Graph theory-based planetary gear mechanism configuration synthesis method for floor grinding machine

Abstract

The invention provides a construction synthesis method for a planetary gear mechanism of a floor grinding machine based on a graph theory, which comprises the following steps of: firstly, deeply analyzing the planetary gear mechanism, determining a quantitative relationship of basic elements, and obtaining a new quantitative relationship of the elements through higher pairs and lower generations; next, adjacency matrixes of all configurations are generated through adjacency matrix interpolation, and various unreasonable monochromatic topological graphs are removed on the basis; configuration synthesis of monochromatic topological graphs is carried out according to a standard adjacency matrix set method, monochromatic topological graphs which do not meet conditions are eliminated by combining the length requirement of the longest transmission chain of the planetary gear mechanism, and finally position points suitable for higher pairs and composite hinges are selected to generate a multi-color topological graph. And performing standard point numbering and one-to-one correspondence verification on the multi-color topological graph to complete the whole configuration comprehensive process. According to the method, the configuration integration of the transmission coefficient of the complex planetary gear is realized, and the comprehensiveness and high efficiency are ensured.

Description

Technical Field

[0001] The present invention relates to the technical field of planetary gears, and in particular to a configuration synthesis method of a planetary gear mechanism of a floor grinder based on graph theory, which is particularly suitable for configuration design and optimization of a complex planetary gear transmission system. Background Art

[0002] Planetary gear mechanisms are widely used in automotive transmissions, wind turbines, aerospace equipment, industrial robots and other fields due to their compact structure, large transmission ratio and strong load-bearing capacity. However, with the increasing complexity of application scenarios, the configuration design of planetary gear mechanisms faces more and more challenges. Traditional configuration design methods mainly rely on empirical formulas and trial and error methods, lacking systematic theoretical support, resulting in low design efficiency, lack of innovation, and difficulty in meeting high complexity and high performance requirements.

[0003] In recent years, graph theory, as a powerful mathematical tool, has been gradually applied in the design of mechanical system configurations. Graph theory can abstract complex mechanical structures into a combination of nodes and edges, thus providing a systematic and visual analysis method.

[0004] In view of this, the inventor of this case conducted in-depth research on the above-mentioned issues, which led to the emergence of this case. Summary of the invention

[0005] The present invention aims to solve the technical problems existing in the prior art. The present invention provides a configuration synthesis method of a planetary gear mechanism of a floor grinder based on graph theory. The method realizes the configuration synthesis of complex planetary gear transmission coefficients and ensures comprehensiveness and high efficiency.

[0006] The present invention is implemented as follows: A comprehensive method for the configuration of a planetary gear mechanism of a floor grinding machine based on graph theory: (1) Analyze the planetary gear mechanism to obtain the basic element quantity relationship, obtain the new element quantity relationship based on the high-pair low-generation, and use the adjacency matrix interpolation to obtain the adjacency matrix of all configurations; (2) Delete the monochrome topological graphs containing discrete points, and delete the unreasonable monochrome topological graphs based on high pairs and compound hinges; (3) Calculate all loops of each monochrome topology graph, that is, each monochrome topology graph corresponds to a loop set, and delete the separable monochrome topology graphs through the loop set; (4) Calculate independent loops through the loop sets of the remaining monochrome topological graphs, and determine and delete the monochrome topological graphs containing rigid subchains; (5) Perform configuration synthesis of monochrome topological graphs based on the method of canonical adjacency matrix sets; (6) Based on the length requirement of the longest transmission chain of the planetary gear mechanism, the single-color topological diagrams that do not meet the requirements are deleted, and the position points suitable for the high pair and compound hinge are selected to generate a multi-color topological diagram; (7) According to the standard labeling method, the multi-color topological diagram is configured and synthesized.

[0007] Furthermore, in step (1), the planetary gear mechanism is first analyzed to determine the number of components, low pairs, high pairs and compound hinges, and then the high pair is substituted for the low pair, and then the number of components, low pairs and compound hinges is determined, and the components and compound hinges are drawn as vertices and the low pairs are drawn as edges; according to the number of components, low pairs, high pairs and compound hinges, the corresponding zero matrix is ​​generated, and then 1 is inserted into the longest side of the upper triangular matrix, and 1 is inserted into the remaining positions of the upper triangular matrix, which is related to the number of components, low pairs, high pairs and compound hinges, and then the lower triangular matrix is ​​symmetric according to the upper triangular matrix, thereby obtaining the adjacency matrix of all configurations.

[0008] Furthermore, in step (2), the graphs containing vertices with degree 1 or less are excluded, that is, according to the adjacency matrix, the number of non-zero elements in each row (or column) is searched row (or column). If the number of non-zero elements in a row (or column) is 0 or 1, the monochrome topology graph is deleted, and the monochrome topology graphs whose points do not meet the degree requirements are deleted according to the number of high pairs and compound hinges.

[0009] Furthermore, in step (3), all loop sets are obtained through the adjacency matrix, and separable monochrome topological graphs are deleted, that is, monochrome topological graphs in which two loops are related by only one point or one edge are deleted.

[0010] Further, in step (4), the loop set is used to utilize the loop The calculation is independent of the loop, and the rigid subchain is used to identify if There is no rigid subchain, where is the activity factor group, , is the number of independent loops. Further, in step (5), the perimeter monochrome topology map of the kinematic chain is drawn, and the canonical perimeter monochrome topology map set is obtained through the perimeter monochrome topology map, and then the canonical adjacency matrix set corresponding to the canonical perimeter monochrome topology map is obtained, and it is determined whether the intersection of the canonical adjacency matrix sets of the two kinematic chains is non-empty. If it is non-empty, they are isomorphic.

[0011] Furthermore, in step (6), the length of the longest transmission chain of the planetary gear transmission mechanism is determined, and a reasonable single-color topology diagram is screened out by length, and then coloring is performed according to the existence relationship between the high pair and the compound hinge, that is, the high pair is a point with a degree of 2, and the compound hinge is a point with a degree of a few according to the actual planetary gear transmission mechanism, to obtain a multi-color topology diagram.

[0012] Furthermore, in step (7), the above-mentioned standard labeling method is continued to be used to standardize the labeling of the multi-color diagrams to determine whether the position points of the high pairs and compound hinges of the two diagrams correspond one to one. If so, they are isomorphic, otherwise they are not.

[0013] Compared with the existing technology, the present invention has the following effective effects: the configuration synthesis method proposed in the present invention can systematically, comprehensively and effectively solve the problem of configuration synthesis of complex planetary gear transmission mechanisms, and can realize the automation of configuration synthesis by means of computer programming, thereby reducing the low efficiency and easy omissions in manual mechanism synthesis, and improving the efficiency and accuracy of configuration synthesis. BRIEF DESCRIPTION OF THE DRAWINGS

[0014] The present invention will be further described below in conjunction with embodiments with reference to the accompanying drawings.

[0015] Figure 1 It is a flow chart of the comprehensive method for the configuration of the planetary gear mechanism of the floor grinder based on graph theory of the present invention.

[0016] Figure 2 2 is a simplified motion diagram of an embodiment of the present invention.

[0017] Figure 3 It is a monochrome topological diagram corresponding to the motion diagram of the embodiment of the present invention.

[0018] Figure 4 It is a modified monochrome topological diagram corresponding to the motion diagram of the embodiment of the present invention.

[0019] Figure 5 It is a monochrome topological map containing discrete points of the present invention.

[0020] Figure 6 It is a separable monochrome topological map of the present invention.

[0021] Figure 7 It is a monochrome topological diagram containing rigid subchains of the present invention.

[0022] Figure 8 It is a monochrome topological diagram of the independent loop calculation case of the present invention.

[0023] Fig. 9 It is the original monochrome topological diagram of the case of calculating the canonical adjacency matrix set of the present invention.

[0024] Fig.10 It is a canonical monochrome topological graph of the case of calculating the canonical adjacency matrix set in the present invention.

[0025] Fig.11 It is the comprehensive result of the monochrome topological graph configuration.

[0026] Fig.12 It is the result of monochrome topology map filtered according to the transmission chain.

[0027] Fig.13 This is a multi-color topological diagram after adding high pairs and compound hinges.

[0028] Fig.14 It is the comprehensive result of the multi-color topological graph configuration. DETAILED DESCRIPTION

[0029] In order to better understand the technical solution of the present invention, the technical solution of the present invention will be described in detail below in conjunction with the accompanying drawings and specific implementation methods.

[0030] Reference Figure 1-14 As shown, the present invention provides a comprehensive method for the configuration of a planetary gear mechanism of a floor grinding machine based on graph theory: (1) The planetary gear mechanism is analyzed to obtain the quantitative relationship of the basic elements. A new quantitative relationship of the elements is obtained based on the high-pair low-generation method, and the adjacency matrix of all configurations is obtained by adjacency matrix interpolation.

[0031] (2) Delete the monochrome topological graphs containing discrete points, and delete the unreasonable monochrome topological graphs based on high pairs and compound hinges.

[0032] (3) All loops of each monochrome topology graph are calculated, that is, each monochrome topology graph corresponds to a loop set, and the separable monochrome topology graphs are deleted through the loop set.

[0033] (4) Calculate independent loops through the loop set of the remaining monochrome topological graphs, and determine and delete the monochrome topological graphs containing rigid subchains.

[0034] (5) Perform configuration synthesis based on the method of canonical adjacency matrix sets.

[0035] (6) Based on the length requirement of the longest transmission chain of the planetary gear mechanism, the single-color topological diagrams that do not meet the requirements are deleted, and the position points suitable for the high pair and compound hinge are selected to generate a multi-color topological diagram.

[0036] (7) According to the standard labeling method, the multi-color topological diagram is configured and synthesized.

[0037] Preferably, in step (1), the planetary gear mechanism is first analyzed to determine the number of components, low pairs, high pairs and compound hinges, and then the high pair is replaced by the low pair, and then the number of components, low pairs and compound hinges is determined, and the components and compound hinges are drawn as vertices and the low pairs are drawn as edges. The embodiment of the present invention takes a planetary gear mechanism with 5 components, 8 low pairs, 3 high pairs and 1 compound hinge as an example, and refers to Figure 2-4 As shown in the figure, the motion diagram and monochrome topology diagram of the embodiment of the present invention are generated, a 9*9 zero matrix is ​​generated, and then the longest side of the upper triangular matrix is ​​inserted 1, insert in the remaining positions of the upper triangular matrix 1, The range is from 6 to 11, and then the lower triangular matrix is ​​symmetric according to the upper triangular matrix, thus obtaining the adjacency matrix of all configurations.

[0038] Preferably, refer to Figure 5 As shown, in step (2), the monochrome topological graphs containing vertices with a degree of 1 or less are excluded, that is, according to the adjacency matrix, the number of non-zero elements in each row (or column) is searched row (or column). If the number of non-zero elements in a row (or column) is 0 or 1, the monochrome topological graph is deleted, and the monochrome topological graphs whose points do not meet the degree requirements are deleted according to the number of high pairs and compound hinges.

[0039] Preferably, refer to Figure 6 As shown, in step (3), all loop sets are obtained through the adjacency matrix, and the separable monochrome topological graphs are deleted through the relationship between the loops in the loop set, that is, the monochrome topological graphs in which two loops are related by only one point or one edge are deleted.

[0040] Preferably, in step (4), the loop set is used to utilize the loop Operation and calculation independent loop.

[0041] Assume that a certain configuration monochrome topology graph contains loop 1 as 1, 2, 3, 8, 7, 1; then ; Representation ring The number of vertices in ; loop " The operation is defined as The result of the operation is recorded as .

[0042] “ "Operation rules: Ring Each element of the array and the ring The corresponding elements of are algebraically calculated. If the difference is greater than 0, then The value of the corresponding bit is 1; otherwise, its value is 0, that is, .

[0043] In the formula For the ring The first Bit element; For the ring The first Bit element; For the ring and “ After the operation The value of the bit element.

[0044] Loop The operation is defined as The result of the operation is recorded as .

[0045] “ "Operation rules: comparison ring Each element of the array and the ring If the corresponding bit elements of are all 1, then The value of the corresponding bit is 1; otherwise, its value is 0, that is, .

[0046] In the formula For the ring The first Bit element; For the ring The first Bit element; For the ring and “ After the operation The value of the bit element.

[0047] Loop The operation is defined as The result of the operation is recorded as .

[0048] “ "Operation rules: Ring Each element of the array and the ring The corresponding bit elements of are algebraically summed. If the sum is greater than 0 and less than the local degree of the corresponding bit vertex, then The value of the corresponding bit is 1; otherwise, its value is 0, that is, In the formula For the ring The first Bit element; For the ring The first Bit element; For the The local degree of the vertices. If the array after the operation is also a ring, then it can be used Indicates that there is a ring and ring A combined loop.

[0049] Local degree of vertex: After the corresponding monochrome topological graph is processed as follows, the degree of the vertex obtained is its local degree.

[0050] 1. In the monochrome topology graph, remove the vertices that are not in any operation ring and the connection relationship corresponding to this vertex.

[0051] 2. Remove the inner connection relationship of each ring participating in the operation.

[0052] Loop "The existence conditions of the operation: 1. The two loops participating in the operation have at least two common points, that is, .

[0053] 2. During the operation, the following situation occurs and only occurs twice, the corresponding bit element values ​​of the two loops are 1 and " "After the operation, the corresponding bit element of the resulting array is still 1.

[0054] Loop and of" The operation is only possible if the two conditions for its operation are met. It makes sense, at this time ;otherwise It doesn't make sense.

[0055] Loop "Nature of Operation: Loop" The operation satisfies the commutative law, that is, .

[0056] Loop The operation satisfies the associative law, that is, .

[0057] Loop The operation satisfies the self-elimination law, that is, ; It is an n-dimensional zero array, where n is the number of vertices of the monochrome topology graph.

[0058] Loop The operation satisfies the absorption law, that is, .

[0059] For a monochrome topological graph with v vertices, e edges, and L independent loops, the following Euler relationship exists among the three: , embodiments of the present invention , using loops through loop sets The independent loop can be calculated by operation.

[0060] Reference Figure 7 As shown, select the loop set , , , The remaining loop sets are , , , .

[0061] .

[0062] .

[0063] .

[0064] Knowable Cycle Set It is the independent loop of this monochrome topology diagram.

[0065] Reference Figure 8 As shown, the independent loop of the known monochrome topology , , , , through the rigid subchain judgment, if , then there is no rigid subchain, where is the activity factor group, , is the number of independent loops. The steps are as follows: Step 4.1. Determine the number of independent loops , , , .

[0066] Step 4.2. Select the ring , and obtain its activity factor ; , .

[0067] Step 4.3. Select the ring , and obtain its activity factor relative to the first ring, ,in ; , , .

[0068] Step 4.4. Select the ring , and obtain its activity factor relative to the first two rings, ,in , , , .

[0069] judge , and then determine whether it contains a rigid subchain.

[0070] , so refer to Figure 5 Contains rigid subchains.

[0071] Preferably, refer to Fig. 9As shown, in step (5), the canonical sequence string of the kinematic chain loop is obtained. The canonical sequence string of the loop is only related to the degree of the vertex. According to the loop sequence number, the degree corresponding to the vertex is written at the corresponding position of the vertex. The one with the largest degree is placed first, which is the canonical sequence string. Figure 6 The configuration monochrome topology graph contains loop 1 of 2,7,6,5,1,4,3,2, and the corresponding degrees are 3,3,2,2,4,2,2, so the standard sequence string is 4223322.

[0072] The longest canonical sequence string is defined as the canonical degree sequence of the circumference ring of the kinematic chain, and the corresponding ring is the circumference ring of the kinematic chain.

[0073] Monochrome topology of the perimeter of a kinematic chain: When making a representation of the monochrome topology of a kinematic chain, follow these rules: 1. Place the perimeter ring at the outermost edge to form a regular polygon; 2. Place the remaining vertices that are not on the perimeter ring inside the regular polygon.

[0074] The above can obtain the monochrome topological map of the circumference of the kinematic chain.

[0075] Canonical vertex labeling of perimeter monochrome topological graphs: canonical vertex labeling of perimeter rings and canonical vertex labeling of internal subchains.

[0076] The canonical vertex numbering of the perimeter ring: 1. Numbering from small to large according to the canonical degree sequence of the perimeter ring; Figure 6 The standard sequence string of the perimeter ring of the configuration monochrome topological graph is 4223322, and the corresponding vertex numbers are 1, 2, 3, 4, 5, 6, 7.

[0077] 2. If the starting point corresponding to the canonical degree sequence is not unique, then the vertex connected to the largest inner subchain is selected from these vertices as the starting vertex for canonical labeling.

[0078] Description of the inner subchain: If an inner subchain is marked with m and n on the perimeter ring ( ) are connected, then use the array To represent this inner subchain, this inner subchain is also called Sub-chain.

[0079] Level of inner subchains: For two inner subchains and , we stipulate that the inner subchain with more vertices has a higher rank than the inner subchain with fewer vertices. If the two inner subchains contain the same number of vertices, then if , then the inner subchain The level is high, and vice versa; if ,and , then the inner subchain The level is high and vice versa.

[0080] Standard vertex numbering of inner sub-chains: numbering according to the level of the inner sub-chains.

[0081] The monochrome topological graph obtained after canonical vertex labeling is the canonical perimeter monochrome topological graph of the kinematic chain, and its corresponding adjacency matrix is ​​the canonical adjacency matrix.

[0082] Reference Fig.10 Shown for reference Fig. 9 The monochrome topological map obtained after the canonical vertex labeling is the canonical perimeter monochrome topological map of the kinematic chain.

[0083] The set consisting of all canonical adjacency matrices of a kinematic chain as elements is defined as the canonical adjacency matrix set of the kinematic chain.

[0084] Isomorphism judgment based on canonical adjacency matrix sets: The necessary and sufficient condition for two kinematic chains A and B to be isomorphic is that the intersection of their canonical adjacency matrix sets is non-empty. Fig.11 As shown, the comprehensive result of the monochrome topological graph configuration of an embodiment of the present invention.

[0085] Preferably, in step (6), the monochrome topological diagrams that do not meet the requirements are deleted in combination with the length requirement of the longest transmission chain of the planetary gear mechanism, and the position points suitable for the high pair and the compound hinge are selected to generate a multi-color topological diagram; the length of the longest transmission chain in the embodiment of the present invention is 7, there is only one compound hinge and its degree is 3 and there are 3 high pairs, that is, the monochrome topological diagrams whose length is not 7 are deleted in the monochrome topological diagram (refer to Fig.12 As shown), then a multi-color topology is generated based on the degree of high pairs and compound hinges, and the fact that no edges can be directly connected in the topology (refer to Fig.13 ), where red dots represent compound hinges and green dots represent high pairs.

[0086] Preferably, in step (7), the standard labeling method mentioned in step (5) is continued to be used to standardize the multi-color topological diagram, and the configuration synthesis is performed according to the one-to-one correspondence between the high pair and the composite hinge. If there is a one-to-one correspondence, it is isomorphic, otherwise it is not, refer to Fig.13 As shown, the comprehensive result of the multi-color topological diagram configuration of an embodiment of the present invention.

[0087] The present invention is based on the planetary gear mechanism configuration synthesis method of graph theory. The planetary gear mechanism is first analyzed to obtain the basic element quantity relationship. The new element quantity relationship is obtained according to the high pair and low generation. The adjacency matrix of all configurations is obtained by interpolation of the adjacency matrix. Based on these adjacency matrices, the graphs containing discrete points are first deleted. All loops contained in each graph are calculated so that each graph has a loop set. The separable graphs are deleted according to the loop set and the independent loops of the remaining graphs are calculated. It is judged whether the rigid subchain is contained by the independent loop, and the graph containing the rigid subchain is deleted. The configuration synthesis is performed according to the method of the standard adjacency matrix set, and then the monochrome topological graph that does not meet the conditions is eliminated in combination with the length requirement of the longest transmission chain of the planetary gear mechanism. Finally, the position points suitable for the high pair and the compound hinge are selected to generate a multi-color topological graph. After the multi-color topological graph is numbered and the one-to-one correspondence relationship is verified, the entire configuration synthesis process is completed. It can systematically, comprehensively and effectively solve the problem of complex planetary gear transmission mechanism configuration synthesis. With the help of computer programming, it can realize the automation of configuration synthesis, reduce the low efficiency and easy omissions in manual mechanism synthesis, and improve the efficiency and accuracy of configuration synthesis.

[0088] Although the specific implementation modes of the present invention are described above, those skilled in the art should understand that the specific implementation modes described are only illustrative and are not intended to limit the scope of the present invention. Equivalent modifications and changes made by those skilled in the art in accordance with the spirit of the present invention should be included in the scope of protection of the claims of the present invention.

Claims

1. A graph theory-based method for synthesizing the configuration of a planetary gear mechanism for a floor grinding machine, characterized in that: (1) Analyze the planetary gear mechanism, obtain the basic element quantity relationship, obtain the new element quantity relationship based on high-pair low-generation, and use adjacency matrix interpolation to obtain the adjacency matrix of all configurations; (2) Delete the monochrome topological graphs containing discrete points, and delete the unreasonable monochrome topological graphs based on high pairs and compound hinges; (3) Calculate all loops of each monochrome topology graph, that is, each monochrome topology graph corresponds to a loop set, and delete the separable monochrome topology graphs through the loop set; (4) Calculate independent loops through the loop sets of the remaining monochrome topological graphs, and determine and delete the monochrome topological graphs containing rigid subchains; (5) Perform configuration synthesis of monochrome topological graphs based on the method of canonical adjacency matrix sets; (6) Based on the length requirement of the longest transmission chain of the planetary gear mechanism, the single-color topological diagrams that do not meet the requirements are deleted, and the position points suitable for the high pair and compound hinge are selected to generate a multi-color topological diagram; (7) According to the standard labeling method, the multi-color topological diagram is configured and synthesized.

2. The graph-theory-based configuration synthesis method for a planetary gear mechanism of a floor grinding machine according to claim 1 is characterized in that: In step (1), the planetary gear mechanism is first analyzed to determine how many components, low pairs, high pairs, and compound hinges there are, and then high pair low generation is performed to determine how many components, low pairs, and compound hinges there are. The components and compound hinges are drawn as vertices and the low pairs are drawn as edges. According to the number of members, lower pairs, upper pairs and compound hinges, the corresponding zero matrix is ​​generated, and then inserted into the longest side of the upper triangular matrix 1, insert in the remaining positions of the upper triangular matrix 1, It is related to the number of components, low pairs, high pairs and compound hinges. The lower triangular matrix is ​​symmetric according to the upper triangular matrix to obtain the adjacency matrix of all configurations.

3. The graph-theory-based configuration synthesis method for a floor grinding machine planetary gear mechanism according to claim 1 is characterized in that: In step (2), exclude graphs containing vertices with degree 1 or less, that is, according to the adjacency matrix, search for the number of non-zero elements in each row or column according to the row or column. If the number of non-zero elements in a row or column is 0 or 1, delete the monochrome topological graph, and delete the monochrome topological graph whose points do not meet the degree requirements according to the number of high pairs and compound hinges.

4. The graph-theory-based configuration synthesis method for a floor grinding machine planetary gear mechanism according to claim 1 is characterized in that: In step (3), all loop sets are obtained through the adjacency matrix, and separable monochrome topological graphs are deleted, that is, monochrome topological graphs in which two loops are related by only one point or one edge are deleted.

5. The graph-theory-based configuration synthesis method for a floor grinding machine planetary gear mechanism according to claim 1, characterized in that: In step (4), the loop set is used to utilize the loop The independent loop is calculated and judged by the rigid subchain. , then there is no rigid subchain, where is the activity factor group, , is the number of independent loops.

6. The graph-theory-based configuration synthesis method for a floor grinding machine planetary gear mechanism according to claim 1, characterized in that: In step (5), draw the perimeter monochrome topology map of the kinematic chain, obtain the canonical perimeter monochrome topology map set through the perimeter monochrome topology map, and then obtain the canonical adjacency matrix set corresponding to the canonical perimeter monochrome topology map, and determine whether the intersection of the canonical adjacency matrix sets of the two kinematic chains is non-empty. If it is non-empty, they are isomorphic.

7. The graph-theory-based configuration synthesis method for a floor grinding machine planetary gear mechanism according to claim 1, characterized in that: In step (6), the length of the longest transmission chain of the planetary gear transmission mechanism is determined, and a reasonable single-color topology diagram is screened out by length, and then coloring is performed according to the existence relationship between the high pair and the compound hinge, that is, the high pair is a point with a degree of 2, and the compound hinge is a point with a degree of a few according to the actual planetary gear transmission mechanism, and a multi-color topology diagram is obtained.

8. The graph-theory-based configuration synthesis method for a floor grinding machine planetary gear mechanism according to claim 1, characterized in that: In step (7), continue to use the standard labeling method to standardize the polychromatic diagrams and determine whether the position points of the high pairs and compound hinges in the two diagrams correspond one to one. If so, they are isomorphic, otherwise they are not.

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