Available online at www.sciencedirect.com
ScienceDirect Procedia CIRP 33 (2015) 550 – 555
9th CIRP Conference on Intelligent Computation in Manufacturing Engineering
Selective rotor assembly using fuzzy logic in the production of electric drives Daniel Coupek*, Aybike Gülec, Armin Lechler, Alexander Verl Institute for Control Engineering of Machine Tools and Manufacturing Units (ISW), Seidenstraße 36, 70174 Stuttgart, Germany * Corresponding author. Tel.: +49 711 685 84523; fax: +49 711 685 74523. E-mail address:
[email protected]
Abstract
The growing sector of sustainable mobility requires an increasing number of electric drives for automobiles, thus increasing the need for their efficient mass production. In current production scenarios, End-Of-Line (EOL) inspection is applied to determine the quality of the assembled final product. The work presented in this paper is developed in the project MuProD, which aims at developing an innovative quality control system to change the current concept of EOL quality control. The case under investigation is the production of electric drives where the rotor is composed of several magnetized stacks. The magnetic properties of each stack differ due to variations of the single magnets and the magnetization process itself. In addition, handling of the magnets within the production line may cause cracks that decrease the strength of the magnetic field. This paper proposes a new solution for deviation compensation in the production of electric drives by selective assembly based on a crisp classification and a Mamdani style fuzzy inference system. The magnetic field of each stack is measured after the magnetization stage, yielding a discrete space-resolved magnetic profile. This magnetic profile is transformed into another feature space through a combination of feature selection and feature extraction to reduce the dimension. Based on these new features, the stacks can be classified into crisp sets. In the second part, an appropriate fuzzy rule base for the matching of the stacks is developed to obtain a uniform magnetic field of the rotor. By applying this assembly strategy, the rotor and consequently the final motor reaches desired quality targets although deviations in the single stacks are present. The benefits of the approach are validated within an industrial context. © 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license © 2014 The Authors. Published by Elsevier BV. Selection and/or peer-review under responsibility of Professor Roberto Teti. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Selection and peer-review under responsibility of the International Scientific Committee of “9th CIRP ICME Conference” Keywords: Assembly; Fuzzy logic; Drive; Selecvtive assembly; Feedforward Control
1. Introduction The increasing need for electric drives in the automotive sector requires an efficient mass production, considering that optimized methods and machines from current production lines for combustion engines cannot be transferred to electric drive production directly. The challenges of the electric drives production and strategies to overcome them are part of the research carried out in the European funded project MuProD, while the focus is on the rotor magnetization and assembly [1]. A previous work presented selective and sequential assembly concepts for this kind of electric
drives [2]. This paper extends the concept of selective assembly by combining it with fuzzy logic in order to choose the optimal combination of rotor stacks for compensating deviations in the magnetic field. Section 1 of this paper introduces briefly the production line, the composition of the electric drive and the space-resolved measurement developed within MuProD. Furthermore, it provides a literature review about selective assembly and fuzzy logic. The features of a rotor stack and feature reduction are discussed in Section 2. Section 3 describes the fuzzy system, its components and experimental results. A brief summary and outlook is given in Section 4.
2212-8271 © 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Selection and peer-review under responsibility of the International Scientific Committee of “9th CIRP ICME Conference” doi:10.1016/j.procir.2015.06.074
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1.1. Electric Drives Production The assembly line of electric drives is composed of two main branches, respectively dedicated to the assembly and magnetization of the rotor and to the production of the stator. The focus of the MuProD activity is the rotor line. In detail, this line is composed of seven main stages, dedicated to the following operations Mi: x M1: loading of the stacks on the pallet. x M2,1,M2,2: assembly of the magnets on the stacks. x M3: stack magnetization and total flux measurement. x M4: heating station. x M5: rotor assembly machine. x M6: rotor balancing station. x M7: rotor marking station. After assembling the rotor and the stator, the completed motor undergoes the EOL inspection. At this stage, motor characteristics as well as customer requirements such as torque, speed, etc. are tested. Though some of the previously carried out production stages already include subordinated testing steps, defects due to chain-linking or super positioning of errors are only detected at the EOL inspection. Since failures in the magnetic circle have a considerable effect on the performance of the whole electric machine, a continuous high quality of the permanent magnet rotor is necessary.
Due to variability of the magnetization process and the material properties, the magnetization of the magnets differs from the target value. Additionally, handling of the magnets within the production line can cause cracks and decrease the strength of the magnetic field. In order to increase the motor quality, a uniform magnetic field must be achieved for the complete rotor. 1.3. Space-resolved Measurement In the current production line, the stack undergoes a total flux measurement after the magnetization process stage, resulting in one cumulative value for each stack. This value indicates whether the corresponding stack is outside a tolerance band and to be classified as defect. However, the magnet(s) responsible for causing this deviation cannot be identified. Within MuProD, a new space-resolved inspection strategy was developed. A hall sensor is located next to the rotating stack yielding a space-resolved (0°-360°) distribution of the magnetic field B(φ) with φ=0…2Π (Fig. 2, left). For the optimization method described in this paper, this highresolution signal is transformed, so that we obtain one value per magnet mij resulting in the discrete signal B(k) with k=1,…,24 (Fig. 2, right). The value B(k) is an indicator for the magnetic strength of the corresponding magnet k. 0°
1.2. Composition of the Rotor
SP = 4
In the production line a number Ptotal of different rotors p can be manufactured, with p = 1, …, Ptotal. One rotor p is composed of SP laminated rotor stacks, which is the size of the batch of stacks to be assembled. The part is called stack, as it consists of several metal sheets that are stacked. Each stack contains MP magnets, where MP is an even number as positive and negative poles alternate. In order to apply optimization methods to find the optimal assembling policy, the rotor is represented as a two dimensional matrix (Fig. 1), where the rows represent the stacks and each matrix element mij stands for one magnet (column i, row j). Columns of the matrix contain magnets of the same polarity. Consequently the matrix dimension is SP x MP. By definition, odd indexes stand for positive and even indexes for negative polarity of the corresponding magnet. + + + +
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Fig. 1. Rotor consisting of SP = 4 stacks and MP = 12 magnets (matrix dimension is 4x12).
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Fig. 2. Space-resolved magnetization measurement B(φ) of one rotor stack (left) and the resulting discrete signal B(k) (right).
1.4. Literature Review State of the art quality control in the production of electric drives is the end of line (EOL) inspection [2]. This means that the defect is detected at the final inspection stage [3]. The main drawback of EOL inspection is the late and off-line inspection at the final stage of the manufacturing chain, where already all possible defects of the production chain have been accumulated. Thus, a defective workpiece is machined wasting time, money and energy resources for creating a final product, which is out of tolerances and has to be recycled or scrapped. Therefore, methods for feedforward control are considered in order to
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compensate deviations generated at previous process stages [4-5]. Selective assembly has traditionally faced the problem of obtaining high precision assemblies from relatively low precision components by mating compounds according to their geometry. Typically, there are two different compounds that mate, like shafts and holes. The application of selective assembly strategies in the automotive industry has also been investigated. In [6] a General Selective Assembly approach is presented, which extends the classical approach of selective assembly. Selective assembly can be split up into grouping similar parts and into combining parts from corresponding groups. Existing approaches use clustering algorithms to create groups of mating parts. In [7] and [8] the impact of reducing the dimension of the feature vector on clustering and classification processes is analysed and methods are proposed to reduce the dimension of crisp features. Fuzzy set theory, introduced by Zadeh in 1965 [9], provides means to perform calculations on systems that are imprecisely defined and can be used to mimic human reasoning when calculating similarity of complex and multivariate compounds. Within the wide application field of grouping technology, various investigators have used fuzzy sets to form groups. In [10] a method to combine fuzzy and crisp features for grouping is presented. Several groups like [11] and [12] used fuzzy clustering algorithms. In order to combine the parts, there are few approaches, which apply for multivariate compounds and fuzzy features. In [13] a fuzzy evolutionary programming method was developed to find the best combination of selective groups for complex assembly, using the example of piston and cylinder assembly. In [14-15] Mamdani and Assilian proposed a fuzzy decision making process using linguistic variables. This Mamdani type Fuzzy Inference System has become an acknowledged method in
controller design, especially when a model is not available or too complex. It has not been used in selective assembly so far. 2. Feature Reduction Selective rotor assembly requires the identification of all relevant stack features. Before matching the stacks, it is necessary to reduce the dimension of the feature space, which is MP=24 (number of magnets), in order to reduce the complexity of the data set. Furthermore, an efficient compression of the features enhances comprehension of the data set and enables generalizing upon the stack (see also [7-8]). This reduction strategy consists of two steps, Feature Selection and Feature Extraction. First, redundant information is removed, by neglecting all magnets within a tolerance band (Fig. 3). Only the magnets out of tolerance are considered to characterize the deviation of the stack. In the second step, new features are created based on the remaining magnets. These new features extract latent information and contain the essential characteristics of the stack. In case of the rotor stacks, the two most dominant structures out of the 24 magnet values are extracted. A structure is defined as a section of the stack that only contains magnets of the same deviation (positive or negative). Each structure is characterized by four features, namely its sum of deviations (Ai), the height of its peak (Pi) and the number of included magnets (Li). In addition, we use the distance (D12) between the balance points of each structure. The feature reduction W maps the original vector v containing the 24 magnetization values to the feature vector w:
ܹǣ ݒԦ ՜ ݓǡ ݒԦ אԹଶସ ǡ א ݓԹ ݒԦ ൌ ݉ଵ ݉ ڮଶସ ் ݓൌ ܣଵ
ܲଵ
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(1) (2) ்
(3)
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Fig. 3 summarizes the steps of feature reduction described above exemplarily for one stack taken from the production line. In the last step, the whole stack is shifted to start with the balance point of the first structure β1. In order to sustain the convention of odd and even positions with the polarity, β1 is denoted as magnet number one, if it is an odd number; otherwise β1 is denoted as magnet number two. Within this feature space a first classification can be performed. Following upper classes are defined: PP, NN, PN, NP, N, P. Thereby, P indicates that the feature characterizes a positive deviation and N a negative one. PP means that both structures have positive deviations from the tolerance; the same applies for the other classes. In the classes P and N, there are stacks, which have only one positive or negative structure.
computational effort and time, a trained human operator could find matching stacks in an intuitive way. A controller based on fuzzy logic is an effective way to mimic the reasoning of an expert. Thus, a fuzzy inference system (FIS) is used to match the stacks of the six upper classes. Here, the mechanism is exemplary performed for the seven stacks of the upper classes PN and NP shown in Fig. 5. 3
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Table 1. Inputs of stack combinations PN-NP.
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where w1 and w2 are the feature vectors of the corresponding stacks. For the stacks in Fig. 5, the input vectors are given in Table 1. The fuzzifier converts the crisp inputs to degrees of match with linguistic variables using the membership functions stored in the database. With use of the fuzzy If-Then-Rules from the rule base,
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A widespread method, that suits the human mind well, is the FIS of Mamdani and Assilian [14-15]. The three main parts are a fuzzifier, an inference block with rule base and a defuzzifier. The FIS calculates a suitability index for two stacks based on the similarity of their features. The input x to the FIS is given by
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Fig. 5. Stacks S1,S2,S3 in class NP (left) and S4,S5,S6,S7 in PN (right).
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Given a number of different stacks in the upper classes, those two stacks from opposed upper classes (PP-NN, PN-NP, P-N), that have the maximum percental compensation should be matched together in order to minimize the deviation of the magnetic field. Experiments gave evidence to the assumption that the two dominating features are the length and the height of a structure. The exact interaction between field lines of different magnets and stacks is unknown, as the relation is highly nonlinear and complex in the assembled rotor. However, models derived from experiments allow the prediction of the basic behaviour. The optimal compensation occurs, when two stacks form the exact mirror image of each other: B1(k)=-B2(k). In real applications, this case can be only approximated. In order to find the optimal matching policy several features have to be compared, which results in a huge amount of possible permutations. While an optimization algorithm requires solving medium
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3.00 3.43 0.10 5.00 1.00] 4.00 4.67 1.34 5.00 5.00] 9.00 4.30 1.77 3.00 7.00] 8.00 1.72 1.72 0.00 4.00] 1.00 0.48 0.48 0.00 0.00] 14.00 0.85 0.05 2.00 12.00] 4.00 0.70 0.98 1.00 3.00] 3.00 0.54 0.26 1.00 1.00] 10.00 0.17 0.69 1.00 11.00] 7.00 0.84 1.54 3.00 6.00] 14.00 2.08 0.30 3.00 2.00] 1.00 1.71 0.13 1.00 14.00]
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Fig. 7. Aggregated Output Distribution and defuzzification with centroid method for input of S1 and S3.
fuzzy reasoning is performed in the inference block. Finally, a defuzzifier is applied to obtain a crisp output, which is a measure of suitability. The Matlab® Fuzzy Logic Toolbox offers an environment to create and tune Mamdani style FIS, which has been used for the implementation. Trapezoidal-shaped membership functions have been used for inputs and outputs (Fig. 4). Each input is a linguistic variable and has three linguistic terms. The curves are given by
generate a single output distribution (Fig. 7). The centroid method is used to convert the output distribution into a crisp value and is given by
(5) where the parameters a,b,c,d have been adjusted in an iterative process by considering the ranges of values, that the features can reach. These were estimated by statistical analysis of measured data. 3.2. Fuzzy Rule Base
ݕ ൌ
ݕ݀ ݕ ߤ ڄ ݕ ݕ݀ ݕ ߤ
(6)
where yD is the defuzzified output variable, μ(y) is the aggregated membership function and y is the output variable. Table 3 ranks the 12 combinations according to the suitability value that was calculated by the FIS and the plot shows the corresponding stacks. The combination of the two stacks ʹ and S5 achieved the highest suitability value and the squared sum of the deviations decreased by 65% compared to a noncompensated assembly. It can be seen that selective assembly based on fuzzy logic decreases the variability of the magnetic field compared to random assembly currently applied in industry. Table 2. Fuzzy rule base for selective rotor assembly.
Knowledge about the mechanism of compensation is introduced into the FIS through the rules. The fuzzy rule base (FRB) should cover all the possible combinations of linguistic variables and terms, reaching 37=2.187 rules, increasing the computational effort and consequently the calculation time. There are possibilities for reducing the FRB, especially in non periodic systems with a relatively small number of extreme points [16]. In the case of selecting two matching stacks out of a limited number of stacks, the FRB can be reduced. The resulting FRB for selective rotor assembly is given in Table 2. 3.3. Implication and Defuzzification Given a certain input, the FIS calculates the qualified consequences for each rule depending on its activation strength and weight (Fig.6). The consequences of each rule are aggregated by the maximum method and
Rule Antecedent dA1 is small dA1 is medium dA1 is large dP1 is small dP1 is medium dP1 is large dL1 is small dL1 is medium dL1 is large dA2 is small dA2 is medium dA2 is large dP2 is small dP2 is medium dP2 is large dL2 is small dL2 is medium dL2 is large dS is small dS is medium dS is large
Rule Consequence good suitability medium suitability low suitability good suitability medium suitability low suitability good suitability medium suitability low suitability good suitability medium suitability low suitability good suitability medium suitability low suitability good suitability medium suitability low suitability good suitability medium suitability low suitability
Rule Weight 1 1 1 0.5 0.5 0.5 0.75 0.75 0.75 0.5 0.5 0.5 0.25 0.25 0.25 0.5 0.5 0.5 0.8 0.8 0.8
Daniel Coupek et al. / Procedia CIRP 33 (2015) 550 – 555 Table 3. Corresponding suitability values calculated by the FIS. Rank
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Plot of Sx from class PN (blue) and Sy from class NP (red)
Acknowledgements The research leading to these results has received funding from the European Union Seventh Framework Program (FP7/2007-2011) under grant agreement n° 285075 in the project MuProD, which is part of the 4ZDM Cluster. References
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4. Conclusion In this paper, the design of a novel method for selective assembly in the production of electric drives using a Mamdani FIS is presented. Based on a spaceresolved measurement discrete values of the magnetizations of the 24 magnets in the laminated stacks are calculated. The essential features are extracted and the stacks are grouped into six upper classes. A Mamdani FIS selects the best combination of two stacks from corresponding classes for achieving minimum variation in the magnetic field of the rotor. Compared to the current random assembly, the proposed method can control and successfully reduce variation in the magnetic fields of the permanent magnet rotors. The work presented in this paper will be extended from pairs of stacks to combine several matching parts stacked on each other. Furthermore, it was assumed that corresponding classes have similar frequency distributions. Methods to prevent mismatching and deadlocks, should be implemented if statistical analysis of the production line shows that corresponding classes have distinct frequency distributions. Another future research area is the development of automated online optimization of the fuzzy parameters by a feedback from the quality inspection of the assembled rotor, instead of using manually tuned parameters, resulting in adaptive fuzzy selective assembly.
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[8] [9] [10]
[11]
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[13]
[14]
[15]
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