Computational Intelligence and Neuroscience

Volume 2015, Article ID 606734, 11 pages

http://dx.doi.org/10.1155/2015/606734

## A Novel Mittag-Leffler Kernel Based Hybrid Fault Diagnosis Method for Wheeled Robot Driving System

^{1}School of Control Science and Engineering, Shandong University, Jinan 250061, China^{2}School of Computer Science and Technology, Shandong University, Jinan 250101, China

Received 3 April 2015; Accepted 22 June 2015

Academic Editor: Michele Migliore

Copyright © 2015 Xianfeng Yuan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The wheeled robots have been successfully applied in many aspects, such as industrial handling vehicles, and wheeled service robots. To improve the safety and reliability of wheeled robots, this paper presents a novel hybrid fault diagnosis framework based on Mittag-Leffler kernel (ML-kernel) support vector machine (SVM) and Dempster-Shafer (D-S) fusion. Using sensor data sampled under different running conditions, the proposed approach initially establishes multiple principal component analysis (PCA) models for fault feature extraction. The fault feature vectors are then applied to train the probabilistic SVM (PSVM) classifiers that arrive at a preliminary fault diagnosis. To improve the accuracy of preliminary results, a novel ML-kernel based PSVM classifier is proposed in this paper, and the positive definiteness of the ML-kernel is proved as well. The basic probability assignments (BPAs) are defined based on the preliminary fault diagnosis results and their confidence values. Eventually, the final fault diagnosis result is archived by the fusion of the BPAs. Experimental results show that the proposed framework not only is capable of detecting and identifying the faults in the robot driving system, but also has better performance in stability and diagnosis accuracy compared with the traditional methods.

#### 1. Introduction

In recent years, the wheeled robots have received a wide range of applications and developments [1–3]. Particularly, in home service area, various kinds of wheeled service robots have become members of our family, such as the elderly companion robot [4] and the sweeping robot [5]. However, robot users are usually nonexpert in robot technology, which means that the faults which occurred in the wheeled robot system may cause serious damage to their life and property. The increasing demand of safety, reliability, and the necessity of low cost have become the bottleneck of wheeled robot applications with current technology. Therefore, it is meaningful to focus on novel fault diagnosis methods, particularly for the man-robot coexistent environments.

Generally speaking, the existing fault diagnosis methods can be classified as the model based and the data driven ones [6, 7]. In the earlier days, the research of model based fault diagnosis methods drew much attention and constituted the mainstream of this field [8, 9]. In [10], based on the mathematical model of the robotic manipulator, Caccavale et al. presented a discrete-time framework for diagnosis of sensors and actuators of robotic manipulators. Using particle filter, Yu et al. [11] proposed a fault-proneness prediction method for robot dead reckoning system. Besides the abovementioned methods, the adaptive observer and some other model based methods have also been designed for fault diagnosis of robot platform or robot manipulator [12, 13]. Those model based fault diagnosis methods are effective and suitable for the diagnosis problem of robot manipulator or robot arm, because robot arm usually works in a structured environment and it is relatively easier to get the accurate mathematical model. While, for wheeled robots, firstly, their working environments are usually dynamic and unstructured, secondly, wheeled robots are usually equipped with various kinds of equipment that are more complex in both hardware and software aspects compared with manipulators. Thus it is hard to get an accurate mathematical model of a wheeled robot working in an unstructured environment, which becomes a restriction of those model based methods. Moreover, the wheeled robots are well equipped with multiple sensors which implies that large data volumes containing robot running status information are available. Those large data volumes imply difficulties in system modeling, while they provide the required information for data driven based fault diagnosis method.

Principal component analysis (PCA) is a typical representative of the data driven fault diagnosis method. PCA is more suitable for fault detection rather than diagnosis, because it does not use the input-output relationships [14]. Therefore, in order to improve the diagnosis ability, PCA is often used by combining the classifiers, such as the neural network (NN) and the support vector machine (SVM). This hybrid method has been applied in the fault diagnosis of rotating machinery [15], power transmission systems [16], and some other aspects [17, 18]. Applications of PCA could be useful in extracting and interpreting process information from massive data sets, and the pattern recognition techniques could also be used to diagnose the specific running status of the robot.

Nevertheless, there are mainly two problems that exist in the above hybrid diagnosis methods. On the one hand, most of the studies adopted the existing classical kernel (e.g., Gaussian kernel and polynomial kernel) as the kernel function of SVM in their diagnosis methods, while new kernel functions with better classification performance need to be proposed, proved, and applied to the robot fault diagnosis fields. On the other hand, the diagnosed objects are usually complex and with varying degrees of uncertainties. A single PCA model cannot achieve full and complete awareness of the diagnosed object so that the information fusion in data level or decision level is needed to reduce the existing uncertainties.

Mittag-Leffer functions [19, 20] play fundamental roles in fractional calculus, which exhibit intermediate process among exponential function, power function, and polynomial function. Nowadays, fractional calculus has been successfully applied in many aspects, such as the application of fractional Fourier transform in signal processing [21] and the application of fractional order PI controllers [22]. Inspired by fractional calculus, a novel fractional Gaussian kernel named ML-kernel is proposed in this paper, which is a generalization of the traditional Gaussian kernel. The proposed ML-kernel is proved to be positive definite and its diagnosis performance is discussed in this paper. Besides, a hybrid fault diagnosis framework is discussed for robot driving system based on Dempster-Shafer (D-S) fusion and ML-kernel support vector machine (SVM). Multiple PCA models are established to do fault feature extraction and the fault feature vectors are used as the inputs of the ML-kernel SVM classifiers. The ML-kernel SVM classifiers output the preliminary fault diagnosis results which are fused by D-S fusion and the fusion result is taken as the final diagnosis result. Two sets of comparative experiments are carried out to validate the proposed method.

The remainder of this paper is organized as follows. Section 2 briefly introduces the SVM method and the positive definiteness of the presented ML-kernel is also proved in this section. In Section 3, the proposed fault diagnosis framework is described in detail. Section 4 illustrates the architecture of the experimental wheeled robot driving system and the application studies for various fault conditions. Section 5 is devoted to conclusions.

#### 2. SVM Algorithm and the Presented ML-Kernel Function

##### 2.1. Conventional SVM Algorithm

In the past few years, SVM has been one of the most highly studied topics in the machine learning fields, and it has been successfully applied in practice, especially for classification problems (e.g., fault diagnosis) [23, 24]. Based on the statistical theory of VC dimension and structural risk minimization inductive principle, SVM reaches the best compromise between the complexity of modeling and the leaning ability and hunts the best generalization ability. The basic SVM [25] deals with linearly separable two class cases and it can cope with nonlinear problems by introducing kernel functions and slack penalty. Given a training set , where is the th training input vector, is the number of training data for SVM, is the dimension of the input data, and is the set of classification tag for training. The optimal hyperplane separating the data can be obtained as a solution to the following optimization problem:where is the slack penalty, is the adjustable weight vector, is the offset of the hyperplane, and is the distance between the margin and the lying on the wrong side. The equivalent Lagrangian dual problem can be described as follows:where is the Lagrangian coefficient, from which we can obtain , , to solve (1).

The kernel function can map the input vector into feature space and returns a dot product of the feature space. The linear discriminant function with kernel is given by the following:where is the signum function.

The fault diagnosis of a robot driving system is a multiple class classification problem, while the conventional SVM was designed for the binary classification problem, so it is not suitable for the fault diagnosis in its original form. A few types of methods for multiclass SVM have been proposed [26]: one against one, one against others, direct acyclic graph, and so forth. This study employs the “one against one” multiclass SVM. In order to construct the BPAs, we need the probabilistic outputs of the SVM classifiers and the “pairwise coupling” method [27] is used to solve this problem.

##### 2.2. Kernel Function

The nonlinear pattern recognition problem in fault diagnosis can be transformed into the linear problem in some very high-dimensional feature spaces. The kernel function is able to handle any dimension feature spaces without the accurate calculation of and . It has been proven that any function that satisfies the Mercer theorem can be used as a kernel function [28]. Currently, there are three typical kinds of kernel functions:

(1) Polynomial kernel function

(2) Radial basis kernel function (RBF)

(3) Sigmoid kernel function

##### 2.3. Proof of the Positive Definiteness of the ML-Kernel

As the core of SVM, kernel function and the parameters of the model determine the performance of the SVM algorithm applied to the fault diagnosis system. In this paper, we employ the Mittag-Leffler function as a novel kernel function named as ML-kernel. The Mittag-Leffler function [29] is defined as follows:where is the Gamma function and is an arbitrary positive constant. For , (7) becomes . The presented ML-kernel function can be defined aswhere . When , (8) becomes , and the ML-kernel is identical to the Gaussian RBF kernel in this condition.

Given a kernel, it is in general straightforward to verify its continuity and symmetry, while the positive definiteness is more important and essential for a kernel. Thus, the proof of the positive definiteness of the proposed ML-kernel is given as below.

For convenience, letting , the ML-kernel (8) can be written as . In Laplace domain, we can get [30]where is a real number that keeps the contour path of integration which is in the region of convergence of , , and denotes the inverse Laplace transformation.

The integration path in (9) can be bended into the equivalent Hankel contour , which contains three parts: one line that starts from , an arc that encircles the circular disc counterclockwise, and the other line that ends at . So, (9) can be written as

Along , we have , , and as goes from to , goes from to ,

Along , we have , , and as goes from to , goes from to :

Along , we have , , and

Hence, we can obtain

Letting , we have

Therefore , and, for all , we have . In other words, the proposed ML-kernel is symmetrical and positive definite. Therefore, the proof is complete.

#### 3. Fault Diagnosis Method Based on ML-Kernel SVM and D-S Fusion

As shown in Figure 1, there are two main processes of the proposed approach, namely, the training process and the fault diagnosis process. Before the application of the proposed approach, the initial samples should be obtained from the laboratory experiments. In the training process, multiple PCA models are set up based on the data sampled in the normal and faulty states. Then, those models are used to do fault feature extraction and the ML-kernel SVM classifiers are trained. In the diagnosis process, new sampled data are normalized firstly. Secondly, the PCA models established in the training process are applied to do fault feature extraction. The fault feature vectors are then served as the inputs of the trained ML-kernel SVM classifiers, respectively, and the probabilistic outputs of the classifiers are taken as the preliminary fault diagnosis results. The BPAs are constructed based on the preliminary fault diagnosis results and the confidence values calculated from the confusion matrix. To reduce the uncertainties of the preliminary diagnosis results, the D-S fusion algorithm is introduced for decision fusion and the final diagnosis results are given based on the fusion results. The proposed approach is elaborated in detail as follows.