MODELING AND ISOMORPHISM VERIFICATION OF EIGHT LINKS BASE KINEMATIC CHAINS USING MATLAB

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1 International Journal of Advanced Research in Engineering and Technology (IJARET) Volume 9, Issue 6, November - December 2018, pp , Article ID: IJARET_09_06_018 Available online at ISSN Print: and ISSN Online: IAEME Publication MODELING AND ISOMORPHISM VERIFICATION OF EIGHT LINKS BASE KINEMATIC CHAINS USING MATLAB Vipin Kaushik Research Scholar, Department of Mechanical Engineering, Faculty of Engineering and Technology, Jamia Millia Islamia Delhi, India Aas Mohammad Professor, Department of Mechanical Engineering, Faculty of Engineering and Technology, Jamia Millia Islamia Delhi, India ABSTRACT The present study deals with the development of all possible kinematic base chains for 8-link single degree of freedom mechanism and to verify the method of Eigen values and Eigen vectors [4] for the identification of isomorphic kinematic chains. Currently for determining the identical kinematic chains methods such as link's adjacent-chaintable [11], Adjacency Matrix [8], Hamming number technique [2], Distance Concept [5] etc. are used that are mostly time consuming and rather more difficult to compute using a program on a computer. In the current study the Eigen values as well as their corresponding Eigen vectors of various kinematic chain assortments are used for comparison. In this the matrix corresponding to the numbering given to the links of the kinematic chain is made and the Eigen values and the Eigen vectors of the same matrix are found using a programme code on MATLAB. These values are then compared to the values and vectors of a differently assorted kinematic chain. If the corresponding Eigen vector and value are found to be same then the kinematic chains are isomorphic. Key words: Isomorphism, Eigen values, Kinematic chains Cite this Article: Vipin Kaushik and Aas Mohammad, Modeling and Isomorphism Verification of Eight Links Base Kinematic s Using Matlab. International Journal of Advanced Research in Engineering and Technology, 9(6), 2018, pp INTRODUCTION Kinematic chains play a vital role in development of mechanisms especially in the field of Robotics and Manufacturing machinery. The study of kinematic chains is thus very important in order to develop newer and better mechanisms. Among the vast field of study of kinematic chain, Isomorphism of kinematic chains is an important topic for study. In a kinematic chain if one link is fixed and a kind of constrained motion is obtained by moving other links and also on fixing some other link other than the first link the same motion is obtained then it is known as isomorphism of the kinematic chains or identical chains. Since the number of different chains editor@iaeme.com

2 Vipin Kaushik and Aas Mohammad that can be developed with any number of links is a lot therefore scholars and researchers have been looking for an efficient and an easy method to determine the isomorphism in kinematic chains as it will save time and also help in developing much more efficient machines. In present work the method of Eigen value& eigenvector approach is studied and its validity is examined for 8 link base kinematic chains. The kinematic chain structure provides vital information such as which link is connected to which other links and this information can be used to form a matrix for the kinematic chain and determine the Eigen values and Eigen vectors of the matrix using basic computer algorithms in MatLab. The main motive of present work is to identify the efficacy of the method of finding Eigen vectors & Eigen values and also to prove if it is valid for 8 link base kinematic chains or not along with trying to understand how efficient is it for analysis of chains of higher number of links and the time consumed in order to analyze them. 2. METHODOLOGY The above-mentioned method is applied on 8 link Single Degree of freedom, Base kinematic chains. These 8 link Single D.O.F kinematic chains can be classified into 16 base kinematic chains that were tested for isomorphism using the method of Eigen values and vectors. These 16 base kinematic chains have further assortments which may also be tested for possibility of isomorphism Formation of matrix: The matrix is the mathematical representation of the kinematic chain. Let L be the matrix of a kinematic chain l and be represented as {lij}= link between the two links i and j a) If the two links i, j are linked i.e. are adjacent to each other, then the value of lij=1 b) Whereas if the two links are not linked to each other i.e. they are not adjacent then the value of lij= 0 The general matrix of the kinematic chain as: 2.2. Program used on MATLAB The following programme may be used to determine the Eigen values and Eigen vectors of the matrix of the kinematic chain: L= [l11 l12 l13 l14 l15 l16 l17 l18; l21 l22 l23 l24 l25 l26 l27 l28; l31 l32 l33 l34 l35 l36 l37 l38; l41 l42 l43 l44 l45 l46 l47 l48; l51 l52 l53 l54 l55 l56 l57 l58; l61 l62 l63 l64 l65 l66 l67 l68; l71 l72 l73 l74 l75 l76 l77 l78; l81 l82 l83 l84 l85 l86 l87 l88] [q,r]=esig(l) editor@iaeme.com

3 Modeling and Isomorphism Verification of Eight Links Base Kinematic s Using Matlab On solving the matrix using the following programme, the output is obtained in form of two matrices, q & r respectively, where: q= matrix of Eigen vectors r= matrix of Eigen values The Eigen values and Eigen vectors for the matrices of two different kinematic chains are calculated and if the Eigen values and their corresponding Eigen vectors of both the matrices are found to be same then we may say that the two kinematic chains are isomorphic. 3. ILLUSTRATIVE EXAMPLE: Step 1: Selection of kinematic chain Select a multiple jointed kinematic chain with eight links and give a number to each of its links. We consider the kinematic chain given in Figure 1: Figure 1 Step 2: Formation of matrix for the kinematic chain Matrix for the kinematic chain given in figure1 Step 3: Run the code for the Eigen values and vectors The code to form the matrix p & to determine its Eigen values and Eigen vectors is as follows: P = [ ; ; ; ; ; ; ; ] [U, V] = eig (P) Where U= matrix of the corresponding Eigen vectors editor@iaeme.com

4 Vipin Kaushik and Aas Mohammad V= matrix of Eigen values Step 4: Summarizing the Eigen values and the corresponding Eigen vectors The Eigen values and the Eigen vectors so obtained for the matrix p are as follows: Table 1. Summarized table of Eigen values & Eigen vectors EIGEN VALUES EIGEN VECTORS Step 5: Check for Isomorphism Repeat the steps 1 to 4 for all the other multiple joint kinematic base chains and compare the results in order to determine isomorphism. 4. RESULTS Table 2: s S.No Kinematic Eigen Values Type editor@iaeme.com

5 Modeling and Isomorphism Verification of Eight Links Base Kinematic s Using Matlab editor@iaeme.com

6 Vipin Kaushik and Aas Mohammad 5. CONCLUSION The test for isomorphism for 8 link multiple joint base kinematic chains of single degree of freedom has been performed satisfactorily and depicted in the results in a tabular form. Since it is theoretically known that 16 base kinematic chains cannot be isomorphic and as per the result obtained, it can be established that the method is valid and stands on the proof. This method is also very easy to perform and since the code is to be run on MATLab, it stated to be efficient as it founds to be less time consuming. The efficacy of the method can further be tested by applying it the chains of higher order. It gives us a scope for further evolution of the method and find whether any further changes need to be made in any of the steps of methodology. As for far, the steps are pretty easy to understand and the test can be performed with ease. REFRENCES [1] Uicker J. J., Raicu A., A method for the identification and recognition of equivalence of Kinematic s, Mech. Mach. Theory, vol.10, , 1975 [2] Rao A.C., Kinematic s, Isomorphism, Inversions, and Type of Freedom using the concept of Hamming distances, Indian Journal of Tech., vol.26, pp , [3] Mruthyunjya T.S and Raghvan M.R, Computer aided analysis of the structure of kinematic chains, Mechanism and Machine Theory, vol. 19, No 3 pp , [4] Zongyu chang ce Zhang, Yuhu Yang, Yuxin Wang, A new method to mechanism kinematic chain isomorphism identification, Mechanism and Machine Theory, vol. 37, Issue 4, pp , [5] J. N. Yadav, C. R. Pratap, V. p. Agrawal, Detection of Isomorphism Among Kinematic s Using the Distance Concept, Journal of Mechanical Design, December 1995 Vol.117/611 [6] Vipin kaushik, Aas mohammad, A Mathematical Approach to Identify the Number of Distinct Mechanisms of Multiple Jointed Kinematic s using Concepts of MATLAB, International Journal of Control Theory and Applications, vol.10, Number 4, pp1-9, 2017 [7] Sethi and Agrawal, Hierarchical classification of kinematic chains, Mechanism and Machine Theory, vol. 28, No 4, pp ,1993 [8] Aas Mohammad, Khan I.A. And M.K. Lohumi, Interactive weighted distance approach for Identification of distance Mechanism derived from the kinematic chain, International Journal of Applied Engineering Research, ISSN vol. 3, pp , [9] Aas Mohammad, Agrawal V.P., Identification and Isomorphism of Kinematic s and Mechanisms, 11 th ISME Conference, pp [10] Mruthyunjya T.S and Raghvan M.R., Structural analysis of kinematic chains and mechanisms based on representation, Transaction of the ASME, Journal of Design, vol. 101, pp , 1979 [11] CHU JIN-KUI and CAO WEI-QING, Identification of isomorphism among kinematic chains and inversions using link's adjacent-chain-table, Mech. Mach. theory Vol. 29, No. I, pp , editor@iaeme.com