Comparison Of Tests For Isomorphism In Planetary Gear Trains

There are plenty of ways available for synthesis and analysis of Planetary Gear Trains (PGTs) of one DOF. However, every method has its own shortcomings. In this paper a comparison is made between Characteristic polynomial, Eigenvalues and Eigenvectors, Hamming Number Method and Modified Path Matrix (MPM) method. There are many methods available to test isomorphism in PGTs, out of which these four methods wasanalyzed in this paper. For a given PGT with a number of links and a single Degree of Freedom (DOF), adjacency matrix is enough to find out the Eigenvalues and Eigenvectors. Isomorphism of PGT is determined using the Eigenvalues. If Eigenvalues are similar then the PGTs are isomorphic. Similarly, if the characteristic polynomials of two PGTs are samethen it represents the isomorphic PGTs. Characteristic polynomials are determined from the Adjacency matrix. Hamming method also uses adjacency matrix to generate Hamming matrix. Hamming strings are developed from Hamming Matrix. Uniform hamming strings of two PGTsindicates isomorphism in PGTs. Along with isomorphism, Symmetry also known from the Hamming method. Modified path matrix method uses a connectivity matrix to generate MPM. If train values of two PGTs are same then they will be isomorphic otherwise non isomorphic. As per literature as the number of links increases the results may not be accurate with Characteristic coefficients and Eigenvalues methods, though all these methods are used to detect isomorphism among a group of PGTs. Whereas with the Hamming number approach, one can detect isomorphism, symmetry and number of possible level combinations of an PGT with a single hamming matrix.


Introduction
Researchers using Graph theory for synthesis and analysis of PGTs from the decades. A kinematic chain is converted as a graph for easy analysis. Buchsbaum and Freudenstein are pioneers [1] developed conditions to be satisfied by a graph of aPGT. The correspondence between the graph and displacement equations was developed by Freudenstein [2]. In a labeled graph lower pairs or turning pairs are represented by thin lines and geared pairs by double line or Thick line. Levels are denoted on lower pairs which trace the location of the rotational axis. In a 'n' link PGT there must be (n-1) turning pairs and (n-2) gear pairs. Transfer vertex exists in each fundamental circuit [3]. Characteristic polynomial method was developed by Uicker and Raicu [4] which is the fundamental research on kinematic chains to determine isomorphism. This concept is adapted to geared kinematic chains up to 4 links by Lung-Win ICRAEM 2020 IOP Conf. Series: Materials Science and Engineering 981 (2020) 042023 IOP Publishing doi: 10.1088/1757-899X/981/4/042023 2 Tsai [5]. A computerized analysis of geared kinematic chains using characteristic polynomials was done by Ravi Shankar and Mruthyunjaya [6]. Canonical graph representation using Pseudo isomorphic graphs was developed by Chatterjee and Tsai [7]. Canonical graph is another way to represent the geared kinematic chains in which each PGT is divided into a number of basic entities. Zongyu Chang [8] proposed the Eigenvectors and Eigenvalues method to detect isomorphism in PGTs. Complicated computations required if the Eigenvalues of PGTs are identical to confirm the isomorphism. Efficiency of this method is very low. Fuzzy adjacency matrix is used in fuzzy logic method [9]. Nomographs concept was introduced by E. L. Ismail to determine the velocity of an PGT [10]. Rajasri et. all used the Hamming number approach to detect isomorphism of an PGT [11,12]. Ali Hasan et al. [13] used Modified Path Matrix (MPM) to identify the isomorphism inPGTs. Identifying isomorphic PGTs is very important to reduce the duplication of PGTs in the generation process [14]. From a group of PGTs, one best isomorphic PGT is selected for further process of generation. All the isomorphic PGTs are having different structural properties i.e. they may not generate equal number of next level graphs, they may vary in velocity ratios, they may vary in structural arrangement etc. This should be identified by the structural aspects like symmetry of an PGT, which is only possible with the hamming matrix.

Characteristic Polynomial Method
This method was introduced by Uicker and Raicu [4] and is further developed by Tsai [5]. The connectivity of the links will be known from the Distance matrix and it is same as adjacency matrix. Using matrix algebra, characteristic polynomial coefficient is determined. If the coefficient of characteristic polynomial is the same for two PGTs then the two PGTs are said to be isomorphic. The condition|‫ܫݔ‬ − ‫|1ܣ‬ = ‫ܫݔ|‬ − ‫|2ܣ‬must be satisfied in case if the PGTs are isomorphic.One need to calculate the determinant of the matrix (XI-A), to determine the linkage characteristic Polynomial ofa PGT. Where X represents the variable, I represent Identity Matrix and A is Adjacency Matrix.Characteristic coefficient of a PGT is calculated using the MAT lab program. For finding isomorphism in a PGT the characteristic polynomial test is sufficient. Coefficient must be unique for a given topology.

Eigenvalues and Eigenvectors Method
Eigenvalues and Eigenvectors of an PGT are computed using a MATLAB Program. These are derived through adjacency matrix. Isomorphism is confirmed if the two PGTs have unique Eigenvalues otherwise not. To confirm the isomorphism after identical PGTs, one needs to find Eigenvectors which requires a lot of computational efforts. Rows should be interchanged in a sequential manner to calculate the eigenvectors. A row transformation matrix (R) is generated after interchanging the rows.

Hamming Number Method:
Information and communication theory first used hamming distances to know the planar [10] Kinematic chains (KCs) and PGTshaving 'n' links withsingle DOF. Adjacency matrix is generated first based on the link connectivity then hamming matrix is calculated using adjacency matrix. Hamming strings are generated from hamming matrices. Unique hamming string for the two PGTs is the sign for isomorphism. Symmetry is identified with hamming strings of a PGT [15, 16].

Modified Path Matrix Method
MPM method is used to identify the isomorphism in PGTs and also to compare the geared kinematic chains by its train values. If the train values of two PGTs are identical then the two PGTs are said to be isomorphic otherwise non isomorphic. MPM is a square symmetric matrix [a ij ] calculated based on the connectivity of the links [13]. It is calculated as follows In this method joint value is identified to each joint. Then the least path value is calculated based on the shortest path of the link using joint values. Elements in the Modified Path Matrix are generated from least path value. Summation of elements in each row generates a pair value. The summation of all pair values results in a train value of aPGT. If train values are similar then the PGTs are isomorphic otherwise non isomorphic.

Characteristic Polynomial Method
Four Six link PGTs with one DOF are shown in Figs. 1,2,3 and 4. These PGTs are considered to compare the above four methods to identify isomorphism. Table: 1 consists of four PGTs along with their Adjacency matrices:

Eigenvalues and Eigenvectors Method
Considerthe Adjacency matrix of the PGT in Figure 1 Eigenvalues and Eigenvectors are given as Eigenvalues of both the PGTs are identical; hence both the PGTs are isomorphic. As a second case, Figure 3 and Figure 4 are two PGTs to be considered for further evaluation, Eigenvalues and Eigenvectors are determined from the adjacency matrix. From the results we can conclude that the Eigenvalues of both the PGTs are same.

Hamming Number Method:
To test isomorphism in PGTs using hamming approach, consider two 6-link PGTs with single DOF shown in Fig: 1  Hamming strings of PGTs shown in Figure 3 and Figure4 are same. It means both the PGTs are isomorphic. Along with isomorphism, symmetry is also known from Hamming matrix.

Modified Path Matrix Method
As an example consider two 6-link PGTs with single DOF shown in Figure 1 and Train values of above two PGTs are identical; hence both the PGTs are isomorphic. As a second example consider two PGTs shown in Figure 3 and Figure 4, the corresponding Modified Path Matrix and train values are given as follows