CN111104563B - Method for determining isomorphism of kinematic chain based on prime number asymmetric adjacency matrix - Google Patents
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Abstract
The invention relates to a kinematic chain isomorphism judging method based on a prime asymmetric adjacency matrix, which comprises the following steps: numbering each component in the motion chain, assigning corresponding prime numbers to the components according to the number of the motion pairs of the components to construct a feature code M of the motion chain, comparing whether the feature codes of the motion chains are identical, if not, isomerism is realized, and if so, performing the next step; constructing an asymmetric adjacency matrix A for the motion chains with the same feature codes; constructing a discrimination matrix D for each matrix A, calculating the sum array of the discrimination matrix D, comparing whether the sum array of the discrimination matrix D of each motion chain is identical or not, if not, carrying out isomerism, and if so, carrying out the next step; and calculating the eigenvalue and the eigenvector of the discrimination matrix D of the motion chains which are the same as the array, comparing whether the eigenvalue and the eigenvector of each motion chain are the same or not, if so, carrying out isomerism, and if so, carrying out isomorphism. The invention has the advantages of visual expression and discrimination, and is very simple and efficient.
Description
Technical Field
The invention relates to the technical field of mechanical kinematic chains, in particular to a kinematic chain isomorphism judging method based on a prime asymmetric adjacency matrix.
Background
In 1964, graph theory was introduced into the field of kinematic chain topology structure research, and the relationship between a mechanism diagram and a topology diagram is established by using a vertex to represent a connecting rod and a side to represent a joint. Because of the convenience of the computer in matrix calculation, graph theory provides a powerful mathematical tool in the research and development of institutions. In the motion chain type synthesis, whatever the method, it is very important to solve the uniqueness and comprehensiveness of the motion chain description, especially the motion chain with double twisting, which is a bottleneck problem in the field of mechanism topology research. To describe the compound hinges in the kinematic chain, students propose a method of bi-color topology, matrix identification and conversion. The size of the matrix constructed in this way changes.
In the kinematic chain analysis, the description of the kinematic chain plays an important role. The information contained in the adjacency matrix describing the motion chain construction is limited for conventional methods and may fail for some highly symmetric motion chains. Other methods may have the disadvantages of application range, insufficient visual expression, complex judgment method, and the like. Therefore, a motion chain isomorphism determination method with simpler and more efficient determination needs to be found.
Disclosure of Invention
The invention aims to provide a method for determining isomorphism of a moving chain based on a prime number asymmetric adjacency matrix, which can uniquely describe the structure of the moving chain and determine whether the moving chain is isomorphic or not, and is simpler and more efficient.
The invention solves the technical problems by adopting the following scheme:
a kinematic chain isomorphism judging method based on prime asymmetric adjacency matrix comprises the following steps:
s1: numbering each component in each moving chain, assigning corresponding prime numbers to each component in each moving chain according to the number of kinematic pairs of each component, and constructing feature codes M, M= [ M ] of each moving chain 1 ，m 2 ，m 3 …m n ]Wherein m is n Comparing whether the feature codes of all the motion chains are identical or not for the prime numbers corresponding to the number of the motion pairs of the component with the number of n, if not, isomerism is achieved, and if so, performing the next step;
s2: constructing an asymmetric adjacency matrix A for the kinematic chains with the same feature codes in the step S1, assigning corresponding prime numbers to the kinematic pairs of each component in each kinematic chain according to the number of the kinematic pairs, and determining corresponding element values in the matrix according to whether the two components are adjacent to each other or not to obtain an n multiplied by n asymmetric adjacency matrix of each kinematic chain:
wherein i, j represents the number of members in the kinematic chain, n represents the number of members in the kinematic chain, and the element a of the diagonal of the matrix i，j (i=j) is 0; other elements a of the matrix i，j (i+.j, i= … n, j= … n) is the prime number corresponding to the number of j kinematic pairs constructed when the component i and the construction j are contiguous; when building block i and build j are not contiguous then a i，j The value of (i+.j, i= … n, j= … n) is zero;
s3: constructing a corresponding discrimination matrix D for each motion chain asymmetric adjacent matrix A constructed in the step S2, wherein D=A' and calculating the sum array of the discrimination matrices D of each motion chain, and then comparing whether the sum array of the discrimination matrices D of each motion chain is identical or not, if not, the discrimination matrices D of each motion chain are heterogeneous, and if so, the next step is carried out;
s4: and (3) calculating a discrimination matrix D of the motion chains with the same discrimination matrix D as the array in the step S3, calculating the eigenvalue and the eigenvector of the discrimination matrix D, comparing whether the eigenvalue and the eigenvector of each motion chain are the same or not, if different, the motion chains are heterogeneous, and if the motion chains are the same, the motion chains are isomorphic.
Further, when comparing whether the feature codes of the motion chains are identical, firstly, according to M in the feature codes M of the motion chains n The value size is relative to M in each feature code M n Reordered and then compared.
Further, according to m n The value of M in the feature code M of each motion chain is from small to large or from large to small n And (5) rearrangement is carried out.
Further, in comparing whether the sum arrays of the discrimination matrices D of the respective kinematic chains are identical, the respective values in the sum arrays of the discrimination matrices D are rearranged from small to large or from large to small according to the respective values in the sum arrays of the discrimination matrices D, and then compared.
Further, when comparing whether the eigenvalues and eigenvectors of the motion chains are the same, the eigenvalues of the discrimination matrix D of each motion chain are rearranged in 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 uses prime numbers to construct feature codes and corresponding adjacency matrix to clearly describe the adjacency relation difference between components in topological graph. The method realizes the one-to-one correspondence between the kinematic chain diagram and the components, is visual in expression, can judge whether isomorphism is realized without complex operation, and is very simple and efficient in judgment.
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FIG. 1 is a schematic view of an exercise training a1 according to an embodiment of the present invention;
FIG. 2 is a schematic diagram of an exercise training a2 according to an embodiment of the present invention;
fig. 3 is a schematic view of the structure of the exercise amount a3 according to the embodiment of the present invention.
Detailed Description
For a better understanding of the present invention, the following examples are further illustrative of the present invention, but the contents of the present invention are not limited to the following examples only.
The invention provides a kinematic chain isomorphism judging method based on a prime asymmetric adjacency matrix, which comprises the following steps:
s1: three ten-bar kinematic chains with one degree of freedom in the double twist of 1 as shown in fig. 1-3 were analyzed. Firstly, the components in each motion chain are numbered, the number of the components is not limited, and the components only need to be numbered in sequence. And then, according to the number of kinematic pairs of each component in the three kinematic chains, corresponding prime numbers are given to the kinematic pairs, so that the feature codes Ma1, ma2 and Ma3 of the three kinematic chains are obtained, and the feature codes corresponding to the kinematic chain a1, the kinematic chain a2 and the kinematic chain a3 are 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 components in the three kinematic chains are arbitrarily numbered, rearranging the values in the feature codes does not affect the feature codes. In order to facilitate comparing whether the feature codes of the three motion chains are the same, the values in the feature codes of the three motion chains can be rearranged from small to large or from large to small, and in this embodiment, the feature codes are rearranged from large to small to obtain the following feature codes corresponding to the three motion 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]；
by comparison, it can be found that the feature codes of the three motion chains are the same, and whether isomorphism exists cannot be determined, so that the next determination is needed.
S2: the 3 kinematic chains containing the compound strands are directly converted into an asymmetric matrix, corresponding prime numbers are given to the kinematic pairs of each component in each kinematic chain according to the number of the kinematic pairs, and corresponding element values in the matrix are determined according to whether every two components are adjacent to each other, so that an n multiplied by n asymmetric adjacent matrix of each kinematic chain is obtained:
wherein i, j represents the number of members in the kinematic chain, n represents the number of members in the kinematic chain, and the element a of the diagonal of the matrix i，j (i=j) is 0; other elements a of the matrix i，j (i+.j, i= … n, j= … n) is the prime number corresponding to the number of j kinematic pairs constructed when the component i and the construction j are contiguous; when building block i and build j are not contiguous then a i，j The value of (i+.j, i= … n, j= … n) is zero;
in this embodiment, the asymmetric adjacency matrix of exercise a1 is:
asymmetric adjacency matrix A a1 Element a of (2) ij (i.noteq.j) represents the structure in the motion chain a1Connection between member i and member j, e.g. element a 12 The value of (2) is 3, which means that the first member is adjacent to the second member and the element value of the second member is 3 (i.e., the number of kinematic pairs of the second member). Element a 21 A value of 5 indicates that the member two is adjacent to the member one and the element value of the member one is 5 (i.e., the number of kinematic pairs of the member 1).
Similarly, the asymmetric adjacency matrix of the kinematic chain a2 is:
the asymmetric adjacency matrix of the kinematic chain a3 is:
s3: constructing a discrimination matrix D for the matrix A constructed by each moving chain in the step S2, wherein D=A, calculating the sum array of the discrimination matrices D of each moving chain, comparing whether the sum arrays of the discrimination matrices D of each moving chain are identical or not, if not, isomerism is realized, and if so, the next step is carried out;
in this embodiment, the discrimination matrices of a1, a2 and a3 and the sum array of the discrimination matrices are respectively as follows:
in order to facilitate comparison of whether the sum arrays of the three motion chains are identical, the numerical values in the sum arrays of the discrimination matrices D of the 3 motion chains are rearranged from large to small to obtain a sum array H as follows. Of course, the values in the sum array of the 3 kinematic chains can also be rearranged from small to large.
H a1 ＝[136 128 120 108 83 79 72 44 39 36]
H a2 ＝[136 128 120 108 83 79 72 44 39 36]
H a3 ＝[136 124 112 108 79 79 72 55 44 36]
Thus, it can be seen from the above information that H a1 And H is a2 Identical to H a3 Different. Thus, both kinematic chains a1 and a2 are different from kinematic chain a 3. Due to H a1 And H is a2 It is therefore impossible to determine whether the moving chains a1 and a2 are isomorphic, and next the next step is to determine whether the moving chains a1 and a2 are isomorphic.
S4: and (3) calculating the discrimination matrix D of the motion chain which is the same as the array in the step (S3), calculating the eigenvalue and the eigenvector of the discrimination matrix D, comparing whether the eigenvalue and the eigenvector of the discrimination matrix D of each motion chain are the same or not, if different, the discrimination matrix D is heterogeneous, and if the eigenvalue and the eigenvector are the same, the discrimination matrix D is isomorphic.
In the present embodiment, the discrimination matrix D of the motion chains a1 and a2 is calculated a1 And D a2 Can be calculated by matlab to obtain the following characteristic value Ta1 and characteristic vector Sa1 of the motion chain a1 and the following characteristic value Ta2 and characteristic vector Sa2 of the motion chain a2:
as can be seen from the table, the eigenvalues and eigenvectors of the discrimination matrices D of the kinematic chain a1 and the kinematic chain a2 are the same, so the kinematic chain a1 and the kinematic chain a2 are isomorphic. So far, the determination of whether the kinematic chain is isomorphic is ended.
While the invention has been described with respect to the preferred embodiments, it will be understood that the invention is not limited thereto, but is capable of modification and variation without departing from the spirit of the invention, as will be apparent to those skilled in the art.
Claims (5)
1. A kinematic chain isomorphism judging method based on a prime number asymmetric adjacency matrix is characterized by comprising the following steps:
s1: numbering each component in each moving chain, assigning corresponding prime numbers to each component in each moving chain according to the number of kinematic pairs of each component, and constructing feature codes M, M= [ M ] of each moving chain 1 ，m 2 ，m 3 …m n ]Wherein m is n Comparing whether the feature codes of all the motion chains are identical or not for the prime numbers corresponding to the number of the motion pairs of the component with the number of n, if not, isomerism is achieved, and if so, performing the next step;
s2: constructing an asymmetric adjacency matrix A for the kinematic chains with the same feature codes in the step S1, assigning corresponding prime numbers to the kinematic pairs of each component in each kinematic chain according to the number of the kinematic pairs, and determining corresponding element values in the matrix according to whether the two components are adjacent to each other or not to obtain an n multiplied by n asymmetric adjacency matrix of each kinematic chain:
wherein i, j represents the number of members in the kinematic chain, n represents the number of members in the kinematic chain, and the element a of the diagonal of the matrix i，j (i=j) is 0; other elements a of the matrix i，j (i+.j, i= … n, j= … n) is the prime number corresponding to the number of j kinematic pairs constructed when the component i and the construction j are contiguous; when building block i and build j are not contiguous then a i，j The value of (i+.j, i= … n, j= … n) is zero;
s3: constructing a corresponding discrimination matrix D for each motion chain asymmetric adjacent matrix A constructed in the step S2, wherein D=A' and calculating the sum array of the discrimination matrices D of each motion chain, and then comparing whether the sum array of the discrimination matrices D of each motion chain is identical or not, if not, the discrimination matrices D of each motion chain are heterogeneous, and if so, the next step is carried out;
s4: and (3) calculating a discrimination matrix D of the motion chains with the same discrimination matrix D as the array in the step S3, calculating the eigenvalue and the eigenvector of the discrimination matrix D, comparing whether the eigenvalue and the eigenvector of each motion chain are the same or not, if different, the motion chains are heterogeneous, and if the motion chains are the same, the motion chains are isomorphic.
2. The method for determining isomorphism of moving chains based on prime number asymmetric adjacency matrix as recited in claim 1, wherein M in the feature code M of each moving chain is first determined according to whether the feature codes of each moving chain are identical or not n The value size is relative to M in each feature code M n Reordered in a certain order and compared.
3. The method for determining isomorphism of a moving chain based on a prime number asymmetric adjacency matrix as claimed in claim 2, characterized in that according to m n The value of M in the feature code M of each motion chain is from small to large or from large to small n And (5) rearrangement is carried out.
4. The method for determining the isomorphism of a moving chain based on a prime number asymmetric adjacency matrix according to claim 1, wherein, when comparing whether the sum array of the discrimination matrix D of each moving chain is the same, the values in the sum array of the discrimination matrix D are rearranged from small to large or from large to small according to the values in the sum array of the discrimination matrix D, and then compared.
5. The method for determining the isomorphism of a moving chain based on a prime number asymmetric adjacency matrix according to claim 1, wherein when comparing whether the eigenvalues and eigenvectors of the moving chains are identical, the eigenvalues of the determination matrix D of each moving chain are rearranged in order from small to large or from large to small, and then compared.
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