Numerical Control Interpolation Algorithm of Aspheric Surface Based on the Genetic Algorithms and Neural Network

When the Neural Network model is used to interpolate the non-circular curves, there are shortcomings of converging slowly and getting into the local optimum easily. A novel numerical control interpolation algorithm based on the GA (Genetic Algorithms) and NN (Neural Network) was introduced for the ultra-precision machining of aspheric surfaces. The algorithm integrated the global searching of GA with the parallel processing of NN, enhanceed the convergence speed and found the global optimum. At the end, the quintic non-circular curve was taken as an example to do the emulation and experiment. The results prove that this algorithm can fit the non-circular curve accurately, improve the precision of numerical control interpolation and reduce the number of calculating and interpolation cycles. Introduction In the numerical control system of the aspheric surface ultra-precision machining, the interpolation is the most essential subprogram to produce machining path, and decides the precision and the maxim feed rate of numerical control machine. Recently, with the ability of parallel processing and simulating any non-circular curves, the NN is often used to interpolate aspheric surface [1, 2]. But during the NN training, there are the problems of converging slowly and getting into the local optimum easily, so a novel numerical control interpolation algorithm, which integrates the GA with NN to interpolate the non-linearity curves, is introduced to solve the problems. Because the GA has the global searching ability, it can find the optimum quickly in the complicated, having many peak values, or non-differentiable large vector space. The problems of converging slowly and getting into the local optimum easily of NN are solved accordingly. GA and NN Model Because the NN can map the non-linearity curves, and a three-layer feed-forward neural network can map any curves theoretically [2], this project has created a three-layer BP (Error Back Propagation) Neural Network to fit the aspheric curve. The common training method of BP NN is BP algorithm, but as a gradient descent searching method, it usually has the slow convergence speed, and can’t find the global optimum in searching the complicated, having many peak values or non-differentiable space. In order to solve these problems, the method of taking the GA to optimize the weights and thresholds of BP NN firstly and then taking the BP algorithm to train them is adopted. The NN Model of the Numerical Control Interpolation. For the non-linear curves, after the shape of curves and the feed rate of the numerical control machine having been decided, the interpolation information of coordinate values and figures in every interpolation cycle can be expressed as a non-linear function of the former cycle information, so the non-linear model can be created with the NN model. The structure of the NN is 2-6-4 BP NN, as shown in Fig. 1; it includes the input layer, the hidden layer and the output layer. The input layer has two nodes xk-1, yk-1, which are the interpolation coordinate values of the former interpolation cycle, the output has four nodes xk, yk, θk, Key Engineering Materials Vols. 364-366 (2008) pp 25-29 online at http://www.scientific.net © (2008) Trans Tech Publications, Switzerland Online available since 2007/Dec/06 All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of the publisher: Trans Tech Publications Ltd, Switzerland, www.ttp.net. (ID: 130.203.133.33-14/04/08,12:56:07) rk; xk, yk are the interpolation coordinate values in the current interpolation cycle, θk is the included angle between the symmetry axis and the normal vector of the workpiece, rk is the curvature radius of the curve. The excitation function of the hidden layer is the sigmoid function f(x)=1/(1+exp(-x)), the excitation function of the output layer is the linear function. Fig.1 The structure of the NN GA Optimizing the Weights and Thresholds of the BP NN. The GA is a global search procedure that searches from one population (the first generation from which the genetic algorithm will begin its search for the optimum) of the optimal values to another. The evolvement steps of the GA are as follows: (1) Coding: The above built structure of the NN must be coded according to the requirements of the GA, the weights between input layer and hidden layer, the weights between hidden layer and output layer, and the thresholds in the hidden layer and the thresholds in the output layer are expressed as an one-dimension array, as shown in Eq. 1: { } , , , , 1,2 i i i i i M W V b r i p = = ⋅⋅⋅ (1) Where Wi is the weight matrix between the input layer and the hidden layer; Vi is the weight matrix between the hidden layer and the output layer; bi is the threshold matrix of the hidden layer; ri is the threshold matrix of the output layer. The Mi is a chromosome in the GA, the number of the chromosome is p, also is the size of the population. The above 4 matrixes are changed into chromosome cluster in order to use the GA to optimize the weights and thresholds of the BP NN. The created chromosome cluster is shown in Fig. 2. Fig.2 The chromosome cluster of weights and thresholds The coding method is real number coding instead of binary valued coding. The chromosome codes are used directly without any transformation in the real number coding. It is more available to the function optimization, so the error is not produced during the decoding, and the real number coding can also increase the calculation precision and speed. (2) Fitness Calculation: The object function of NN can be expressed as the whole error based the weighting vector and the threshold vector, as shown in Eq. 2. 2 -1 ( ) ( ) l l 1 0 1 ( ) 2 p m p p p l sse d y = = = − ∑∑ (2) Where dl (p) is the standard output, yl (p) is the fact output. P is the input pattern number; m is the nerve cell number of the output. Because the intent of the GA is to minimize the objective function, the fitness function of the GA can be decided by the objection function of the NN, and built fitness function is shown as Eq. 3. fit(Qi)=sse(Qi) i=1,2,...P (3) Where Qi is the population i. 26 Optics Design and Precision Manufacturing Technologies