Elsevier

Applied Soft Computing

Volume 69, August 2018, Pages 610-624
Applied Soft Computing

Using robust generalized fuzzy modeling and enhanced symbolic regression to model tribological systems

https://doi.org/10.1016/j.asoc.2018.04.048Get rights and content

Highlights

  • Application of fuzzy and symbolic regression modeling for friction systems.

  • Enhanced fuzzy modeling algorithm for datasets with many binary and noisy variables.

  • Multi-objective symbolic regression to optimize complexity and prediction errors.

  • Comparison of fuzzy and symbolic regression models with state-of-the-art techniques.

Abstract

Tribological systems are mechanical systems that rely on friction to transmit forces. The design and dimensioning of such systems requires prediction of various characteristic, such as the coefficient of friction. The core contribution of this paper is the analysis of two data-based modeling techniques which can be used to produce accurate and at the same time interpretable models for friction systems. We focus on two methods for building interpretable and potentially non-linear regression models: (i) robust fuzzy modeling with batch processing and an enhanced regularized learning scheme, and (ii) enhanced symbolic regression using genetic programming. We compare our results of both methods with state-of-the-art methods and found that linear models are insufficient for predicting the coefficient of friction, temperature, wear, and noise-vibration-harshness rating of the tribological systems, while the proposed robust fuzzy modeling and the enhanced symbolic regression approaches, as well as the state-of-the-art regression techniques, are able to generate satisfactory models. However, robust fuzzy modeling and enhanced symbolic regression lead to simpler models with fewer parameters that can be interpreted by domain experts.

Introduction

Friction models have been studied for more than a hundred years [1] and are essential to understanding and accurately describing tribological systems, which occur in almost all mechanical systems. The main difficulty in modeling friction is that it is a complex phenomenon that depends on a large variety of parameters, including mechanical properties (e.g., surface roughness and hardness, lubrication), load (e.g., pressure, energy and sliding speed), and environmental conditions (e.g., humidity and temperature) [1]. Further, friction is a dynamic phenomenon, as abrasive effects, material deterioration, and temperature changes induced by friction strongly influence friction characteristics [2].

Mathematical models of friction systems that are derived solely from mechanical principles are drastically simplified and therefore limited [2]. The forces acting in tribological systems at the micro- and nano-levels depend on the surface characteristics of the friction materials and on lubrication, and are difficult to describe mathematically [1], [3].

Here we focus on friction systems in which oil is used both for lubrication and for cooling the friction system. In particular, we study friction systems as applied in clutches and automatic transmission systems employed in power-trains [4], [5], [6], [7]. Relevant factors include the composition and mechanical properties of the friction material, the oil, and the geometry of groovings on the friction material. Many of these factors are hard to capture in a mathematical model that is based solely on physical principles [8].

The overall objective is to improve the design and dimensioning process of tribological systems by using accurate models for predicting the most important characteristics of these systems. These models are intended for inclusion in an expert system [9] for the virtual design and dimensioning of friction systems [8], [10]. Using and combining the resulting models should enable fast exploration of viable design alternatives and thus reduce the number of tests of physical prototypes on expensive test benches [11], [12].

The literature on friction models and tribological systems is extensive, and new articles appear regularly in multiple journals focusing specifically on tribology and wear. Friction models can be roughly categorized into analytical models derived from physical principles and purely empirical models [1]. Analytical models, such as that introduced by De Wit et al. in [2], are usually sophisticated and describe friction forces based on surface characteristics of the friction materials. Drawbacks of these models are their high complexity and their limited applicability to real world scenarios, mainly because important aspects such as non-linear dynamics, abrasive effects, or the deterioration of oil are not captured. It is difficult, highly demanding, and time-intensive task to adapt or extend these models such that they accurately capture additional dynamics that are relevant in practical applications. Models based on finite element simulation [3] are computationally expensive and also have the drawback that it is almost impossible to accurately model relevant effects that occur in practical applications. The main challenge is that “friction and structural modeling for dynamic simulations must include careful consideration of not only the information required as output of the simulation, but also the simulation objectives themselves for computational efficiency, accuracy, and fidelity.” [1]

Empirical modeling has advantages when the primary goal to estimate key characteristics of tribological systems in real-world applications. Empirical models have predictive capabilities, but in general they do not provide a detailed, physically correct and general description of friction. Depending on the simulation objectives, these empirical models can be relatively simple, such as the non-linear numerical model presented in [13], which predicts the coefficient of friction based solely on load parameters. An example of a more complex numerical model that also includes effects of surface characteristics was introduced in [14]. In both of these cases, the model structure was manually defined based on intuition and experience of the main effects, and the parameters of the model were been optimized to fit the model to measurements.

Artificial neural networks (ANN) have been recently applied successfully in data-based modeling of tribological systems [8]. For example, wear of brake friction materials has been predicted by neural networks that use the complete formulation of the friction material, important manufacturing conditions, load parameters, sliding velocity and the temperature as input [15]. Extension of this work has resulted in a so-called “neuro-genetic model” — a dynamic model based on a recurrent neural network for the optimization of brake pressure “to achieve the maximum and stable brake performance during a braking cycle” [16]. ANNs have also been employed to predict (i) friction behavior (especially the coefficient of friction) in sliding acceleration motions as they occur in brake and clutch systems [6] and (ii) wear and mechanical properties such as the compressive strength and modulus of polymer-matrix composite materials. Another successful application of ANNs predicting the friction coefficient of ceramic compounds was described in [17].

Hosenfeldt and colleagues presented an approach using data mining: They trained an ANN model that “can predict the tribological behaviour of camshaft and bucket tapped systems” [10], and stated that they achieved “a deviation of 8%”, which “is a very good result, especially when considering that the measurement error with reference to friction is 5%” [10]. As input variables for the ANN they used, among others, the type of coating, its hardness, the surface quality, lubricant oil additives and their concentrations, the base oil and its viscosity, and the material of the counterbody and its structure. The network was trained to predict the friction and wear values of the camshaft and bucket system. Generally, ANNs produce black-box models with high structural complexity that cannot be inspected and validated by domain experts. The complex structure of neural networks makes it difficult to discuss the general behavior of the model; rather, one must rely on empirical estimates such as the cross-validation error or partial dependency plots [18].

We have previously shown that fuzzy modeling methods and symbolic regression using genetic programming can be used successfully to model key metrics of tribological systems [19]. In terms of modeling performance and model accuracy, these methods are better than linear modeling and comparable to state-of-the-art methods for non-linear modeling. Here we describe in detail these algorithms, which are based on fuzzy modeling and symbolic regression, and present an elaborate analysis of the resulting models and the most relevant variables for predicting each of the target variables.

The virtual design of clutch and transmission systems requires prediction models that can subsequently be used to optimize design parameters. In this context, it is essential that models are valid and in line with the experience of experts. Therefore, it must be possible to inspect and interpret the models. The main research questions that we focus on in this work are:

  • (RQ1) whether state-of-the-art data-based modeling techniques (aside from ANNs) are able to produce satisfactory models,

  • (RQ2) whether and how robust fuzzy modeling and enhanced symbolic regression methods can be used, and how the predictive accuracy of such models compares to models produced by state-of-the-art methods, and

  • (RQ3) to which extent these models can offer interpretable insights into system behavior.

Our methodological approach is based on the following assumptions.

  • We assume a representative dataset that contains measurements of all relevant parameters for friction performance, and that the quality of measurements is sufficient to identify relevant correlations.

  • We assume that there are no unknown or not measurable factors that affect friction performance.

  • We assume that in the data collection process, parameters are changed independently of each other. This is a necessary precondition for the identification of relevant parameters for friction performance with data-based methods.

  • We further assume that in the data collection process, parameters are varied over a wide range of possible settings. In particular, the parameters used in the data collection must span the complete space of all possible parameter configurations for which the model is later used for prediction. Data-based models often have weak extrapolation properties. Collecting data over the complete space of possible values prevents extrapolation.

  • We assume that friction performance depends non-linearly on design and load parameters, and that parameter changes are not completely separable.

  • We assume that the most important effects can be approximated using simple functional expressions or IF-THEN rules.

  • We assume that the number of numeric parameters of a data-based model is representative for its complexity. In particular, we assume that a model with few parameters is easier to read and understand than a model with many parameters.

We built prediction models for nine performance characteristics that must be predicted when designing and dimensioning clutch and transmission systems. ANNs have previously been used successfully in similar scenarios [10]. We also applied several other highly-tuned state-of-the-art techniques for non-linear function approximation: support vector regression [20], random forests [21], and gradient boosted trees [18].

Additionally, we focused on two architectures for data-driven regression models: namely Takagi-Sugeno-type (TS) fuzzy systems [22], as recently introduced in generalized form [23], and symbolic regression models [24], [25]. Both offer some sort of interpretability; fuzzy systems contain rules which are representable in linguistic form and symbolic regression models can be read as mathematical expressions. However, both approaches require using model simplification techniques such as linguistic enhancements and post-modeling complexity reduction after the modeling process [26], [27], [28].

To cope with the large number of binary indicator variables and significant noise levels which we encountered in this application, we improved two learning engines used for these model architectures:

  • 1.

    In (generalized) TS fuzzy systems, the main challenges arise from the large number of binary indicator variables and the fact that the remaining variables are affected by significant noise levels. Therefore, we integrated a convex combination of Lasso and ridge regression for optimizing the linear consequent parameters. This approach is similar to the idea of elastic net regularization [29], [30], but tuned to the specific (locally weighted) optimization problem for generalized TS fuzzy systems (inducing local learning of rule consequent parameters). For learning the rule structure and the rules’ antecedent parts, we employ the Gen-Smart-EFS technique, originally developed for streaming data [31], which we adapted for batch off-line use. We call this new method Robust-GenFIS (short for Robust Generalized Fuzzy Inference Systems).

  • 2.

    In the case of symbolic regression models, the main challenge lies in producing models that are accurate, compact and interpretable. Therefore, we augmented the genetic programming (GP) approach described in [25] with multi-objective optimization of complexity and predictive accuracy [32]. We also included gradient-based optimization of numeric constants as a memetic extension of GP [33], [34], which has been shown to improve both accuracy and parsimony of symbolic regression solutions [34].

Concentrating on prediction accuracy and model complexity, we compared the new fuzzy modeling method Robust-GenFIS to related and widely used fuzzy system extraction algorithms such as LoLiMoT [35], genfis2 (a modified version, [36]) and FLEXFIS [37].

In the following section, we outline the methods, specifically the applied data acquisition and preparation methods, data-driven fuzzy modeling and its enhancements to robust fuzzy models, and symbolic regression techniques and their enhancements, and give an overview of the state-of-the-art methods used for comparing the performance of these modeling techniques.

Section 3 describes the experimental setup, that is test protocols and evaluation strategy, and parametrization of the newly developed and standard methods. Section 4 provides a detailed description of the modeling results, focusing on model accuracy, compactness of the models, variable relevance, model interpretation and a discussion of the results.

Section snippets

Data acquisition and preprocessing

The data used for modeling were acquired through extensive testing of numerous friction plates using various friction materials, oils, and groovings on test benches specifically designed for evaluating the friction characteristics of automatic transmission plate clutches (wet friction systems). Two types of testing procedure were used. The first focused on friction characteristics and included static (breakaway) and dynamic tests, and the second focused on measuring NVH

Test protocol and evaluation strategy

For techniques that require tuning of hyper-parameters, we performed grid searches using a customized validation scheme similar to traditional cross-validation [84]. In particular, we allocated rows to folds deterministically rather than randomly. This was necessary because a single test run produces multiple data rows of measurements — one row for each cycle. Since we had no information from an actual (partial) test run for the deployment of models, we had to guarantee that all data from a

Model accuracy

Table 3 shows the results for all modeling methods, which are grouped into linear SoA methods, non-linear SoA methods, and specifically enhanced symbolic regression and various fuzzy modeling variants including the newly proposed robust generalized fuzzy systems training method (Rob-GenFIS).

The values shown in Table 3 are the relative mean of absolute errors (MAE in percent) on the test data set achieved by the models trained on the whole training data sets using the tuned parameter settings.

Summary and conclusions

We used a real-world data set recorded on a test bench for friction plates for wet clutch or transmission applications to analyze if and how key characteristics that are important in the design and dimensioning of such systems can be predicted and interpreted using regression models. In particular, we applied a number of regression techniques, including highly tuned state-of-the-art techniques for non-linear function approximation and state-of-the art fuzzy modeling techniques, and analyzed the

Acknowledgements

The work described in this paper was done within the COMET Project “Heuristic Optimization in Production and Logistics (HOPL)”, #843532, funded by the Austrian Research Promotion Agency (FFG) and within the project “Smart Factory Lab” funded within the EU programme IWB 2020 by the state government of Upper Austria.

References (91)

  • E. Lughofer et al.

    Integrating new classes on the fly in evolving fuzzy classifier designs and its application in visual inspection

    Appl. Soft Comput.

    (2015)
  • F. Serdio et al.

    Fault detection in multi-sensor networks based on multivariate time-series models and orthogonal transformations

    Inf. Fusion

    (2014)
  • E. Lughofer et al.

    Autonomous data stream clustering implementing incremental split-and-merge techniques – towards a plug-and-play approach

    Inf. Sci.

    (2015)
  • K. Tabata et al.

    Data compression by volume prototypes for streaming data

    Pattern Recogn.

    (2010)
  • E. Lughofer et al.

    Handling drifts and shifts in on-line data streams with evolving fuzzy systems

    Appl. Soft Comput.

    (2011)
  • F. Bauer et al.

    Comparing parameter choice methods for regularization of ill-posed problems

    Math. Comput. Simul.

    (2011)
  • J. Enríquez-Zárate et al.

    Automatic modeling of a gas turbine using genetic programming: an experimental study

    Appl. Soft Comput.

    (2017)
  • C. Cernuda et al.

    NIR-based quantification of process parameters in polyetheracrylat (PEA) production using flexible non-linear fuzzy systems

    Chemom. Intell. Lab. Syst.

    (2011)
  • E. Berger

    Friction modeling for dynamic system simulation

    Appl. Mech. Rev.

    (2002)
  • C.C. De Wit et al.

    A new model for control of systems with friction

    IEEE Trans. Autom. Control

    (1995)
  • K.A. Snima

    Kenngrößen und Belastungsgrenzen von nasslaufenden Lamellenkupplungen unter Dauerschlupfbeanspruchung, Ph.D. thesis

    (2005)
  • G. Rao

    Modellierung und Simulation des Systemverhaltens nasslaufender Lamellenkupplungen, Ph.D. thesis

    (2010)
  • D. Aleksendric et al.

    Soft Computing in the Design and Manufacturing of Composite Materials: Applications to Brake Friction and Thermoset Matrix Composites

    (2015)
  • E. Castillo et al.

    Expert Systems: Uncertainty and Learning

    (2007)
  • T. Hosenfeldt et al.

    Friction tailored to your requirements

  • S. Franklin et al.

    The implementation of tribological principles in an expert-system (‘precept’) for the selection of metallic materials, surface treatments and coatings in engineering design

    Wear

    (1995)
  • C. Brands

    Holistic simulation – the future approach for calculating engine systems?

  • W.-Y. Loh et al.

    Dynamic Modeling of Brake Friction Coefficients, Technical Report

    (2000)
  • V. Ćirović et al.

    Neuro-genetic optimization of disc brake performance at elevated temperatures

    FME Trans.

    (2014)
  • J.H. Friedman

    Greedy function approximation: a gradient boosting machine

    Ann. Stat.

    (2001)
  • E. Lughofer et al.

    Robust fuzzy modeling and symbolic regression for establishing accurate and interpretable prediction models in supervising tribological systems

  • A. Smola et al.

    Support vector regression machines

    Adv. Neural Inf. Process. Syst.

    (1997)
  • L. Breiman

    Random forests

    Mach. Learn.

    (2001)
  • T. Takagi et al.

    Fuzzy identification of systems and its applications to modeling and control

    IEEE Trans. Syst. Man Cybern.

    (1985)
  • A. Lemos et al.

    Multivariable Gaussian evolving fuzzy modeling system

    IEEE Trans. Fuzzy Syst.

    (2011)
  • J.R. Koza

    Genetic Programming: On the Programming of Computers by Means of Natural Selection

    (1992)
  • M. Affenzeller et al.

    Genetic Algorithms and Genetic Programming: Modern Concepts and Practical Applications

    (2009)
  • G. Kronberger et al.

    Knowledge discovery through symbolic regression with heuristic lab

  • T. Hastie et al.

    Regularized paths for generalized linear models via coordinate descent

    J. Stat. Softw.

    (2010)
  • H. Zou et al.

    Regularization and variable selection via the elastic net

    J. R. Stat. Soc. Ser. B

    (2005)
  • E. Lughofer et al.

    Generalized smart evolving fuzzy systems

    Evol. Syst.

    (2015)
  • M. Kommenda et al.

    Evolving Simple Symbolic Regression Models by Multi-Objective Genetic Programming

    (2016)
  • A. Topchy et al.

    Faster genetic programming based on local gradient search of numeric leaf values

  • M. Kommenda et al.

    Effects of constant optimization by nonlinear least squares minimization in symbolic regression

  • O. Nelles

    Nonlinear System Identification

    (2001)
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