Send Orders for Reprints to [email protected] The Open Automation and Control Systems Journal, 2014, 6, 277-282 277 Open Access Parabola Error Evaluation Based on Geometry Optimizing Approxima- tion Algorithm Lei Xianqing1,*, Gao Zuobin1, Wang Haiyang2 and Cui Jingwei3 1Henan University of Science and Technology, Luoyang, 471003, China 2Heavy Industries Co., Ltd, Luoyang, 471003, China 3Luoyang Bearing Science &Technology Co. Ltd, Luoyang, 471003, China Abstract: A more accurate evaluation for parabola errors based on Geometry Optimizing Approximation Algorithm (GOAA) is presented in this paper. Firstly, according to the least squares method, two parabola characteristic points are determined as reference points to form a series of auxiliary points according to a certain geometry shape, and then, as- sumption ideal auxiliary parabolas are reversed by the parabola geometric characteristic. The range distance of all given points to the assumption ideal parabolas are calculated by the auxiliary parabolas as supposed ideal parabolas, and the ref- erence points, the auxiliary points, reference parabola and auxiliary parabolas are reconstructed by comparing the range distances, the evaluation for parabola error is finally obtained after repeating this entire process. On the basis of 0.1 mm of a group of simulative metrical data, compared with the least square method, while the criteria of stop searching is 0.00001 mm, the parabola error value from this algorithm can be reduced by 77 µm. The result shows this algorithm realizes the evaluation of parabola with minimum zone, accurately and stably. Keywords: Error evaluation, geometry approximation, minimum zone, parabola error. 1. INTRODUCTION squares techniques, but the iterations of the calculations is relatively complex. Ahn's method [3] improved the efficien- In the Image Processing or the Computer Graphics, the cy of iteration by processing the weighted fitting coefficient fitting of conic plays a foundational and important role in the matrix. Zhang [4] and Strum [5] proposed the objective func- curve profile matching [1]. Now that, parabola is difficult to tion based on the bias of geometry distance. But, the compu- fit as its geometry has conic feature. In recent years, parabola tation process contains a 4th order equation. It is not stable to is applied in many fields of Mechanical design and parts construct an ideal conic in some conditions. Liu [1] used six measurement algorithm. For example, among the aviation different constraints to get six basic conics, and generate the and aerospace industry, the parabola type of radial flow im- best conic by adding certain weights to the coefficients of the peller is adopted widely in the radial flow impeller machine six basic conics. Different fitting methods with constrain of propulsion system in order to enhance the aerodynamic matrices were listed for specific types of conic in Harker’s performance and strength of the impeller [2]. At present, a paper [6]. Chernov [7] posted a weighted gradient fitting mature and representative algorithm to evaluate parabola method to the scattered data points, and pointed out that the profile error is not used in most of coordinate measuring method can relieve the complexity in some conditions. Gao machine (CMM). Therefore, to evaluate parabola profile [8] combined the Genetic Algorithm and least square method error accurately has important application value for design together to solve the conic fitting problems. Li [9] mainly and processing of aerospace parts and precision measure- talks about how to select the coordinate system and the con- ment methods. straints to make the fitting curve approximately closer with the implicit conic equation. But her method is needed to So far, there is not a unified standard for parabola profile transfer the coordinate system first. error at home and broad. GB/T 1184-1996 does not have a specialized criterion to judge parabola profile error. Parabola All in all, the works just mentioned discussed the objec- fitting methods can be divided into two types: the methods tive functions and the different constraints after transferring based on algebraic distance and the methods based on geo- the coordinates. In this paper, we discuss two important metric distance. In 2001, the parabola fitting of least-squares problems. The one is to fit parabola in optional position of orthogonal distance (LOD) method was presented by Sung rectangular coordinate system, another is to evaluate the pro- Joon Ahn, and the method measured better than the least file error of given data. 2. THE PRINCIPLE OF THE GOAA Firstly, we can obtain two feature points (focus and inter- section between axis and directrix) and initial profile error d0 1874-4443/14 2014 Bentham Open 278 The Open Automation and Control Systems Journal, 2014, Volume 6 Xianqing et al. Y The least squares parabola Assumption ideal parabola Real measurement parabola Auxiliary focus Measurement point Nj(xnj,ynj) Wt(xt,yt) Reference focus Auxiliary vertex N0(xn.yn) Mi(xmi,ymi) Reference point M0(xm,ym) 0 X Fig. (1). The evaluation principle. by least squares method (LSM) with constraints [10]. Then, directrix point N0 (xn, yn ) ) of parabola as initial reference the two feature points are treated as initial reference points, points according to plane analytic geometry knowledge. The and then, 4 auxiliary points (vertexes of searching square) length of searching square is LSM error or estimated para- are arranged around them respectively (see Fig. 1), so we can get 16 groups of auxiliary feature points. If we assume arbi- bolic error d0 . The precision parameter can be changed ac- trary groups of auxiliary feature points are ideal feature cording to the effective number of measurement data, preci- points of ideal parabola 16, assuming ideal parabolas are sion of instrument and accuracy requirement of measuring reserved with geometry knowledge. Afterwards, calculating object, when the precision parameter is under micron level, the shortest distances between given points and assuming δ = 0.0001 mm. ideal parabolas, the minimum distance is named d1. If d0 ≤ d1, then the reference feature points remain and the length of 3.2. Constructing Auxiliary Feature Points searching square is reduced by half. Meanwhile, the new auxiliary feature points are reconstructed and repeat the pro- Two searching squares are constructed whose lengths l cess of solving d1. If d0 > d1, the feature points of assuming have initial error d0 . At same time, their centers are initial ideal parabola corresponding to d1 are the new reference fea- ture points and the length of searching square is remains the reference points (that are feature points) M0 (xm, ym ) and same, 16 groups of new auxiliary feature points are N (x , y ) (see Fig. 1). The vertexes of searching squares achieved. Repeating above processing, when the length of 0 n n searching square reduces a certain value, we could consider are named auxiliary points Mi (xmi , ymi ) and N j (xnj , ynj ) that the assuming ideal parabola approach ideal parabola (i = j = 1,2,3,4) , they are calculated below: very closely and stop the searching process. So the minimum between d0 and d1 is the profile error of parabola. "x = x = x + l / 2 m1 m2 m "xn1 = xn2 = xn + l / 2 $ $ x = x = x ! l / 2 3. THE STEPS OF THE GOAA $ m3 m4 m $xn3 = xn4 = xn ! l / 2 # # (1) y y y l / 2 y y y l / 2 $ m1 = m4 = m + $ n1 = n4 = n + 3.1. Selecting the Initial Reference Points, Initial Length $y = y = y ! l / 2 $y = y = y ! l / 2 and Accuracy Parameter % m2 m3 m % n2 n3 n Assuming the measurement points are expressed by 3.3. Constructing Auxiliary Parabolas Wt (xt , yt )(t =1,2,3,...,N). In order to reserve 6 parameters of parabola expressed by conic equation, we set LSM feature The 16 groups of auxiliary feature points are obtained points (focus M (x , y ) and intersection between axis and from auxiliary points M (x , y )(i = 1,2,3,4) and 0 m m i mi mi Parabola Error Evaluation Based on Geometry Optimizing The Open Automation and Control Systems Journal, 2014, Volume 6 279 N (x , y )( j = 1,2,3,4) according to permutation and com- The minimum range distance is named d1 with equation j nj nj (7). bination knowledge. Next, the 16 groups of main parameters of parabolas are achieved by the equation (2). d = min(d ) (7) 1 ij # p = M ! N % ij i j 3.5. Approximation Searching Method % xc x + x / 2 % ij ( mi nj ) If d ! d , searching squares are reconstructed with con- $ = (2) 0 1 yc y y / 2 % ij ( mi + nj ) stant reference feature points and half of length, repeat steps % 3.3-2.5, If d ! d , the feature points of assuming ideal pa- %tan " = y ! y / x ! x 0 1 ( ij ) ( mi nj ) ( mi nj ) & rabola corresponding to d1 are the new reference feature points and the length of searching square is the error right Where, p are focal length, are vertexes, ! ij xcij , ycij ij ( ) now l = d1, repeating steps 3.3-3.5, when the length of are the rotation angles of assuming ideal parabolas. searching square reduces a certain value (general condition l ! 0.0001 mm ), we could consider that the assuming ideal The measured parabola in any position of plane can be 2 2 parabola approaches ideal parabola very closely and stop the expressed by Ax + Bxy + Cy + Dx + Ey + F = 0 . The coef- searching process. So the minimum value between d0 and d1 ficients of the equation are gained with equation (3). is the profile error of parabola Δd . #A = cos2 ! % ij ij Δd = min(d0,d1) (8) %B = 2sin! cos! % ij ij ij 2 4. EXAMPLE %Cij = sin !ij % % 2 There is a series of measurement points belonging to the $Dij = 2( pij sin!ij " xcij cos !ij " ycij sin!ij cos!ij ) (3) % optional position parabola in the plane, added some noise E 2(xc sin cos yc sin2 p cos ) % ij = " ij !ij !ij + ij !ij + ij !ij points whose pre-set error is 0.1mm and are then simulated % 2 to test the algorithm in Table 1.