graph is set as AM0 with its corresponding vertexes numbered in the counterclockwise direction. The new adjacency matrixes are set as AMa(i) and AMb(i) during the sequential numbering processes respectively in the counterclockwise and clockwise direction. Thus, the initial matrix AMa(1) and AMb(1) satisfy AMa(1) = AMb(1) = AM0, thereby, AMa(i + 1) and AMb(i + 1) are derived as Equations (8)  In order to perform a traversal search on the perimeter loop with loop number r, the vertex numbered with i in perimeter loop will be renumbered with i − 1 (i = 2, 3, . . . , r) or r (i = 1) in the sequential numbering process. The vertex numbered with i in perimeter loop will be renumbered with r + 2 − i (i = 2, 3, . . . , r) or 1 (i = 1) in a reverse numbering process. The corresponding adjacency matrix generated from each step in Figure 10 can be described with transformation matrixes STM and RTM as Equations (6) and (7) to respectively express the steps of sequential numbering and reverse numbering: where The given adjacency matrix of a topological graph is set as AM 0 with its corresponding vertexes numbered in the counterclockwise direction. The new adjacency matrixes are set as AM a (i) and AM b (i) during the sequential numbering processes respectively in the counterclockwise and clockwise Symmetry 2020, 12, 474 9 of 16 direction. Thus, the initial matrix AM a (1) and AM b (1) satisfy AM a (1) = AM b (1) = AM 0 , thereby, AM a (i + 1) and AM b (i + 1) are derived as Equations (8) and (9), respectively: The new adjacency matrix after each step is derived as: We can obtain that AM b (5) = AM b (1) by computing each step in Figure 10 with Equations (8) and (9). The corresponding vertex numbers of AM b (5) are marked in red, which indicates that vertexes 1 and 3 are equivalent adjacency symmetry.
Four-bar and six-bar linkage mechanisms both fulfill conditions 2 and 3 in their topological structures. As for condition 1, the equivalent adjacency symmetry method can identify two vertexes both in a canonical perimeter loop, while an equal adjacency symmetry method can identify two vertexes, respectively, in a sub-chain and canonical perimeter loop. Therefore, the two symmetry identifications above are workable. In addition, this symmetry identification is not only effective for topological graphs that satisfy the above conditions, but also feasible to identity symmetry for schematic diagram of mechanisms due to less concern about the process from a topological graph or mechanical chain to sundry schematic diagrams.

Computational Identification
Symmetry identification of mechanism diagrams is to distinguish the adjacent relations between sliding mechanical parts, revolute mechanical parts and their mechanical rack. In order to obtain all mechanism diagrams distinct in their kinematic joint types, we need to decide which vertex number is the mechanical rack firstly because two mechanism diagrams, with the same joint types and the same adjacent relation in topological graphs but two different label numbers of mechanical rack, may be two different structures. Then, the number of prismatic joints is the next step to be determined. The identification procedure to obtain an asymmetrical extended adjacent matrix is shown in Figure 11 and detailed steps and explanations are described as follows: (i) Determine label number of mechanical rack: Two vertexes with topological symmetry form a vertex pair. All vertex pairs can be determined by symmetry identification of a topological graph. Among these symmetrical vertex pairs, the vertices between which have topological asymmetry, are chosen as labels of mechanical rack. The vertex labels of alternative rack are put into row matrix a with the length of N a . (ii) Determine the number of prismatic joints: Since revolute joints are generally used in DSM and more prismatic joints will lead to more different mechanism diagram, one or two prismatic joints are both considered in the process to get mechanism diagrams. N p is used to denote the number of prismatic joints. (iii) Obtain all possible extended adjacent matrixes: All extended adjacent matrixes of mechanism diagrams with one or two prismatic joints are put into the matrix RecordAM for the following symmetry identifications. Corresponding rack vertex label and the labels of vertexes connected by prismatic joints are stored into matrix LocP.
[RecordAM_LocP] = RestoreAM_Loca (AM 0 , N p , a, n, r), where RestoreAM_Loca () is a user-defined MATLAB function. (iv) Symmetry identification of mechanism diagrams: One extended adjacent matrix EAM is taken out from RecordAM to compare with another extended adjacent matrix EAMx one by one. If EAM EAMx, record EAM in matrix R x and record the corresponding rack and prismatic joints information from LocP in matrix Lp x ; otherwise, abnegate EAM and identify the next one.
[R x LP x ] = SymmetryIdentP (n, r, N p , a, RecordAM, LocP), where SymmetryIdentP () is a user-defined MATLAB function. (v) Discard the mechanism diagrams with two consecutive prismatic joints in a six-bar mechanism except those both connected with mechanism rack: Two consecutive prismatic joints mean a mechanical part is connected with other two parts both by prismatic joints, which leads to a more complex structure and more difficult diagram design relative to that in a four-bar mechanism. Since a mechanism rack is immovable, its position can be quickly determined according to required movement. The number of alternative mechanisms is N PLx = length (L Px )/N p , where length () is a MATLAB function to obtain the matrix length. If the common vertex of two consecutive prismatic joints is not equal to rack label, a corresponding extended adjacent matrix from matrix R x and a matrix of rack and prismatic joints information from Lp x will be recorded in matrixes f R and f LP , respectively. (vi) Abandon six-bar mechanism diagrams with all prismatic joints in four-bar sub-loops and the mechanisms for which any prismatic joint exist in a sub-loop without a mechanism rack: From objective topological graphs, a sub-loop of a six-bar topological graph is four vertexes and will be substituted with four mechanical parts in a conversion process to the mechanism diagram. A six-bar mechanism with all prismatic joints existing in a four-bar sub-loop is almost equivalent to four-bar mechanisms, due to the following movement of the other sub-chain. In addition, if prismatic joints exist in this sub-loop without mechanism rack, there is little possibility to achieve locking in a limited position because the function is achieved by setting prismatic joints and a mechanism rack in the same sub-loop. Therefore, the above conditions can be neglected.
Symmetry 2020, 12, x FOR PEER REVIEW 9 of 16 We can obtain that AMb (5) = AMb (1) by computing each step in Figure 10 with Equations (8) and (9). The corresponding vertex numbers of AMb (5) are marked in red, which indicates that vertexes 1 and 3 are equivalent adjacency symmetry.
Four-bar and six-bar linkage mechanisms both fulfill conditions 2 and 3 in their topological structures. As for condition 1, the equivalent adjacency symmetry method can identify two vertexes both in a canonical perimeter loop, while an equal adjacency symmetry method can identify two vertexes, respectively, in a sub-chain and canonical perimeter loop. Therefore, the two symmetry identifications above are workable. In addition, this symmetry identification is not only effective for topological graphs that satisfy the above conditions, but also feasible to identity symmetry for schematic diagram of mechanisms due to less concern about the process from a topological graph or mechanical chain to sundry schematic diagrams.

Computational Identification
Symmetry identification of mechanism diagrams is to distinguish the adjacent relations between sliding mechanical parts, revolute mechanical parts and their mechanical rack. In order to obtain all mechanism diagrams distinct in their kinematic joint types, we need to decide which vertex number is the mechanical rack firstly because two mechanism diagrams, with the same joint types and the same adjacent relation in topological graphs but two different label numbers of mechanical rack, may be two different structures. Then, the number of prismatic joints is the next step to be determined. The identification procedure to obtain an asymmetrical extended adjacent matrix is shown in Figure  11 and detailed steps and explanations are described as follows:  Figure 11. Identification procedure to obtain EAM. According to vertexes' symmetry identification in Section 2, the computational identification procedures from step i to detect equal and equivalent symmetrical vertexes are demonstrated in Figure 12, where R G with dimension of n × n is defined to display the information about two symmetrical vertexes and R G (p, q) represent the element in the pth row and the qth column of R G . Thus, R G (p, q) = 1 indicates that vertexes p and q have topological symmetry.
A-B is used to describe that vertexes A and B have topological symmetry. It can be easily concluded that three symmetrical vertex pairs A-B, B-C, and A-C denote that A, B, and C are mutually symmetrical. Therefore, matrix R1 denotes that vertex pairs with topological symmetry are 1-2, 1-3, 1-4, 2-3, 2-4, 3-4, and vertexes 1, 2, 3, 4 have mutual symmetry. According to this principle, symmetrical vertex pairs and the determined rack positions can be derived as shown in Table 1. According to link adjacency and joint incidence theory and the method in Ref. [16], if two vertex family strings of all their corresponding cross matrixes are the same, two vertexes are proved to be isomorphic. {Fi}1−L denotes the family string of the ith vertex with L generation matrixes in a topological graph.
A-B is used to describe that vertexes A and B have topological symmetry. It can be easily concluded that three symmetrical vertex pairs A-B, B-C, and A-C denote that A, B, and C are mutually symmetrical. Therefore, matrix R 1 denotes that vertex pairs with topological symmetry are 1-2, 1-3, 1-4, 2-3, 2-4, 3-4, and vertexes 1, 2, 3, 4 have mutual symmetry. According to this principle, symmetrical vertex pairs and the determined rack positions can be derived as shown in Table 1. According to link adjacency and joint incidence theory and the method in Ref. [16], if two vertex family strings of all their corresponding cross matrixes are the same, two vertexes are proved to be isomorphic. {F i } 1−L denotes the family string of the ith vertex with L generation matrixes in a topological graph.
The vertex family strings of four-bar mechanism shown in Figure 3a are: Vertex family strings of six-bar mechanism with r = 5 shown in Figure 4a are: Vertex family strings of four-bar mechanism with r = 6 shown in Figure 5a are: The results calculated by the method in Ref. [16] show that four-bar mechanism family strings of vertexes 1-4 are the same, which represents that vertexes 1-4 in Figure 3a are isomorphic or symmetric. Similarly, vertexes between 2 and 6, vertexes between 1 and 3, and vertexes between 4 and 5 in Figure 4a are symmetric, respectively. In Figure 5, Vertexes 1 and 3 are symmetric, and vertexes 2, 3, 5, 6 are mutually symmetric. These results are consistent with that in Table 1 and indicate the validity of symmetry identification of this paper.
According to the steps from (i) to (vi) and mechanism rack label in Table 1, the number of all mechanisms N R , N lpx , N f and N z can be obtained as Table 2, where N R , N lpx , and N f are calculated from matrixes LocP, LP x and f LP , N z is obtained from step (vi). (A, B) is used to describe a pair of prismatic joint between vertexes A and B. When N p = 2, two pairs of prismatic joint are separated with ".". Therefore, the final alternative mechanisms in Table 2 correspond to prismatic joints as shown in Table 3.

Alternatives for DSM and Results
In order to describe these schematic diagrams numbered by the perimeter loop, a new description is proposed as their types of mechanisms. The sequence for each schematic diagram of mechanism is expressed as NB.NPL-RP-NP-PSP, where NB, NPL, RP, NP, and PSP denotes the number of bars, the number of perimeter loop, rack label, the number of prismatic joint, and the sequence of prismatic joints in Table 3, respectively. All schematic diagrams of DSM are shown in Figures 13-15. It can be seen that all schematic diagrams fulfill the previous motion requirements: swinging in a certain angle and being locked in two limited positions.

Alternatives for DSM and Results
In order to describe these schematic diagrams numbered by the perimeter loop, a new description is proposed as their types of mechanisms. The sequence for each schematic diagram of mechanism is expressed as NB.NPL-RP-NP-PSP, where NB, NPL, RP, NP, and PSP denotes the number of bars, the number of perimeter loop, rack label, the number of prismatic joint, and the sequence of prismatic joints in Table 3, respectively. All schematic diagrams of DSM are shown in Figures 13-15. It can be seen that all schematic diagrams fulfill the previous motion requirements: swinging in a certain angle and being locked in two limited positions.
Analyzing from the kinematic structure in Figure 13, a four-bar mechanism can be divided into two categories, respectively with one P-joint and two P-joints. Four-bar mechanisms with one P-joint are simple in structure and easy to design, while those with two P-joints need more elaborate design for the direction and position of initiative force, especially the bar as mechanical rack with a P-joint. Moreover, a mechanical rack with a P-joint in Figure 13 may have problems of insufficient locking force in their limited position because one of the prismatic joints without a spring only provides sliding motion and its motion direction is limited.

Alternatives for DSM and Results
In order to describe these schematic diagrams numbered by the perimeter loop, a new description is proposed as their types of mechanisms. The sequence for each schematic diagram of mechanism is expressed as NB.NPL-RP-NP-PSP, where NB, NPL, RP, NP, and PSP denotes the number of bars, the number of perimeter loop, rack label, the number of prismatic joint, and the sequence of prismatic joints in Table 3, respectively. All schematic diagrams of DSM are shown in Figures 13-15. It can be seen that all schematic diagrams fulfill the previous motion requirements: swinging in a certain angle and being locked in two limited positions.
Analyzing from the kinematic structure in Figure 13, a four-bar mechanism can be divided into two categories, respectively with one P-joint and two P-joints. Four-bar mechanisms with one P-joint are simple in structure and easy to design, while those with two P-joints need more elaborate design for the direction and position of initiative force, especially the bar as mechanical rack with a P-joint. Moreover, a mechanical rack with a P-joint in Figure 13 may have problems of insufficient locking force in their limited position because one of the prismatic joints without a spring only provides sliding motion and its motion direction is limited.   Comparing schematic diagrams in Figure 14 with that in Figure 15, six-bar mechanisms are more stable in structure when their rack degree is three, while they will lose this advantage when their rack degree is two. In general, six-bar mechanisms with two prismatic joints in Figures 14 and 15 are stable against resisting the rope displacement, but the failure rate will be higher than that of four-bar mechanism and six-bar mechanism with single P-joint. When n = 6, r = 5 and Np = 1 as shown in Figure  14a with single prismatic joint, since there are five minimum sub-loops with prismatic joints while the others have four, there are more mechanical bars to swing, and they need a great driving force. When n = 6, r = 5, and rack label is vertex 4 in Figure 14b, these types of mechanisms have a smaller swing angle, owing to a closed four-bar sub-loop in each of them. In this case, we can choose fourbar mechanisms with single prismatic joints for first priority in the shaft, and then the six-bar mechanism with two prismatic joints if more stable resistant forces are needed.
Based on the above analysis about Figures 13-15, we can find that the four-bar mechanisms are simple in structure, easy to design, have less resistance force, and occupy a small space compared with the six-bar mechanism. Thus, four-bar mechanisms are suitable for mine shafts with small space and small swing. Six-bar mechanisms with two prismatic joints and three mechanical rack degree are applicable for wide shaft space in deep shaft, due to their stable structure and double resistant force. This development is helpful for DSM dimension synthesis design in the future.

Conclusions
We determine possible topological graphs and the kinematic joints type of schematic diagrams, according to three working condition requirements of DSMs in this paper. Then, isomorphism identification by two steps of schematic diagrams is proposed, and the computational process is presented to obtain the alternative schematic diagrams of DSMs in rope-guided mine shaft.
Two concepts of equal symmetry and equivalent symmetry are proposed to identify the symmetrical vertexes and isomorphic schematic diagrams by changing the vertex label order. It is a simple and efficient way to obtain schematic diagrams of DSM, with 1-DOF four-bar and six-bar mechanisms, by two steps of symmetric identification in rope-guided mine shaft in three working conditions. Four-bar mechanisms with simple structure, few kinematical parts but less resident force are suitable for mine shafts with small space and small swing. Six-bar mechanisms with two prismatic joints and three mechanical rack degree are applicable for wide shaft space in deep shaft.
It is worth noting that alternative schematic diagrams for 4-bar and six-bar mechanisms are sufficient. Therefore, 8, 10, 12, or more bar mechanisms are not taken into consideration, due to multiple bar mechanisms leading to the complexity for the required simple motion of DSM in mine shafts. In addition, each alternative schematic diagram of DSM presented in this paper can satisfy different mine shaft space, wideness, or straightness. No matter which structure we choose, further dimension synthesis to each structure is still required to be solved. Thus, it is important for further choice and design of DSM in rope-guided mine shaft.
Author Contributions: L.Y. and G.C contributed equally to conceptualization and methodology of this paper. L.Y. and W. P.: references and literature collection and organization. L.Y. and N.W.: validation and main analysis, G.C; supervision, project administration and funding acquisition. All authors have read and agreed to publish this version of the paper.  Analyzing from the kinematic structure in Figure 13, a four-bar mechanism can be divided into two categories, respectively with one P-joint and two P-joints. Four-bar mechanisms with one P-joint are simple in structure and easy to design, while those with two P-joints need more elaborate design for the direction and position of initiative force, especially the bar as mechanical rack with a P-joint. Moreover, a mechanical rack with a P-joint in Figure 13 may have problems of insufficient locking force in their limited position because one of the prismatic joints without a spring only provides sliding motion and its motion direction is limited.
Comparing schematic diagrams in Figure 14 with that in Figure 15, six-bar mechanisms are more stable in structure when their rack degree is three, while they will lose this advantage when their rack degree is two. In general, six-bar mechanisms with two prismatic joints in Figures 14 and 15 are stable against resisting the rope displacement, but the failure rate will be higher than that of four-bar mechanism and six-bar mechanism with single P-joint. When n = 6, r = 5 and N p = 1 as shown in Figure 14a with single prismatic joint, since there are five minimum sub-loops with prismatic joints while the others have four, there are more mechanical bars to swing, and they need a great driving force. When n = 6, r = 5, and rack label is vertex 4 in Figure 14b, these types of mechanisms have a smaller swing angle, owing to a closed four-bar sub-loop in each of them. In this case, we can choose four-bar mechanisms with single prismatic joints for first priority in the shaft, and then the six-bar mechanism with two prismatic joints if more stable resistant forces are needed.
Based on the above analysis about Figures 13-15, we can find that the four-bar mechanisms are simple in structure, easy to design, have less resistance force, and occupy a small space compared with the six-bar mechanism. Thus, four-bar mechanisms are suitable for mine shafts with small space and small swing. Six-bar mechanisms with two prismatic joints and three mechanical rack degree are applicable for wide shaft space in deep shaft, due to their stable structure and double resistant force. This development is helpful for DSM dimension synthesis design in the future.

Conclusions
We determine possible topological graphs and the kinematic joints type of schematic diagrams, according to three working condition requirements of DSMs in this paper. Then, isomorphism identification by two steps of schematic diagrams is proposed, and the computational process is presented to obtain the alternative schematic diagrams of DSMs in rope-guided mine shaft.
Two concepts of equal symmetry and equivalent symmetry are proposed to identify the symmetrical vertexes and isomorphic schematic diagrams by changing the vertex label order. It is a simple and efficient way to obtain schematic diagrams of DSM, with 1-DOF four-bar and six-bar mechanisms, by two steps of symmetric identification in rope-guided mine shaft in three working conditions. Four-bar mechanisms with simple structure, few kinematical parts but less resident force are suitable for mine shafts with small space and small swing. Six-bar mechanisms with two prismatic joints and three mechanical rack degree are applicable for wide shaft space in deep shaft.
It is worth noting that alternative schematic diagrams for 4-bar and six-bar mechanisms are sufficient. Therefore, 8, 10, 12, or more bar mechanisms are not taken into consideration, due to multiple bar mechanisms leading to the complexity for the required simple motion of DSM in mine shafts. In addition, each alternative schematic diagram of DSM presented in this paper can satisfy different mine shaft space, wideness, or straightness. No matter which structure we choose, further dimension synthesis to each structure is still required to be solved. Thus, it is important for further choice and design of DSM in rope-guided mine shaft.
Author Contributions: L.Y. and G.C. contributed equally to conceptualization and methodology of this paper. L.Y. and W.P.: references and literature collection and organization. L.Y. and N.W.: validation and main analysis, G.C; supervision, project administration and funding acquisition. All authors have read and agreed to publish this version of the paper.
Funding: This research received no external funding.