CN111104563A - Kinematic chain isomorphism judgment method based on prime number asymmetric adjacent matrix - Google Patents

Kinematic chain isomorphism judgment method based on prime number asymmetric adjacent matrix Download PDF

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CN111104563A
CN111104563A CN201911094538.2A CN201911094538A CN111104563A CN 111104563 A CN111104563 A CN 111104563A CN 201911094538 A CN201911094538 A CN 201911094538A CN 111104563 A CN111104563 A CN 111104563A
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杨义
孙伟
左惠妍
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Wuhan University of Science and Engineering WUSE
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Abstract

The invention relates to a method for judging isomorphism of a kinematic chain based on a prime number asymmetric adjacency matrix, which comprises the following steps: numbering each component in the kinematic chain, endowing the component with corresponding prime numbers according to the number of kinematic pairs of the component to construct a feature code M of the kinematic chain, comparing whether the feature codes of the kinematic chains are the same or not, if not, determining the kinematic chains are heterogeneous, and if so, performing the next step; constructing an asymmetric adjacency matrix A for the kinematic chains with the same characteristic codes; constructing a discrimination matrix D for each matrix A, calculating a sum array of the discrimination matrix D, then comparing whether the sum array of the discrimination matrix D of each kinematic chain is the same or not, if not, determining that the kinematic chains are heterogeneous, and if so, performing the next step; and calculating the characteristic value and the characteristic vector of the motion chain discrimination matrix D with the same array as the D, comparing whether the characteristic value and the characteristic vector of each motion chain are the same, if the characteristic value and the characteristic vector are different, determining that the motion chains are heterogeneous, and if the characteristic value and the characteristic vector are the same, determining that the motion chains are homogeneous. The invention is very simple and efficient in visual expression and discrimination.

Description

Kinematic chain isomorphism judgment method based on prime number asymmetric adjacent matrix
Technical Field
The invention relates to the technical field of mechanical kinematic chains, in particular to a kinematic chain isomorphism judgment method based on prime number asymmetric adjacency matrixes.
Background
In 1964, graph theory is introduced into the field of kinematic chain topological structure research, a topological graph represents a connecting rod by a vertex and represents a joint by an edge, and the relationship between a mechanism diagram and the topological graph is established. Due to the convenience of computers in matrix computation, graph theory provides a powerful mathematical tool in research and development of institutions. In the kinematic chain type synthesis, whichever method is used, it is very important to solve the uniqueness and comprehensiveness of the kinematic chain description, especially the kinematic chain containing multiple twists, which is a bottleneck problem in the field of mechanism topology research. In order to describe the composite hinge in the kinematic chain, students propose a two-color topological graph, a matrix identification and a conversion method. The size of the matrix constructed in this way varies.
In kinematic chain analysis, the description of the kinematic chain plays an important role. The information contained in the adjacency matrix describing the kinematic chain construction for conventional methods is limited and may fail for some highly symmetric kinematic chains. Other methods may have the disadvantages of wide application range, insufficient intuition of expression, complex determination method and the like. Therefore, a method for determining the isomorphism of the kinematic chain with simpler and more efficient determination is needed.
Disclosure of Invention
The invention aims to provide a method for judging the isomorphism of a kinematic chain based on a prime number asymmetric adjacency matrix, which can uniquely describe the structure of the kinematic chain and is simpler and more efficient in judging whether the kinematic chain is isomorphism.
The scheme adopted by the invention for solving the technical problems is as follows:
a method for judging isomorphism of a motion chain based on a prime number asymmetric adjacent matrix comprises the following steps:
s1: numbering each member in each kinematic chain, endowing each member in each kinematic chain with corresponding prime number according to the number of kinematic pairs of each member, and constructing a feature code M of each kinematic chain, wherein M is [ M ═ M1，m2，m3…mn]Wherein m isnComparing whether the feature codes of all the kinematic chains are the same or not for prime numbers corresponding to the number of kinematic pairs of the member with the number n, if not, determining that the kinematic chains are heterogeneous, and if so, performing the next step;
s2: constructing an asymmetric adjacent matrix A for the kinematic chains with the same feature codes in the step S1, assigning corresponding prime numbers to the kinematic chains according to the number of kinematic pairs of each member in each kinematic chain, and determining corresponding element values in the matrix according to whether every two members are adjacent to each other to obtain an n × n asymmetric adjacent matrix of each kinematic chain:
Figure BDA0002267902140000021
where i, j denotes the number of members in the kinematic chain, n denotes the number of members of the kinematic chain, the element a of the diagonal of the matrixi，j(i ═ j) is 0; other elements a of the matrixi，j(i ≠ j, i ≠ 1 … n, j ═ 1 … n) is a prime number corresponding to the number of kinematic pairs of construction j when construction i and construction j are adjacent; when member i and build j are not contiguous, then ai，j(i ≠ j, i ≠ 1 … n, j ═ 1 … n) has a value of zero;
s3: constructing a corresponding judgment matrix D, D being A x A', for each kinematic chain asymmetric adjacent matrix A constructed in the step S2, calculating a sum array of the judgment matrices D of each kinematic chain, then comparing whether the sum array of the judgment matrices D of each kinematic chain is the same, if not, determining that the kinematic chains are heterogeneous, and if so, performing the next step;
s4: and (4) operating the discrimination matrix D of the kinematic chain with the same array as the discrimination matrix D in the step (S3), calculating the characteristic value and the characteristic vector of the discrimination matrix D, comparing whether the characteristic value and the characteristic vector of each kinematic chain are the same, if the characteristic value and the characteristic vector are different, determining that the kinematic chains are heterogeneous, and if the characteristic values and the characteristic vectors are the same, determining that the kinematic chains are homogeneous.
Further, when comparing whether the feature codes of the motion chains are the same, firstly, the feature codes M of the motion chains are determined according to MnValue size for M in each feature code MnReordering is performed and comparison is performed.
Further, according to mnM in the feature code M of each kinematic chain is sequentially subjected to the sequence from small value to large value or from large value to small valuenA rearrangement is performed.
Further, whether the sum array of the discrimination matrix D of each kinematic chain is the same or not is compared, and according to the magnitude of each numerical value in the sum array of the discrimination matrix D, the numerical values in the sum array of the discrimination matrix D are rearranged from small to large or from large to small and then compared.
Further, when comparing whether the eigenvalue and the eigenvector of each kinematic chain are the same, the eigenvalue of the discrimination matrix D of each kinematic chain is rearranged in the order from small to large or from large to small and then compared.
Compared with the prior art, the invention has at least the following beneficial effects: the invention can clearly describe the difference of the adjacent relation between the members in the topological graph by using the prime number construction feature code and the corresponding adjacent matrix, and the method can uniquely describe the structure of the kinematic chain. The method realizes the one-to-one correspondence relationship between the kinematic chain diagram and the components, has intuitive expression, can judge whether the components are isomorphic without carrying out complex operation, and is very simple and efficient in judgment.
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FIG. 1 is a schematic view of an exercise apparatus a1 according to an embodiment of the present invention;
FIG. 2 is a schematic diagram of an exercise apparatus a2 according to an embodiment of the present invention;
fig. 3 is a schematic view showing the structure of the motion amount a3 according to the embodiment of the present invention.
Detailed Description
The following examples are provided to further illustrate the present invention for better understanding, but the present invention is not limited to the following examples.
The invention provides a kinematic chain isomorphism judging method based on a prime number asymmetric adjacent matrix, which comprises the following steps:
s1: three ten-bar kinematic chains with one degree of freedom of complex twist 1 as shown in figures 1-3 were analyzed. The method comprises the steps of firstly numbering the components in each kinematic chain, wherein the numbering of the components is not limited at all, and only the components need to be numbered in sequence. Corresponding prime numbers are given to the three kinematic chains according to the number of kinematic pairs of each component in the three kinematic chains, and the feature codes corresponding to the three kinematic chains, namely Ma1, Ma2 and Ma3, the kinematic chain a1, the kinematic chain a2 and the kinematic chain a3, are obtained as follows:
Ma1＝[5，3，2，2，2，2，2，2，2，3]
Ma2＝[5，3，2，2，2，2，2，2，2，3]
Ma3＝[5，3，2，2，2，2，2，3，2，2]；
since the members in the three kinematic chains are randomly numbered, rearranging the values in the signature does not affect the signature. In order to compare whether the feature codes of the three kinematic chains are the same, numerical values in the feature codes of the three kinematic chains may be rearranged from small to large or from large to small, in this embodiment, the feature codes are rearranged from large to small, so as to obtain the following feature codes corresponding to the three kinematic chains:
Ma1＝[5，3，3，2，2，2，2，2，2，2]
Ma2＝[5，3，3，2，2，2，2，2，2，2]
Ma3＝[5，3，3，2，2，2，2，2，2，2]；
through comparison, the feature codes of the three kinematic chains are the same, and whether the three kinematic chains are isomorphic cannot be judged, so that the next judgment is needed.
S2: directly converting the 3 complex-twisted kinematic chains into asymmetric matrixes, endowing the asymmetric matrixes with corresponding prime numbers according to the number of kinematic pairs of each member in each kinematic chain and determining corresponding element values in the matrixes according to whether every two members are adjacent to each other to obtain the n multiplied by n asymmetric adjacent matrixes of each kinematic chain:
Figure BDA0002267902140000041
where i, j denotes the number of members in the kinematic chain, n denotes the number of members of the kinematic chain, the element a of the diagonal of the matrixi，j(i ═ j) is 0; other elements a of the matrixi，j(i ≠ j, i ≠ 1 … n, j ═ 1 … n) is a prime number corresponding to the number of kinematic pairs of construction j when construction i and construction j are adjacent; when member i and build j are not contiguous, then ai，j(i ≠ j, i ≠ 1 … n, j ═ 1 … n) has a value of zero;
in this embodiment, the asymmetric adjacency matrix of exercise line a1 is:
Figure BDA0002267902140000051
asymmetric adjacency matrix Aa1Element a ofij(i ≠ j) denotes the connection relationship between member i and member j in kinematic chain a1, e.g. element a12A value of 3 indicates that the element number of member one adjacent to and building member two is 3 (i.e., the number of kinematic pairs for member 2). Element a21The value of 5 indicates that member two is adjacent to member one and the prime number of member one is 5 (i.e., the number of kinematic pairs for member 1).
Similarly, the asymmetric adjacency matrix of the kinematic chain a2 is:
Figure BDA0002267902140000052
the asymmetric adjacency matrix of kinematic chain a3 is:
Figure BDA0002267902140000053
s3: constructing a discrimination matrix D for the matrix A constructed by each kinematic chain in the step S2, wherein D is A x A', calculating the sum array of the discrimination matrix D of each kinematic chain, comparing whether the sum array of the discrimination matrix D of each kinematic chain is the same or not, if not, determining that the kinematic chains are heterogeneous, and if so, performing the next step;
the decision matrices and the sum-matrix arrays of the decision matrices for a1, a2, and a3 in this example are shown below:
Figure BDA0002267902140000061
Figure BDA0002267902140000062
Figure BDA0002267902140000063
in order to compare whether the sum arrays of the three kinematic chains are the same or not, the numerical values in the sum arrays of the decision matrixes D of the 3 kinematic chains are rearranged from large to small to obtain the sum arrays H. Of course, the values in the sum array of 3 kinematic chains can be rearranged from small to large.
Ha1＝[136 128 120 108 83 79 72 44 39 36]
Ha2＝[136 128 120 108 83 79 72 44 39 36]
Ha3＝[136 124 112 108 79 79 72 55 44 36]
Therefore, it can be seen from the above information that Ha1And Ha2Are identical and are all identical to Ha3Different. Thus, the kinematic chains a1 and a2 are not isomorphic with the kinematic chain a 3. Due to Ha1And Ha2Similarly, it cannot be determined whether the kinematic chains a1 and a2 are homogeneous, and the next step is to determine whether the kinematic chains a1 and a2 are homogeneous.
S4: and (4) operating the discrimination matrix D of the kinematic chain with the same array as the discrimination matrix D in the step (S3), calculating the characteristic value and the characteristic vector of the discrimination matrix D, comparing whether the characteristic value and the characteristic vector of the discrimination matrix D of each kinematic chain are the same, if the characteristic value and the characteristic vector are different, determining that the kinematic chains are heterogeneous, and if the characteristic value and the characteristic vector are the same, determining that the kinematic chains are homogeneous.
In the present embodiment, the discrimination matrix D of the kinematic chains a1 and a2 is calculateda1And Da2The eigenvalue and eigenvector of (a) can be calculated by matlab to obtain the eigenvalue Ta1 and eigenvector Sa1 of the kinematic chain a1 and the eigenvalue Ta2 and eigenvector Sa2 of the kinematic chain a2 as follows:
Figure BDA0002267902140000071
as can be seen from the table, the eigenvalues and eigenvectors of the decision matrix D for the kinematic chain a1 and the kinematic chain a2 are the same, and thus the kinematic chain a1 and the kinematic chain a2 are isomorphic. At this point, the determination of whether the kinematic chain is homogeneous is ended.
While the foregoing is directed to the preferred embodiment of the present invention, other and further embodiments of the invention may be devised without departing from the basic scope thereof, and the scope thereof is determined by the claims that follow.

Claims (5)

1. A method for judging isomorphism of a motion chain based on a prime number asymmetric adjacency matrix is characterized by comprising the following steps:
s1: numbering each member in each kinematic chain, endowing each member in each kinematic chain with corresponding prime number according to the number of kinematic pairs of each member, and constructing a feature code M of each kinematic chain, wherein M is [ M ═ M1，m2，m3…mn]Wherein m isnComparing whether the feature codes of all the kinematic chains are the same or not for prime numbers corresponding to the number of kinematic pairs of the member with the number n, if not, determining that the kinematic chains are heterogeneous, and if so, performing the next step;
s2: constructing an asymmetric adjacent matrix A for the kinematic chains with the same feature codes in the step S1, assigning corresponding prime numbers to the kinematic chains according to the number of kinematic pairs of each member in each kinematic chain, and determining corresponding element values in the matrix according to whether every two members are adjacent to each other to obtain an n × n asymmetric adjacent matrix of each kinematic chain:
Figure FDA0002267902130000011
where i, j denotes the number of members in the kinematic chain, n denotes the number of members of the kinematic chain, the element a of the diagonal of the matrixi，j(i ═ j) is 0; other elements a of the matrixi，j(i ≠ j, i ≠ 1 … n, j ═ 1 … n) is a prime number corresponding to the number of kinematic pairs of construction j when construction i and construction j are adjacent; when member i and build j are not contiguous, then ai，j(i ≠ j, i ≠ 1 … n, j ═ 1 … n) has a value of zero;
s3: constructing a corresponding judgment matrix D, D being A x A', for each kinematic chain asymmetric adjacent matrix A constructed in the step S2, calculating a sum array of the judgment matrices D of each kinematic chain, then comparing whether the sum array of the judgment matrices D of each kinematic chain is the same, if not, determining that the kinematic chains are heterogeneous, and if so, performing the next step;
s4: and (4) operating the discrimination matrix D of the kinematic chain with the same array as the discrimination matrix D in the step (S3), calculating the characteristic value and the characteristic vector of the discrimination matrix D, comparing whether the characteristic value and the characteristic vector of each kinematic chain are the same, if the characteristic value and the characteristic vector are different, determining that the kinematic chains are heterogeneous, and if the characteristic values and the characteristic vectors are the same, determining that the kinematic chains are homogeneous.
2. The method of claim 1, wherein the comparing of the identity codes of the motion chains is performed according to M of the identity codes M of the motion chainsnValue size for M in each feature code MnThe comparison is performed after reordering in a certain order.
3. The method for determining the isomorphism of a kinematic chain based on a prime asymmetric adjacency matrix according to claim 2, wherein m is the numbernM in the feature code M of each kinematic chain is sequentially subjected to the sequence from small value to large value or from large value to small valuenA rearrangement is performed.
4. The method for determining the isomorphism of kinematic chains based on the asymmetric adjacency matrix with prime numbers as claimed in claim 1, wherein the sum matrix of the decision matrix D of each kinematic chain is compared to determine whether the sum matrix is the same, and the values in the sum matrix of the decision matrix D are rearranged from small to large or from large to small according to the values in the sum matrix of the decision matrix D and then compared.
5. The method for determining the isomorphism of a kinematic chain based on a prime number asymmetric adjacency matrix according to claim 1, wherein when comparing whether the eigenvalue and the eigenvector of each kinematic chain are the same, the eigenvalue of the discrimination matrix D of each kinematic chain is rearranged in the order from small to large or from large to small and then compared.
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