Diagnosis method based on hidden Markov model and Weibull mixture model for mechanical faults of in-wheel motors

HMM is a statistical model and uses a Markov process that contains hidden and unknown parameters for predicting functions of uncharacterized sequences [31]. In the model, the observed parameters in each state are used firstly to generate an observation random sequence for identifying the hidden parameters, and secondly, hidden Markov chains can be formed to generate an unobservable state random sequence, and finally, the further analysis will be performed. Here, a Markov chain is a discrete-time process for which the next state is conditionally independent of the past given the current state. Each state has a discrete or continuous probability distribution over possible emissions or outputs. When one state transitions to another state or a specific state is visited, the output will be generated. What guides the state-to-state transition is a set of transition and emission probabilities. The transition probability is the probability of moving from one state to another via a connected edge, and the emission probability is the probability of emitting a especial symbol at a state. The probability of any sequence is computed by multiplying the emission and transition probabilities along the path.

Formally, a complete HMM has a combination of three basic parameters λ = ( π, A, B ). The parameters π and A are used to describe the Markov chain and the stochastic process is described by parameter B [32]. Here π is the probability distribution of operating state in initial time t ₀, it defines the probability distribution in hidden state π = [π₁, π₂, ..., π_i , ..., π_N ]. A is the state transition matrix, where A = [a_ij ]_{N ×N} , the element a_ij in the matrix reflects the probability from one state S_i (t) at time t to another state S_j (t+ 1) at time t+ 1. B is the observation probability matrix, where B = [b_jm ]_{N ×M} , and its element b_jm represents the probability of obtaining the observed value in state S_j . The specific elements in the matrix are described as follows:

$$\begin{align}{\pi _i} = P\left( {{q_1} = {S_i}} \right)\end{align} \tag{ 1 }$$

$$\begin{align}{a_{ij}} = P({q_{t + 1}} = {S_j}|{q_t} = {S_i})\end{align} \tag{ 2 }$$

$$\begin{align}{b_{jm}} = P({\boldsymbol{O}_t} = {\boldsymbol{v}}(m)|{q_t} = {S_j})\end{align} \tag{ 3 }$$

where S_i denotes the ith hidden state and q_t denotes the hidden state at time t, P(·) expresses the probability value. Let N be the number of possible hidden states, then the states set is described by S = [S_1, S₂, ..., S_i , ..., S_N ], and q_t ∈ S. v (m) is the mth observation output and O _t is the observation output at time t. Let M be the number of all possible observation outputs, then the observation matrix is described by V = [ v (1), v (2), ..., v (m), ..., v (M)], and O _t ∈ V .

In practical engineering applications, a time-slice is regarded as an observation phase, then the observation outputs from multiple successive time-slices can be formed an observation sequence O = [O₁, O₂, ..., O_t , ..., O_T ]. Similarly, the corresponding hidden state sequence Q = [q₁, q₂, ..., q_t , ..., q_T ] is labeled. Based on the prior data or expert experience, the initial parameter set λ ₀ = ( π ₀, A ₀, B ₀) is usually obtained [33]. In the follow-up process, there are two essential problems. One is the model parameter estimation that is to determine the most likely values of HMM's parameters according to the observed series of history training data. Another is the condition recognition that to find the most likely hidden value sequences.

For updating new basic parameters λ = ( π, A, B ) of HMM, when there are a lot of prior data that the observation outputs and the corresponding hidden states are known, Baum–Welch algorithm is usually used to train the parameters λ of HMM in actual application [34]. Given an observation sequence O , Baum–Welch algorithm can find a local maximum λ = ( π, A, B ) = $\mathop {\max }\limits_{\overline {\lambda} } P\left( {O\left| {\overline {\lambda} } \right.} \right)$ for observation sequence O with random initial conditions, where $\boldsymbol{\bar \lambda}$ is the estimated value of λ at present. The algorithm can be described as follows:

$$\begin{align}{\pi _i} = \frac{{P({\boldsymbol{{O}}},{q_1} = {S_i}|\boldsymbol{\bar \lambda} )}}{{P({\boldsymbol{{O}}}|\boldsymbol{\bar \lambda )}}}\end{align} \tag{ 4 }$$

$$\begin{align}{a_{ij}} = \frac{{\sum\limits_{t = 1}^{T - 1} {P({\boldsymbol{{O}}},{q_t} = {S_i},{q_{t + 1}} = {S_j}\left| {\boldsymbol{\bar \lambda }} \right.)} }}{{\sum\limits_{t = 1}^T {P({\boldsymbol{{O}}},{q_t} = {S_i}|\boldsymbol{\bar \lambda} )} }}\end{align} \tag{ 5 }$$

$$\begin{align}{b_{jm}} = \frac{{\sum\limits_{t = 1,{{\boldsymbol{{O}}}_t} = {\boldsymbol{{v}}}(m)}^T {P({\boldsymbol{{O}}},{q_t} = {S_j}|\boldsymbol{\bar \lambda} )} }}{{\sum\limits_{t = 1}^T {P({\boldsymbol{{O}}},{q_t} = {S_j}|\boldsymbol{\bar \lambda} )} }}.\end{align} \tag{ 6 }$$

When the error threshold or maximum number of iterations is satisfied, the optimal parameters are obtained and marked as λ ^* = ( π *, A *, B *), then the most basic HMM has constructed. At the condition recognition stage, Viterbi algorithm is used to decide the most probably hidden state sequence on the basis of a new observation sequence [35].

In general, GMM is used to fit the characteristics of the observation data in the process of HMM's construction. However, there are some observation data that disobey a random Gaussian distribution in the real, and then it will highlight approaches and solutions on how to effectively fit the especial observation data.

Weibull distribution is a two-parameter or three-parameter family of curves, which gives an approximate but generally good fit to some especial observation data, especially in reliability and lifetime modeling. Certainly, two-parameter Weibull distribution is a special case of the generalized three-parameter gamma distribution. Compared with Gaussian distribution and exponential distribution, Weibull distribution is more flexible, versatile and relative simplicity, and then is widely used in reliability engineering and predictive data analysis. In the field of condition monitoring, Weibull distribution is often selected to describe the variation trend of any one SP extracted from vibration signals [36]. When many SPs are used to evaluate an analytic target, WMM is performed to exploit their talent.

Let K be the number of observation parameters in HMM mentioned above, each observation output can be denoted as v (m) = [v₁(m), v₂(m), ..., v_k (m), ..., v_K (m)]'. Then the same observation parameter gathers in each row of observation matrix V . For analyzing the same observation parameter, v_k is used to express the kth observation parameter in observation matrix, then v _k = [v_k (1), v_k (2), ..., v_k (m), ..., v_k (M)] and V = [ v ₁, v ₂, ..., v_k,..., v_K ]'. Suppose all observation parameters obey some kinds of Weibull distribution, the Weibull probability density function of v _k may be expressed as follows:

$$\begin{align}f\left( {{v_k};{\eta _k},{\beta _k}} \right) = \frac{{{\beta _k}}}{{{\eta _k}}}{\left( {\frac{{{v_k}}}{{{\eta _k}}}} \right)^{{\beta _k} - 1}}{e^{ - {{\left( {\frac{{{v_k}}}{{{\eta _k}}}} \right)}^{{\beta _k}}}}}\end{align} \tag{ 7 }$$

where η_k > 0, β_k > 0 are the scale and shape parameters, respectively. Accordingly, the distribution of the observation sequence V which is composed of K observation parameters can be described by WHH as follows:

$$\begin{align}f\left( {v;\omega ,\eta ,\beta } \right) = \sum\limits_{k = 1}^K {{\omega _k}}\, f\left( {{v_k};{\eta _k},{\beta _k}} \right)\end{align} \tag{ 8 }$$

where η and β are the sets of the scale and shape parameters. w = {w₁, w₂, ..., w_K } is the set of weight coefficients. w_k is the weight coefficient of the kth Weibull distribution, and satisfies the following conditions:

$$\begin{align}\left\{ \begin{gathered} \sum\limits_{k = 1}^K {{\omega _k} = 1} \\ {\omega _k} \geqslant 0 \\ \end{gathered} \right..\end{align} \tag{ 9 }$$

In general, traditional HMM needs the global observation sequences. However, a fault condition of in-wheel motors is implicit in the real, especially in the earlier fault period, then it is difficult to observe directly the fault states of in-wheel motors. Usually, the condition signal such as vibration, noise or electric signal is collected to detect the anomalous change in the time, frequency or time-frequency domain, then extract the different SPs for reflecting the fault states of in-wheel motors [37, 38]. In that way, SPs in the running process of in-wheel motors can form many observation sequences, and observation sequences in the normal condition are more, but observation sequences in the fault states or transition process from normal state to fault states are lesser even no, especially in construction phase of condition monitoring system. When the limited observation sequences are used to establish a HMM of fault diagnosis, the parameter λ = ( π, A, B ) may be affected. Even an unsupervised learning algorithm such as Baum–Welch algorithm is not limited by the lack of training samples, it is sensitive to select the initial parameters λ ₀ = ( π ₀, A ₀, B ₀) [19]. When expert experience or random assignment is relied on, the iterative accuracy may be reduced to cause the insufficient parameters of HMM, and then reduce the diagnosis accuracy. In practical scientific research, there are different fitting methods applied in many fields to fit the experimental data, predict the global distribution, and obtain the global data. For example, a simple ANN is used to estimate the thickness, roughness and density of organic semiconductor thin films for obtaining various types of global data [39]. The backpropagation neural network is used to fit the true distribution of the original sample data, and the CNN is combined to improve the diagnosis accuracy [40]. Moreover, many SPs extracted from vibration and noise signals of in-wheel motor are selected to obey some kinds of Weibull distribution in the previous research [41]. Therefore, WMM is used to expand the limited data for obtaining the optimal parameters of HMM. The proposed method in the paper is known as WMM-HMM.

The approach of WMM-HMM involves three major steps. First, the limited observation sequences are used to confirm the parameters of WMM. Second, the optimal WMM is applied to obtain the global observation sequences. Third, the global observation data are utilized to optimize the parameters of HMM, and then build a diagnosis system of HMM, as follows:

Step 1: Solving the parameters η, β and w of WMM. It can be seen from the above description that the observation sequence V = [ v ₁, v ₂, ..., v_k,..., v_K ]' is composed of K SPs, and they are independent of each other. For the kth SP, the existing observation sequence v _k can be expressed by [SP_k (1), SP_k (2), ..., SP_k (m), ..., SP_k (M)]. Then the two parameters η_k and β_k of Weibull distribution can be obtained using the sample mean μ_k and sample variance σ_k ² of the kth SP's observation value, the value of β_k is estimated as the solution of:

$$\begin{align}\frac{{{\sigma _k}^2}}{{{\mu _k}^2}} = \frac{{\Gamma (1 + 2/{\beta _k})}}{{{\Gamma ^2}(1 + 2/{\beta _k})}} - 1\end{align} \tag{ 10 }$$

and the value η_k of is given by:

$$\begin{align}{\eta _k} = \frac{{{\mu _k}}}{{\Gamma (1 + 2/{\beta _k})}}\end{align} \tag{ 11 }$$

where Γ(·) is gamma function. In this way, the Weibull distribution of each SP can be determined.

For the weight coefficient set w in WMM, the state differentiation of SPs is analyzed to propose an objective weighting method. It is the basic idea of the method that Euclidean distances in a unit range between single SP and all SPs are accumulated to determine the SP's contribution, then the weight coefficient w_k of the kth Weibull distribution can be expressed as follows.

$$\begin{align}{\omega _k} = \frac{{\sum\limits_{m = 1}^M \left(\frac{1}{R}\sum\limits_{r = 1}^R {E_{\text{d}}}(p({{\mathbf{v}}_k}(m)),{ }p({{\mathbf{v}}_k}(r)))\right)}}{{\sum\limits_{k = 1}^K \sum\limits_{m = 1}^M \left(\frac{1}{R}\sum\limits_{r = 1}^R {E_{\text{d}}}(p({{\mathbf{v}}_k}(m)),{ }p({{\mathbf{v}}_k}(r)))\right)}}\end{align} \tag{ 12 }$$

where p(·) is the probability density function of Weibull distribution, E_d(·) is Euclidean distance. R is the number of the observation outputs in a unit range of the mth observation output. Certainly, the greater the difference, the greater the value of w_k . With the determination of w , WMM of observation matrix is determined.

Step 2: Establishing the training sample set of HMM. Based on the determined WMM of observation matrix V , it is easy to build the fuzzy function of the corresponding hidden state sequence Q , which is denoted as Q = f( V ). Then the method of cubic convolution interpolation is used to extend the training sample set of HMM.

$$\begin{align} & F({{\boldsymbol{v}}_1} - {{\boldsymbol{{d}}}_1},{{\boldsymbol{v}}_2} - {{\boldsymbol{{d}}}_2}, \ldots ,{{\boldsymbol{v}}_K} - {{\boldsymbol{{d}}}_K}) \nonumber\\ &\;\, =\sum\limits_{k = 1}^K \sum\limits_{i = -1}^2 f\left({{\boldsymbol{v}}_1} + i \cdot \Delta {{\boldsymbol{v}}_1},{{\boldsymbol{v}}_2} + i \cdot \Delta {{\boldsymbol{v}}_2}, \ldots ,{{\boldsymbol{v}}_K} + i \cdot \Delta {{\boldsymbol{v}}_K}\right)\nonumber\\&\;\;\;\;\quad\times\prod\limits_{k = 1}^K {p(i \cdot \Delta {{\boldsymbol{v}}_k} - {{\boldsymbol{{d}}}_k})} \end{align} \tag{ 13 }$$

where F(·) is the hidden state function of the corresponding coordinate after interpolation. d _k and Δ v _k are the deviation value and sampling interval of v _k , respectively. When the observation output is composed of multiple SPs, the corresponding hidden state is expressed by positive integer, round-function is used to ratiocinate the hidden state. In this way, the limited observation matrix V and the corresponding hidden state sequence Q can be replenished to obtain the global observation data, then establish the training sample set of HMM. In general, the training sample set is shown as [(S_{i 1}, n₁· v )₁, (S_{i 2}, n₂· v )₂, ..., (S_it, n_t · v )_t , ..., (S_iT, n_T · v )_T ], where S_it and n_t are the hidden states and the number of observation output at the tth time segment, respectively.

Step 3: Optimizing the parameters λ = ( π, A, B ) of HMM. With the increase of observation time, WMM is more realistic, the training sample set of HMM is more intact. For the initial parameters λ ₀ = ( π ₀, A ₀, B ₀) of HMM, the statistical method such as count-function is used to determine tentatively. Let C₁(S_i ) be the frequency of S_i in initial time segment, C₂(S_i, S_j ) is the frequency of transferring from state S_i to S_j in training sample set, C₃(n_m · v ) is the frequency of n_m · v obtained in state S_j , the probability of each state π_i in the initial state vector π ₀ is calculated by ${\pi _i} = {C_1}\left( {{S_i}} \right)/\mathop \sum \nolimits_{i = 1}^N {C_1}\left( {{S_i}} \right)$, each transition probability a_ij from state S_i to S_j in the initial transition matrix A ₀ is computed by ${a_{ij}} = {C_2}\left( {{S_i},{\text{ }}{S_j}} \right)/\mathop \sum \nolimits_{j = 1}^N {C_2}\left( {{S_i},{\text{ }}{S_j}} \right)$, each observation probability b_jm of state S_j in the initial observation probability matrix B ₀ is counted by ${b_{jm}} = {C_3}\left( {{n_m}\cdot v} \right)/\sum\nolimits_{m = 1}^M {C_3}\left( {{n_m}\cdot v} \right).$ Once the initial parameters λ ₀ = ( π ₀, A ₀, B ₀) of HMM are determined, Baum–Welch algorithm as shown in (4)–(6) is performed to obtain the optimal parameters λ = ( π, A, B ), then construct a WMM-HMM diagnosis model.

When a new observation sequence O _test = [ O ₁, O ₂, ..., O _T ] is known, the corresponding hidden state sequence can be concluded as follows:

$$\begin{align}{{\boldsymbol{{S}}}^*} = \arg \max p\left( {{\boldsymbol{{S}}}_1}^*,{{\boldsymbol{{S}}}_2}^*,\ldots,{{\boldsymbol{{S}}}_T}^*\left| {{{\boldsymbol{{O}}}_1},} \right.{{\boldsymbol{{O}}}_2},\ldots,{{\boldsymbol{{O}}}_T},{\boldsymbol{\lambda }} \right).\end{align} \tag{ 14 }$$

In the paper, the running state of in-wheel motor is hidden, but vibration signal can be measured easily. Based on WMM-HMM diagnosis method, multiple SPs are extracted from vibration signal to build the observation sequence O , then the maximum likelihood state sequence S * of in-wheel motor can be inferred. When a fault of in-wheel motor occurs, it is easy to judge the fault condition and identify the fault type. The application of WMM-HMM method in in-wheel motor fault diagnosis will be described in the third chapter.

Figure 2 is the test bench of an in-wheel motor that contains three major sections, that is, electric wheel, road excitation and suspension system. The electric wheel is composed of a wheel and an in-wheel motor (Type: QSWP30030X; Rated power: 3 kW), which is controlled by STM32 single chip microcomputer for simulating the voltage of accelerator pedal. Road excitation is simulated by hydraulic exciter (Type: INSTRON 8800) that the excitation frequency and amplitude can be set on the basis of different road levels. Suspension system is composed of spring component, guides component and dampers, the top is fixed in table frame, a pressure sensor (Type: HD-YB-11LY; Sensitivity: 2.0 ± 0.005 mV V⁻¹) is installed between the suspension bottom and the axle of the electric wheel. When the axle of the electric wheel is lifted to a certain height by the hydraulic exciter, the different loads can be simulated. Moreover, an accelerometer (Type: PCB333B30; Sensitivity: 100 mV g⁻¹) is installed at the axle of in-wheel motor, vibration signals are collected with a 12.8 kHz sampling frequency at a relatively steady speed of 100, 200, 300...700 rpm. The sampling time is 90 s.

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In order to research the operation states of in-wheel motors with mechanical faults, common bearing faults such as outer race fault, inner race fault, rolling element fault (Width: 0.3 mm; Depth: 0.1 mm) were artificially made to build on the stator axis of each in-wheel motor by professionals, respectively. In the experiment, in-wheel motors with mechanical faults are orderly exchanged to run with the other condition fixed.

In order to verify the generalization and applicability of the proposed method, and weaken the influence of different speeds for the diagnosis model, the general standard of a vehicle low-speed, medium-speed and high-speed is used, and the radius of the electric vehicle is considered, the 100, 200, 300, 400 rpm speeds of in-wheel motor are collectively defined as low-speed condition; the 500, 600, 700 rpm speeds are regarded as medium-speed condition. Since the dynamic vibration and output torque of in-wheel motor are more differences under low-speed and medium-speed conditions, two types of fault diagnosis models are constructed, respectively. In practical application, the average value of operating speed during a monitoring period is used to judge whether in-wheel motor is operating in low-speed or medium-speed condition, then the corresponding diagnosis model is selected.

Moreover, the operation state of in-wheel motors is a gradual degradation process during its evolution, that is, in-wheel motor can only transfer from normal state to fault state, but not transfer from the fault state to the normal state, and not transfer from one fault state to another fault state. Certainly, a single fault may evolve into a compound fault. Therefore, the hidden state of in-wheel motors is a non-cross-type Markov chain as shown in figure 3. For example, outer race fault of a bearing can easily lead to rolling element defect, and then form a compound-fault state that includes outer race fault and rolling element fault at the same time [42]. As time goes on, inner race is involved in fault state.

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In the paper, single fault state of in-wheel motors is focused, and a sequential diagnosis system is designed for improving the accuracy of fault identification. Firstly, inner race fault is regarded as the primary task of detection. In the following steps, outer race fault and rolling element fault are diagnosed one by one. Then WMM-HMM diagnosis models from normal state (S₁) to inner race fault (S₂), outer race fault (S₃) or rolling element fault (S₄) are constructed for in-wheel motor, respectively. Certainly, the diagnosis goal is different in each step, the input parameters of each model may vary. Here, the WMM-HMM diagnosis model from S₁ to S₂ under low-speed condition is regarded as an example, the construction process is introduced in detail.

3.2.1. Symptom parameters (SPs).

In general, the sensitivity of each SP is different under different conditions, it is not easy to find effective SPs for some faults [43]. Moreover, too few SPs cannot distinguish multiple fault types at the same system [10], but more SPs must come to being data redundance, data exchange difficulty and work inefficiency [37]. Therefore, there are some methods such as distinguish index (DI) [44] and stable average discrimination rate (SADR) [38] that have been used to select moderate and high-sensitive SPs for fault diagnosis and identification. Here, two methods of DI and SADR are combined to select three SPs for three mechanical faults of in-wheel motors, respectively. As far as the WMM-HMM diagnosis model from S₁ to S₂ under low-speed condition, three SPs such as root mean square, skewness and waveform deviation rate are decided, as follows:

$$\begin{align}{\text{S}}{{\text{P}}_1} = \sqrt {\frac{{\text{1}}}{N}\sum\limits_{i = 1}^N x_i^2} \end{align} \tag{ 15 }$$

$$\begin{align}{\text{S}}{{\text{P}}_2} = \left| {\frac{1}{{N \cdot {\sigma ^3}}}\sum\limits_{i = 1}^N {{\left( {{x_i} - \bar x} \right)}^3}} \right|\end{align} \tag{ 16 }$$

$$\begin{align}{\text{S}}{{\text{P}}_3} = \frac{{\text{1}}}{{\sigma \cdot {N_{\text{p}}}}}\sum\limits_{j = 1}^{{N_{\text{p}}}} \left| {{x_{pj}}} \right|\end{align} \tag{ 17 }$$

where {x_i } (i = 1–N) is the sequence of vibration signal, $\bar x$ and σ are the average value and standard deviation of {x_i }. {x_pj } (j= 1–N_p) is the peak value of {x_i }, N_p is the total number of {x_pj }.

When 2 s are regarded as an observation period, the corresponding SP₁, SP₂ and SP₃ are calculated according to the vibration signals of in-wheel motor, and then the SP sequences of SP₁, SP₂ and SP₃ are obtained with 45 observation time periods. Each SP sequence is analyzed, the statistical results are fitted with Gaussian distribution and Weibull distribution, respectively, as shown in figure 4. It is obvious that each SP is more in line with Weibull distribution than Gaussian distribution. Certainly, the method of statistical test can be used to prove that these SPs conform to Weibull distribution.

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3.2.2. WMM construction.

3.2.3. HMM establishment.

With the construction of WMM, the Weibull probability density function f (SP₁, SP₂, SP₃) is known. Based on the original observation sequences, significance level of 0.01 is set to confirm the confidence interval of each SP, the deviation value d _k and sampling interval Δ v _k , formula (13) is used to extend the original observation data. Then new observation data under each condition is up to four times. Since there are 45 observation time periods of each state under each speed, new observation data are enlarged to 180, respectively. Therefore, half of the data is used to build the training samples.

Here, the training samples of HMM S₁ to S₂ under low-speed condition are regarded as an example to describe the composition and sample capacity. Since low-speed condition includes four kinds of speed, the training sample set is shown as [(S₁, 90· v )_{T speed=100}, (S₁, 90· v )_{T speed=200}, (S₁, 90· v )_{T speed=300}, (S₁, 90· v )_{T speed=400}, (S₂, 90· v )_{T speed=100}, (S₂, 90· v )_{T speed=200}, (S₂, 90· v )_{T speed=300}, (S₂, 90· v )_{T speed=400}]. Where v = [SP₁, SP₂, SP₃], (·) _{T speed=S} is the training sample at the time segment when the speed of in-wheel motor is S (S= 100, 200, 300, 400). Next comes the training parameters of HMM. When the supervised learning method mentioned in section 2 is used to obtain the parameters of HMM from S₁ to S₂ under low-speed condition are shown as follows:

$$\begin{align*}{\pi ^*} = \left[ {\begin{array}{*{20}{c}} {0.9910}&{0.0090} \end{array}} \right]\end{align*}$$

$$\begin{align*}{A^*} = \left[ {\begin{array}{*{20}{c}} {0.9882}&{0.0118} \\ {0.0000}&{1.0000} \end{array}} \right]\end{align*}$$

$$\begin{align*}{B^*} = {\left[ {\begin{array}{*{20}{c}} \cdots &{0.0000}&{0.0118}& \cdots &{0.0353}& \cdots \\ \cdots &{0.0116}&{0.0465}& \cdots &{0.0581}& \cdots \end{array}} \right]_{2 \times 2160}}.\end{align*}$$

Thus, the WMM-HMM diagnosis model from normal state (S₁) to inner race fault (S₂) under low-speed condition is established. Similarly, vibration signals of in-wheel motors with inner race fault under medium-speed condition are analyzed to select three high-sensitive SPs for observing the operating state, the original observation sequences are processed by WMM method to input the HMM, then it is easy to build the WMM-HMM diagnosis model from S₁ to S₂ under medium-speed condition. With the threshold judgment of operating speed, one of the two models will be performed to detect whether in-wheel motor is in inner race fault state. If the probability that the current state has been verified as inner race fault (S₂) is high, the diagnosis system automatically stops. Otherwise, the system will come into the next step for the identification of outer race fault (S₃) and rolling element fault (S₄). In the same way, the operating state of in-wheel motor are diagnosed sequentially.

In order to verify the effectiveness of the WMM-HMM diagnosis system for mechanical faults of in-wheel motors, the remaining new observation data that includes normal state and three faults under low-speed and medium-speed conditions is taken as the test sample to test the recognition rate of each operating state of the in-wheel motor. The recognition rates of normal state (S₁), inner race fault (S₂), outer race fault (S₃) and rolling element fault (S₄) are shown in figure 5.

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Obviously, each operating state has been recognized precisely, and the majority of recognition rates are more than 93.8%. There is only one case exception that 72.2% test cases of rolling element fault under medium-speed condition are diagnosed correctly. The reason is that the single filter with the fixed constraint is used to process the more overlap between the natural frequency of test bench and the characteristic frequency of rolling element fault under medium-speed condition. Overall, the WMM-HMM diagnosis system is available for diagnosing mechanical faults of in-wheel motor.

To verify the effectiveness and applicability of the proposed method in the field of mechanical fault diagnosis of in-wheel motor, the following different comparative strategies are designed for analysis and discussion.

3.4.1. Comparison of the proposed method with different inputs.

For the WMM-HMM diagnosis model from normal state (S₁) to inner race fault (S₂) under low-speed condition, seven different observation parameters as shown in table 2 are planned as the inputs of the diagnosis model to verify the importance of the observation sequence. Among them, plan 1, plan 2 and plan 3 use a single SP to build the observation sequence, plan 4, plan 5 and plan 6 use two SPs, plan 7 uses three SPs that is the proposed method in section 3.2. When the different SPs in each plan are input into the WMM-HMM diagnosis model, the recognition rate of each operating state is obtained as shown in figure 6.

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Table 2. Different plans with the inputs of different SPs.

Plan number	Input SPs
Plan 1	SP1
Plan 2	SP2
Plan 3	SP3
Plan 4	SP1 and SP2
Plan 5	SP1 and SP3
Plan 6	SP2 and SP3
Plan 7	SP1, SP2 and SP3

It is apparent that the recognition rates in plan 1, plan 2 and plan 3 are generally low (less than 70%), the experiment results in plan 4, plan 5 and plan 6 have shown some improvement, but wave up and down, and the diagnosis accuracy in plan 7 is high and stable. Therefore, one or two SPs cannot effectively observe the operating state of in-wheel motor, even if they are high-sensitive. The reasons is that too few observation parameters cannot reflect a comprehensive features of in-wheel motor. In the paper, it is beneficial that three SPs are selected to represent the operating characteristic of in-wheel motor.

3.4.2. Comparison of different diagnosis methods.