Creation of neural network models to solve the problems of forecasting the product geometrical accuracy Vadim Pechenin Michael Bolotov Nikolay Ruzanov Department of manufacturing Department of manufacturing Department of manufacturing technology engines technology engines technology engines Samara National Research University Samara National Research University Samara National Research University Samara, Russia Samara, Russia Samara, Russia vadim.pechenin2015@gmail.com maikl.bol@gmail.com kinform_@mail.ru Ekaterina Pechenina Department of manufacturing technology engines Samara National Research University Samara, Russia ek-ko@list.ru Abstract—The article considers the problems of creating a Direct modelling of mating using numerical models of tool for operational forecasting of quality indicators (assembly mating and finite element models of the assemblies requires parameters) for knowledge-intensive products. The basis of significant computing resources [6] and often has decision forecasting is the creation and use of actual geometrical models coincidence problems. Artificial neural networks can be used of parts containing data on their geometrical deviations, and to improve the forecasting efficiency for the assembly numerical models of part mating. Actual geometrical models are parameters. created based on the data on coordinate measurements of parts. The developed models have been validated using the example of The article considers the option developed to solve the an assembly unit composed of three parts of an aircraft engine problem of the product geometrical accuracy based on the data turbine rotor. To reduce computing resources, the use of a on specific part measurements, neural network models, and radial-basis neural network to calculate assembly parameters digital counterparts of the assemblies. The goal of the article has been considered. Training and test samples have been is to study the estimate of the assembly parameter calculation modelled, the network operating parameters have been error with the help of the neural network model based on a lot optimized, and the obtained results have been generalized. of data obtained using a digital counterpart of the assembly. Keywords—numerical model, actual geometry, assembly II. SUBJECT OF THE RESEARCH parameter, neural network The assembly of three turbine parts is considered as the I. INTRODUCTION subject: shaft, retainer, and disc. Fig. 1 shows a sketch of the The most critical quality indicator for engineering assembly unit under consideration. products is the geometrical accuracy of machines, which has The bases A and B in Fig. 1 form a rotation axis (basic a significant impact on the performance. The geometrical accuracy of products can be increased and their production axis). The requirements for face runout Pt r of the discЗ cost can be decreased by developing and implementing digital surface, and radial runout Pr r of the disc surface П have been technologies into product design and production processes. set in relation to the basic axis. Let’s consider models and The new generation high-tech industry is based on data use. A algorithms that allow virtual forecasting of runouts. promising approach to improve design processes and manufacture high-tech products provides for the development III. DIGITAL COUNTERPART OF THE ROTOR ASSEMBLY of digital counterparts of objects being digital analogues of actual objects [1]. In respect to assembly of engines and power The digital counterpart of the assembly includes the plants, a digital counterpart represents related actual models following: digital models of parts including the actual of parts. geometry containing production deviations; calculation of mating states of parts [7, 8]; calculation of assembly Mathematical models [2] implemented in the form of geometrical parameters. computer models are used to forecast quality indicators (in particular, assembly parameters). The assembly model choice A. Creation of part models with actual geometry depends on the stiffness requirements. Some models are based Information about the actual geometry represented as data on the solid state hypothesis, for example, the T-Map model on the part surface measurements is required for the [3]. Other models, such as the Skin form model and the modelling. The assembly model accuracy mostly depends on Deviation Area Model (DD), canal so simulate a flexible part the accuracy of the actual geometry measurements on or assembly [4]. These models can be either point-based or coordinate inspection machines [9, 10] or scanning devices. feature-based. Compared to the features that simultaneously Part surfaces were measured on a coordinate measuring characterize position and direction information, the position of a point in space is described by its location rather than machine (CMM) of DEA GlobalPerformance. orientation, with variations that vary depending on the choice of different points [5]. Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0) Data Science movement of one mating surface in relation to the other one, with the stress application vector of the surface assembly. D 1 To ensure the best adjustment, the iterative algorithm of nearest points (ICP) is used [14, 15]. According to this algorithm, the rotation and movement angles along the coordinate axes are calculated at each iteration with the non- linear optimization search methods. The system of inequalities presented in the work [16] limiting the gap function is used to exclude the intersection of two surfaces G (V ) . The use of the algorithm results in calculating a rotation matrix and moving part movement vector that determines the conversion of its initial coordinate system into the coordinate system in the assembled state. C. Calculating the assembly geometrical parameters The radial runout between the control surface P and bases A and B (Fig. 1) is calculated in the following order:  The main axis of the coordinate system coincides with the normal vector of the a с rotation axis set using the bases А and B. Fig. 1. Assembly unit and controlled surfaces, 1 – shaft, 2 – retainer, 3 – disc.  The distances from the measured points P to the rotation axis are calculated. The number of points measured on the planes and cylindrical surfaces was 200 points. Part ends were measured  The value of the radial runout  r _ r is in cross-sections. In case of cylindrical surfaces, cross- calculated as the difference between maximum sections represent intersection lines of the surface and planes d m a x and minimum d m in from the measured which are perpendicular to the rotation axes. For face surfaces, cross-sections represent intersection lines of the surface and points of the surface P to the rotation axis. cylindrical surfaces which axis and centre coincide with the The face runout of surface 3 is calculated as the difference normal plane vector. The coordinates of the measured points of maximum and minimum distances from the measured were saved as *.txt files for further analysis in the MATLAB points of face 3 to the plane perpendicular to the rotation axis. system. The coincidence of the modelling results with actual After downloading the point coordinates on the surfaces, parameters obtained during the assembly was estimated by they are processed and brought to a specific structure for calculating absolute deviations: further creation of actual surfaces. Processing of the point  a  Pm e a s  Pm , (1) coordinates lies in smoothing outliers and calculating point coordinates which are not enough to build the data structure. and relative deviations: The coordinates were smoothed with the moving average  rel   a / T  1 0 0 % , (2) method. Calculation of the point coordinates lies in creating cross-sections of the part surfaces by approximating or where Pm – is the parameter calculated as a result of interpolating the measured sets of surface point coordinates modelling; using spline functions in the form of profiles or surfaces [12]. Pm e a s is the measured parameter. The general view shows the complex part surfaces in a portion way, like a patchwork quilt. Complex curves and IV. NEURAL NETWORK MODEL OF GEOMETRIC ACCURACY surfaces in CAD systems and metrology software of FORECASTING measuring equipment are described using spline equations. A 3rd degree normalized cubic spline, namely the Hermite curve, To obtain an adequate forecast using the neural network, was used for mathematical representation of spatial curves the following is required: determine the composition of the [13]. The surfaces created on the basis of the bicubic portions network input parameters; create a sufficiently large quantity were used to describe the part surfaces with geometrical of training samples; select an appropriate architecture of the deviations of the forms (Coons portions [13]). neural network. The sufficient volume of the training sample, as a rule, exceeds the available statistics on measurements. In So digital models of the parts represent a set of the addition, the parts obtained in a certain batch may not cover interconnected part surfaces involved in the assembly and all the potential cases, and the next batch will have control. combinations of deviations absent in the previous one, which will have an effect on the forecast quality. This caused the B. Virtual calculation of the part assembly, result saving selection of artificial modelling of the training set of actual To solve the contact task using the surface models, an models based on the data of the available production statistics. iterative algorithm has been developed; it allows calculating the parts mating without taking into account deformation of A. Creating a set of actual part models the parts in the process of assembly detailed in [7]. The The measured points were modelled using production algorithm for determining the mated state assumes iterative statistics on geometrical deviations of cylindrical and flat parts VI International Conference on "Information Technology and Nanotechnology" (ITNT-2020) 145 Data Science of the assembly parts to create training and test data sets. The  Root-mean-square error (RMSE) of predicted and cylindrical and flat ends of the parts are considered. The point actual parameters. coordinate can be set by formula: Let's specify the order of these values calculation: p m   р n  п   f   R  t , (3)  Calculate the error between the predicted and actual parameters: where p m р n is the vector of (х, у, z) point coordinates of the measured (modelled) and nominal (CAD) surfaces,  п  P p r  Pa . (5) respectively;   The number of errors is counted within the allowable п is the normal vector in point р n ; area N  a d d . The allowable area of errors is calculated  f is the form deviation value in point р n ; as a percent of the maximum value of the predicted parameter, namely 10 %.  R , t is the turn matrix and the vector of coordinate  The forecast accuracy is calculated as the quantity transposition for point р n characterizing the arrangement N  a d d to total sample volume ratio: deviation.  add  N  add / N com . (6) The Fourier’s series were used for the form deviation  f [8].  The root-mean-square error value is calculated by formula: B. Training the neural network, assessing the forecasting errors RSM E   2 / N com . (7) A widely used architecture was selected as the neural network for forecasting tasks, namely fully connected radial V. WORK RESULTS basic networks [17]. The architecture of the generalized The required data on the assembly part deviations were regressive neural network (GRNN) has two layers – hidden obtained as a result of the part measurement. The rotor was radial basic layer and output linear layer. A radial basic neuron assembled. The assembly was installed in a special tool and converts the distance from this input vector into the “center” the measurement was made on the CMM. This stage of corresponding to it by a certain non-linear law (generally, the assembly is performed for four shaft positions. The shaft is Gaussian function). The influence parameters that have an rotated at an angle of 90̊ for each new position. The points of effect on displacements Ps p in neurons and are an adjusted the surfaces Z and P are measured (Fig. 1) in relation to the neuron parameter is the changed parameter of the network. shaft bases. The radial and face runouts are calculated. The The number of neurons in the radial basic layer is equal to the measured data of certain parts were processed and the number of elements in the training set. Figure 2 shows the assembly parameters were calculated virtually in the network architecture when the training sample number is MATLAB system. The results of the assembly parameters 9,500. measured in the experiments and resulted from the virtual modelling are given in Table 1. TABLE I. COMPARISON OF THE ASSEMBLY PARAMETERS OBTAINED IN THE PROCESS OF MODELLING AND MEASUREMENT Paramet Angle, ° Pm e a s Pm  a , mm  от н , % er 0 0.133 0.13 0.003 2.31 90 0.139 0.14 -0.001 -0.71 Pr r Fig. 2. GRNN architecture for parameter forecasting. 180 0.150 0.15 0.000 0.00 270 0.111 0.13 -0.019 -14.62 The data that has a direct correlation dependence on the 0 0.078 0.10 -0.022 -22.00 assembly parameters shall be entered into the network. The 90 0.107 0.09 0.017 18.89 following derived parameters were used as these inputs: Pt r parameters of the harmonic series describing the form 180 0.109 0.10 0.009 9.00 deviation for all the surfaces; radius deviations in case of 270 0.090 0.09 0.000 0.00 cylindrical ends; parameters of surface parallel alignment; displacement of cylindrical end centres. A total of 128 Based on the results in Table 1 it may be concluded that parameters were used for the assembly of three parts under the modelling results are mostly sufficiently close to the consideration. The input data was adjusted within the range experimental data when the developed digital counterpart is [0; 1]. used. The differences are explained by the following: measurement errors and creation of the part surface models; Forecasting errors should be estimated to assess the results necessity of part stiffness consideration; assumptions made in of the assembly parameter forecast and update the structure of the process of the assembly model development. Elimination the selected neural network model. The parameter forecasting of the above reasons to reduce the number of deviations is the errors are estimated by two criteria: task of further development of the digital model.  Share of predicted values within the allowable Various cases of the assembly under consideration were accuracy  a d d . modelled to make a forecast using neural networks. A total of VI International Conference on "Information Technology and Nanotechnology" (ITNT-2020) 146 Data Science 10,000 cases were modelled. Their calculation lasted 72 hours TABLE III. VALUES  r e l FOR THE FORECAST of machine time, in the computer with AMD Ryzen 7 2700 Eight-Core processor, clock rate of 3.2 GHz, and RAM 32 Gb. Nv 500 1000 2500 5000 9500 128 parameters of geometrical deviations of the surfaces and Angle, resultant runouts are saved for each case. The allowable error  r e l for Pr r , % ° field is calculated as a percent of the maximum value of the 0 -10.00 -11.54 -6.15 -10.77 -10.77 predicted parameter and is accepted as equal to 10 %. As to 90 -19.29 -18.57 -15.00 -20.71 -20.71 the parameter Pr r , the error tolerance (on the basis of 10 % of 180 -22.00 -19.33 -16.67 -26.00 -20.67 the maximum parameter value for the assembly) is 270 -3.85 -5.38 -5.38 -8.46 -6.92 Angle, ± 0.047 mm; as to the parameter Pt r , the tolerance is °  r e l for Ptr , % ± 0.049 mm. The value of the parameter Ps p was selected so 0 11.00 10.00 6.00 9.00 15.00 that the total value of the parameter R S M E is minimum and 90 21.11 24.44 21.11 24.44 27.78 the value of the parameter  a d d is maximum. The parameter 180 14.00 15.00 8.00 6.00 13.00 P s p was selected within a range of 0.001–3. The test sample 270 27.78 24.44 18.89 20.00 25.56 M 16.13 16.09 12.15 15.67 17.55 was not changed and amounted to 500 cases. Different M rr 13.78 13.71 10.80 16.49 14.77 volumes of training samples were N v were considered: 500, 1,000, 2,500, 5,000, and 9,500 cases. M tr 18.47 18.47 13.50 14.86 20.33 128 parameters of the measured surfaces, which assembly parameters are given in Table 1 for four positions, were None of the deviations has exceeded the tolerance by 10 % entered after selecting the parameter Ps p and network of the maximum parameter value. The results show that the training. Table 2 contains the results of the network operation selected neural network architecture allows achieving the related to forecasting parameters of radial and face runouts for same accuracy, when the training sample value is 2,500 cases the measured assembly. and the parameter is Ps p =0.5, as the developed digital model based on the direct modelling of the part surfaces and TABLE II. RESULTS OF THE NEURAL NETWORK MODELLING assembly process. Ps p 1 1 0.5 0.5 1 VI. CONCLUSION Nv 500 1000 2500 5000 9500 The article contains the research results that allow Angle, ° Pr r , mm forecasting the resultant assembly geometrical parameters on the basis of the measured data. The problem of creating the 0 0.117 0.115 0.122 0.116 0.116 digital counterpart of the rotor assembly that allows 90 0.113 0.114 0.119 0.111 0.111 reproducing the part assembly process on the actual surfaces 180 0.117 0.121 0.125 0.111 0.119 has been solved. The tasks of modelling the actual surfaces 270 0.125 0.123 0.123 0.119 0.121 using small statistics and modelling the measurement data Angle, ° Pt r , mm itself have been solved. The relative deviations of forecasting 0 0.111 0.110 0.106 0.109 0.115 the assembly of three parts of the turbine rotor do not exceed 90 0.109 0.112 0.109 0.112 0.115 22 % and allow speaking about the adequacy of the proposed 180 0.115 0.108 0.106 0.113 decision. A total of 128 affecting parameters of geometrical 0.114 deviations have been selected. The radial basic neural network 270 0.115 0.112 0.107 0.108 0.113 appropriate for forecasting the assembly parameters, which accuracy is comparable to the direct modelling performed The values of relative deviations  r e l of the data in Table using the digital counterpart of assembly, has been created and trained. The use of the trained neural network to forecast the 2 are considered in Table 3. The measurement results in Table assembly parameters of the assembly under consideration 1 are taken as the basis. Besides, Table 3 includes the allows significantly reducing the labor intensity of arithmetical means of the parameter deviations (overall calculations and using the developed decision immediately average M , M r r average for Pr r , and M tr average for Pt r after the part measurement and measured data processing. In ). addition to the solved tasks, there is a number of other tasks Generalizing the results in Tables 2 and 3 it may be noted (labor intensity of measurements, consideration of the part that the highest accuracy is achieved when the volume of the stiffness during assembly modelling) which will be the focus of further researches. training sample amounts to 2,500 cases. Based on the average and limit values  r e l in Table 3, the number of radial runout ACKNOWLEDGEMENT forecast errors is less than the number of face runout forecast The work was supported by the Russian Federation errors. At the same time the absolute values of the limit errors President's grants (project code СП-262.2019.5). in forecasting with the help of direct modelling and neural Experimental studies were carried out on the equipment of the network are close (results in Tables 1 and 3): for Pr r – (- centre for the collective use of CAM technologies of the 14.62 %) and (-16.67 %), respectively, in case of direct Samara University (RFMEFI59314X0003). forecast and forecast with the help of the neural network; for Ptr – 21.11 % and (-22 %). VI International Conference on "Information Technology and Nanotechnology" (ITNT-2020) 147 Data Science REFERENCES [9] A. Korolev, A. Kochetkov and O. 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