Volume 3, Issue 5, 2016
140 Available online at www.ijiere.com
International Journal of Innovative and Emerging
Research in Engineering
e-ISSN: 2394 - 3343 p-ISSN: 2394 - 5494
Kinematic Chain Isomorphism Identification Using HNN
M. M. Gor
aand Anurag Verma
ba G H Patel College of Engineering & Technology, V. V. Nagar, Gujarat, India
b Chhatrapati Shivaji Institute of Technology, Durg, Chattisgarh, India
ABSTRACT:
The present work is an attempt to develop methodology for the detection of isomorphism which is frequently encountered in the structural synthesis of kinematic chains. Methods reported in the literature for detection of isomorphism using Hopfield neural network has been modified for better implementation. Suggested modification includes selection of initial state and termination criterion of the network. In order to escape from local minima, approximations are suggested out from different initial states instead of taking initial state only once. The proposed modifications have been tested with the help of few examples. The Kinematic chains of 6 link 1-d.o.f, 8 link 1-d.o.f and 12 link 1-d.o.f have been tested and simulation results show the effectiveness of the network that it identifies isomorphic kinematic chains correctly.
Keywords: Mechanism kinematic chain; Isomorphism identification; Hopfield neural Network (HNN)
I. INTRODUCTION
Mechanical design point of view, synthesis of kinematic chains is very important for the invention of new mechanisms. Synthesis of kinematic chain i.e. a procedure by which a mechanism is developed to meet the desired needs, usually involves the generation of a complete list of kinematic chains followed by a time-consuming procedure for the elimination of isomorphs. Two kinematic chains are considered as isomorphic if theire is one-to-one relation among the links of one chain to those of other chain. Undetected isomorphic chains results in duplicate solutions and an unnecessary effort.
Over the past several years much work has been reported in the literature on the isomorphism identification of kinematic chains. There are number methods for the recognition and identification of a given kinematic chain isomorphism which are categorized in graphical methods based on the visual inspection of various forms of simplified systematic diagrams and numerical methods many of which are based on the theory of graphs. Most of the numerical methods are based on the adjacency matrix and the distance matrix. For determining the structurally Distinct Matrix of a Kinematic Chain, The link disposition method [1], The flow matrix method [2], Minimum code method [3], Link path code method [4], Summation polynomial method [5, 6] are used to characterize the kinematic chain.
Identifying isomorphism of kinematic chains using the characteristic polynomial method is simple, but the reliability of this method is in question, as several apparently incorrect results were found [7]. Rao and Varada Raju [8] presented a method for detecting isomorphs based on Hamming numbers of the adjacency matrix. Some new approaches to these problems, such as eigenvector [9], adjacent-chain table method by Chu & Cao [10] and artificial neural-network method by Kong, Li & Zhang [11] were also investigated. Ding H, Huang Z [12] proposed a new loop theory of kinematic chains. Based on this theory, some key problems that hamper computer-based automatic synthesis of mechanisms are solved. Zou and He [13] have used node voltage sequence in circuit analysis method to detect isomorphism. Rizvi et al., [14], presented fuzzy similarity index to detect isomorphism.
In this paper, one of the established methods of kinematic chain isomorphism identification using neural network approach by Kong FG, Li Q & Zhang WJ [11] is discussed; also few problems and required modification are discussed for better implementation of this technique.
II. THE EXISTING CONTINUOUS HOPFIELD NEURAL NETWORK METHOD FOR THE MECHANISM KINEMATIC CHAIN
ISOMORPHISM PROBLEM
Volume 3, Issue 5, 2016
141 to the identification of isomorphic graphs. A line graph can be further represented in a matrix called the adjacency matrix.
The main step to prove that two graphs are isomorphic is to carry out the permutation i.e. row exchange and column exchange on one graph. Such permutation corresponds to a matrix operation on the adjacency matrix. This matrix is called the permutation matrix [PM]. Because this matrix operation reflects the row and column exchange on a graph, it has a feature that on each column or row, there is only one element for 1 and all the others for 0.
A general idea to apply ANN techniques [11] to solve for graph isomorphic comparison is proposed such that NxN neuron matrix is defined to correspond a permutation matrix where N is the number of vertices of a concerned graph. Therefore, it is possible to make the change of neuron matrix correspond to the process of permutation matrix operations on the adjacency matrix of a graph. A particular application of the Hopfield tank network model for implementing this idea is to define an Hopfield Tank [HT] energy function which should be associated with the adjacency matrices of the two concerned graphs and, of course, the state variable of each neuron. At each iteration of the Hopfield Tank network, an NxN neuron matrix is built with `zero' and `one' elements.
The main feature of the HT network is that a stable state could be reached by giving an initial state of the network. In this class of applications, the synaptic or connection weights change during the convergence process of the network. Here the connection weights are completely specified by minimization of the energy function defined on the HT network. The convergence process can be described as follows. Firstly, an initial state of the network is randomly defined. Then, the network evolves according to a prescribed dynamics until it reaches a minimum of the energy function. The HT model is a highly interconnected and synchronous network. Each element (neuron) of the network has a sigmoid continuous function defined.
To identify kinematic chain isomorphism, following steps are performed: (1) Preparation of adjacency Matrix, aij and bxy of kinematic chains.
(2) Initialize NxN neuron matrix, where Uix is the input and Vix is the output of ix neuron. Each neuron of the
network has a sigmoid continuous function.
0 1 1 tanh 2 ix ix U V U
(1)
Where, U0 is a constant which determines the slope of the sigmoid function. Here U0 is taken as 0.01.
(3) Initial state U00 of the network should not be any constant value, a small disturbance can be added to the
initial constant [11], that is,
00 U 1 . 0 ix U 00 U 1 . 0 where ix U 00 U ix U (2)
(4) Calculate Energy
jy ix
x y i j
xy ij 2
i x
ix
i j i x
jx ix
i x j x
ij ix V V b a 2 D N V 2 C V V 2 B V V 2 A E
(3)(5) HT dynamic equation
dt dU U
U n ixn
ix 1 n ix (4)
j y jy xy ij 2 j y jy i j jx x y iy ix ix V b a D N V C V B V A U dt dU (5)Volume 3, Issue 5, 2016
142 III. SUGGESTED MODIFICATION
(1) It has been observed that initial state of neuron plays vital role of the final result. Many times network is trapped in local minima. And network energy does not reach to zero value. In order to escape from local minima, we are suggesting different runs out from different initial states. Instead of taking initial state once, we suggest that after 1000 iteration if network energy does not minimize, then randomly another initial state is taken and once again iteration are carried out 1000 times.
(2) Second modification is related to initial state itself. Kong [11] has suggested initial state as per equation (2). But in that equation about U00, no clarification is given. Here suggested initial state is inspired from the paper
of A.R.Bizzari [15], in which Initial state of ix neuron is given by )
1 . 0 ± 1 ( U =
Uix ixin (6)
Where, the sign is chosen in a random way to prevent the system being trapped in an instable equilibrium and Uixin satisfies the condition
) 1 N ln( 2 / U
Uixin 0 (7)
Here, similar concept is suggested with modification. Bizzari has suggested that in equation (6) sign is chosen in a random way. But we have taken multiplication factor of Uixin in a range of 0.9 to 1.1. That means Uix lies
anywhere between Uixinx0.9 to Uixinx1.1. Even to decide it, we suggest following procedure:
If no of link N=6; Increment: (1.1-0.9)/N=0.033,
So the multiplication factor of Uixin in equation (6) will be in a range from 0.9 to 1.1 with increment of 0.033 so the multiplication factor series will be 0.9, 0.93, 0.96, 0.99, 1.03 and 1.06
Also, we suggest that instead of assigning this factor in ascending or descending order; assign it randomly for each neuron of a row. For that,
Take neuron row i=1 to N, one by one
Decide random series
If random series for row i=1; is 3 1 6 4 2 5
Assign 3rd cell Multiplication factor i.e. 0.96 to first position; 1st cell value i.e. 0.9 to second position, 6th cell
value i.e. 1.06 to 3rd position so on.
So the multiplication factors for equation (6) for row i=1 will be:
0.96 0.9 1.06 0.99 0.93 1.03
Like this way, finalize multiplication factors for each row and then taking Uixin from equation (7) finalize
initial state using equation (6) of complete NxN matrix.
(3) Third modification is regarding termination criterion which is an important part to be determined in any iteration scheme. In kong [11] simulation program, they followed Bizzarri[15]. According to it, if the network does not yield a stable neuron matrix after 1000 iterations, the network is considered not convergent at all. That means two graphs are not isomorphic. As kong [11] point out that, the exact zero and one values of neuron output are not possible to reach because continuous neurons are applied and thus energy can not reach to exact 0.000 value. So here it is suggested that if energy reach to
0.01 it should be taken into consideration that neural network is stable and chains are isomorphic. Because here, energy reach to
0.01 only when on each row and column there is only one neuron which has output nearly equal to one.IV. RESULTS AND DISCUSSION
Considering all above modifications, MATLAB program is prepared and few examples are checked which are discussed below.
A. Example 1:
Volume 3, Issue 5, 2016
143 Example1 concerns 6 link kinematic chain as shown in Figure 1a & 1b. Modified method reports that after 1000 iteration & 25 different initial states also, neuron matrix does not reach to its stable state, so kinematic chain shown in Figure 1a & Figure 1b are non isomorphic chains.
Figure 1. Six-link, single-degree-of-freedom kinematic chain with simple joints.
B. Example 2:
8 link, 1-dof simple jointed planar kinematic chain
Figure 2: Eight-link, single-degree-of-freedom kinematic chains with simple joints.
Example 2 concerns 8 link kinematic chain as shown in Figure 2a & 2b. Modified method reports that Figure 2a & Figure 2b are isomorphic chains. Two different solutions obtained for above isomorphic chains are shown in Figure 3. Black and white squares in Figure 3 indicate 1 and 0 outputs, respectively.
Figure 3: Solutions obtained for the isomorphic chains shown in figure 2a & 2b
C. Example 3:
Volume 3, Issue 5, 2016
144 Figure 4: Twelve-link, single-degree-of-freedom kinematic chain with simple joints.
Example 3 concerns 12 link kinematic chain as shown in Figure 4a and 4b. Modified method reports that Figure 4a and Figure 4b are isomorphic chain. Two different solutions obtained for above isomorphic chains are shown in Figure 5. Black and white squares in Figure 5 indicate 1 and 0 outputs, respectively.
Figure 5. Solutions for the isomorphic chains shown in figure 4a & 4b
V. CONCLUSIONS
While checking above problem it is found that suggested modification works and gives satisfactory solution. In this paper, existing method is simplified. If the mechanical designer effort is of concern, then the proposed modification in neural network algorithm is a good choice to quickly identify the isomorphism of the kinematic chains. This proposed modification in identification of isomorphism of chains can exactly identify isomorphism of chains and be automatically executed by a computer. All examples were run on a conventional computer and program is developed in Matlab. For every one of the three selected test problems, the results were obtained for a total of 25 runs & within 1,000 iterations of the neural algorithm described which not the case with earlier reported methods is.
REFERENCES
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145 [9] Z. Chang, C. Zhang, Y. Yang and Y. Wang, A new method to mechanism kinematic chain isomorphism.
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