Control method of wheel slip rate based on fuzzy algorithm

Abstract

The existing control method of wheel slip rate has a long braking time, the gap between the slip rate and the optimal slip rate is obvious, and there is a defect of poor control effect. In order to solve the above problems, the fuzzy algorithm is introduced to design the wheel slip control method. According to the characteristics of ESC system, the control model of wheel slip rate was built. On this basis, the least square optimization algorithm was used to estimate the best slip rate. Based on the estimated optimal slip rate, the multi-agent system was used to design the slip rate controller, and thus to get the threshold value of slip rate controller. Moreover, the fuzzy algorithm was used to optimize the parameters of slip rate controller, so as to keep the optimal slip ratio. Finally, the control for wheel slip ratio based on fuzzy algorithm was achieved. The simulation results show that the slip ratio of this method is kept between 0.18–0.45, which is less different from the optimal slip ratio. Compared with the existing control method of wheel slip ratio, the proposed control method of wheel slip ratio greatly improves the control effect, which fully shows that the proposed control method of wheel slip ratio has better performance.

Keywords

ESC system least square optimization algorithm controller optimal slip rate

1 Introduction

With the enhancement of synthetic national power, the automobile transportation industry is developing rapidly. Some statistical researches show that the car ownership in China has been more than 300 million by the end of 2018. With the increase of car ownership, the number of traffic accidents is also growing exponentially, which seriously threatens the safety of people and property. In order to reduce the serious damage to the life and property, it is very necessary to research on the driving safety. At present, the effective vehicle safety technology is mainly the Electronic Stability Program (ESP). ESP is a group of programs and systems that can prevent the vehicle from reaching its dynamic limit in driving. As the largest automobile manufacturer in the world, China is also vigorously promoting the application of ESP. Due to the technical problems of ESP, the proportion of vehicles with ESP in China is relatively low. Compared with the USA and Europe, ESP assembly rate in China is only 15%. According to statistics, the brake error is the main reason of traffic accidents. Antilock Brake System (ABS) in ESP is to control the braking force when the car is braked, so as to prevent the tire from sliding with the road and ensure the safety of driving in the process of emergency braking. Therefore, the research on ABS is very important to ensure the safety of life and property [2, 3].

Based on the existing research results, we can find three control methods of wheel slip rate, including the state feedback control method of wheel slip rate, the sliding mode control method of wheel slip rate and the wheel slip rate control method based on the state monitoring of back wheels. The state feedback control method of wheel slip rate estimates the slip rate at first, and then controls the estimated slip rate. The core is to estimate the slip rate of each driving wheel by the measured current of driving motor, and then control the current of motor to achieve the purpose of controlling the slip rate. The sliding mode control method adopts an adaptive method. The core idea is to use the feedback linearization to eliminate the influence of control gain, and thus to adjust the feedback item adaptively. The simulation results show that the controller has good robustness in the case of the change of road adhesion coefficient. According to the corresponding data, the control method of wheel slip rate based on the state monitoring of back wheels estimates the optimal slip rate in the driving process by monitoring the state of back wheel. On this basis, the front wheel is controlled with the target slip rate [13, 14]. These three methods have their own advantages, but they cannot keep the wheel slip rate in the optimal slip rate, and the car needs a long braking time, so there is a defect of poor control effect, which cannot meet the needs of today’s automobile safety.

In order to solve above problems, the fuzzy algorithm is used to design the wheel slip control method. The fuzzy algorithm belongs to the intelligent algorithm. When we don’t understand the system model in depth, or some objective reasons limit the intensive research on the control model of system, the intelligent algorithm often plays an inessential role. At this time, it is necessary to adopt the fuzzy algorithm. The common fuzzy algorithms include the mean fuzzy algorithm and Gaussian fuzzy algorithm. In fact, the fuzzy algorithm is not blurry. The fuzzy algorithm is a process of successive refinement. The application of fuzzy algorithm improves the control effect of method to control wheel slip rate and provides more effective guarantee for the safety of automobile braking. Meanwhile, the simulation and comparison experiment proves the control effect of the proposed method [1].

2 Materials and method

2.1 Establishment of model to control wheel slip ratio

The longitudinal force and lateral force of automobile wheels depend on road conditions, wheel characteristics, vertical load, slip ratio and slip angle. The slip rate of wheel has obvious effect on the wheel force. In addition, the slip rate of wheel can be adjusted by the driving torque and braking torque, so it becomes the controlled variable of dynamics control system [7].

According to Newton’s second law, the motion equation of single wheel and the dynamic equation of the automobile which runs in a straight line are established, as shown in Formula (1) ${\begin{matrix} J_{w} ω = T_{m} - r_{w} (μ (λ) + mg + f_{w}) \\ {mv}_{x} = n_{w} μ (λ) mg - c_{v} v_{x}^{2} \end{matrix}$ (1)

In the formula, J_w denotes the rotational inertia of vehicle body; ω denotes the angular speed of wheel; T_m denotes the braking/traction moment; r_w denotes the diameter of wheel; μ (λ) denotes the function of slip rate; m denotes the mass of vehicle body; g denotes the acceleration of gravity; f_wdenotes the viscous frictional force of wheel. v_x denotes the central speed of wheel; n_w denotes the number of driving forces; c_v denotes the viscous frictional coefficient.

Thus, the wheel slip rate is defined as: $λ = {\begin{matrix} \frac{ω R - v_{x}}{ω R}, ω R ⩾ v_{x} \\ \frac{v_{x} - ω R}{v_{x}}, ω R < v_{x} \end{matrix}$ (2)

In the formula, λ denotes the wheel slip rate; R denotes the rolling radius of wheel.

Under a certain sideslip angle α, the longitudinal adhesion coefficient and lateral adhesion coefficient change with the wheel slip rate λ. The wheel longitudinal force and lateral force can be controlled by controlling the slip ratio λ. Theoretically, the ABS system is to control the slip rate be close to the slip rate corresponding to the peak adhesion coefficient, so as to maximize the longitudinal adhesion coefficient of tire and shorten the braking distance. Meanwhile, it is able to maintain a large lateral adhesion coefficient and ensure the lateral stability. According to the needs of vehicle dynamics control, ESC system can determine the longitudinal force and lateral force of each wheel required by the stability of vehicle, and calculates the target slip ratio corresponding to the wheel force [5]. The, the slip rate controller is to make the slip rate of each wheel be close to the target slip rate, so that the stability control can be achieved [12]. The automobile ESC system is shown in Fig. 1.

In Fig. 1, 1 denotes the speed sensor of wheel. 2 and 3 denote the brake pressure sensor. 4 denotes the yaw-rate sensor. 5 denotes lateral acceleration sensor. 6 denotes the brake hydraulic regulation; 7 denotes the engine management.

Fig. 1

Schematic diagram of ESC system.

The control model of wheel slip rate is built, which can support the estimation for optimal slip rate.

2.2 Estimation of optimal slip rate

Based on the above control model of wheel slip rate, the least square optimization algorithm is used to estimate the optimal slip rate. The slip rate is an important parameter during the emergency braking. When the slip ratio is less than the optimal slip ratio, the friction coefficient between the tire and the road surface is too small and the braking time is too long. When the slip ratio is too large, the vehicle wheel will be locked. In order to ensure the stability of emergency braking, it is necessary to keep the optimal value of slip ratio. It is the first premise to estimate the optimal slip ratio under the condition of road surface. Based on the functional relationship between the slip rate and the longitudinal adhesion coefficient, the least square optimization algorithm is used to fit the functional relationship, and then an estimation algorithm of optimal slip rate for the driving road is given, so that the problem about inaccurate estimation of optimal slip rate can be solved [15, 16]. The estimation process is shown as follows.

2.2.1 Analysis of functional relationship between slip ratio and vehicle adhesion coefficient

The sliding friction and rolling friction between wheels and road surface will appear in the process of automobile braking. The sliding rate is a parameter to describe the sliding amount between wheels and road surfaces. The sliding rate influences the friction coefficient between tires and road surfaces in the process of emergency braking. The functional relationship between the sliding rate of vehicle and the longitudinal adhesion coefficient of road surface is shown in Fig. 2.

Fig. 2

Relationship between longitudinal and lateral adhesion coefficients and slip rates.

When the adhesion coefficient between the vehicle and the road reaches the maximum value, the corresponding slip rate is the optimal slip rate, which is denoted by λ_p. When the slip rate is the best in the process of vehicle braking, the friction between vehicle and road reaches the maximum value, so that the safety of emergency braking can be guaranteed. When the slip ratio is less than the optimal slip ratio λ_p, the friction coefficient between tire and road increases with the increase of slip rate. If the friction coefficient between car and road is too small, the long braking time will affect the braking safety. When the slip ratio is greater than λ_p, the longitudinal adhesion coefficient between the car and road decreases with the increase of slip ratio. This is a dangerous braking state, and the wheels are easy to be locked. Therefore, how to estimate the functional relationship between slip rate and longitudinal adhesion coefficient is the premise of accurately estimating the optimal slip rate [9].

Pacejka first proposed that the functional relationship between the longitudinal adhesion coefficient and the slip rate of vehicle, which is shown in Formula (3). Formula (3) is a trigonometric function formula to simulate the wheel data, which is also called “magic formula”. The functional relationship between slip ratio and friction coefficient under different road conditions is expressed as: $μ (λ) = D sin {C arctan [B (1 - E) λ + \frac{E}{B} arctan (B λ)]}$ (3)

In the formula, B, C, D, E are unknown parameters which are closely related to the road conditions. Different road conditions are simulated by changing these parameters. μ denotes the friction coefficient.

According to Formula (3), we can see that magic formula is a serious nonlinear function. Therefore, if we want to obtain the optimal slip ratio by Formula (3), we must calculate its extreme value. It is difficult to estimate the parameters of B, C, D, E, so it is also difficult to get the optimal slip rate by magic formula.

Burckhardt gives another function expression of slip ratio and friction coefficient. The formula is as follows: $μ (λ) = c_{1} (1 - e^{- c_{2} λ}) - c_{3} λ$ (4)

In the formula, c₁, c₂, c₃ are unknown parameters related to road surface conditions, and the specific values are shown in Table 1.

Table 1

Values of c₁, c₂, c₃ under different road surface conditions

Pavement	c ₁	c ₂	c ₃
Ice	0.05	306.39	0
Snow	0.19	94.13	0.06
Wet Pebble	0.40	33.71	0.12
Dry Cobblestone	1.37	6.46	0.66
Dry Cement	1.19	25.17	0.53
Wet Asphalt	0.86	33.82	0.35
Dry Asphalt	1.28	23.99	0.52

Based on Formula (4), the optimal slip ratio and the maximum friction coefficient under different conditions can be calculated indirectly. The formulas are shown as follows. ${\begin{matrix} λ_{p} = \frac{1}{c_{2}} log \frac{c_{1} c_{2}}{c_{3}} \\ μ_{max} (λ_{p}) = c_{1} - \frac{c_{3}}{c_{2}} (1 - log \frac{c_{1} c_{2}}{c_{3}}) \end{matrix}$ (5)

It is difficult to estimate the parameters (c₁, c₂, c₃) of Burckhardt function, so it is also difficult to get the optimal slip rate and the maximum friction coefficient from Formula (5). Therefore, Kiencke et al put forward an approximate function relationship between slip rate and friction coefficient, which is expressed as: $μ (λ) = \frac{30 λ}{1 + x_{1} λ + x_{2} λ}$ (6)

In the formula, x₁, x₂ are the parameters to be estimated.

Formula (6) is transformed to get a function taking x₁, x₂ as variable, which is expressed as: $30 λ - μ (λ) = [μ (λ) \cdot μ (λ) \cdot λ^{2}] \cdot [\begin{matrix} x_{1} \\ x_{2} \end{matrix}]$ (7)

According to Formula (7), the optimal slip ratio and the maximum friction coefficient in the process of automobile braking are shown as follows: ${\begin{matrix} λ_{p} = \frac{1}{\sqrt{x_{2}}} \\ μ_{max} = \frac{30}{x_{1} + 2 \sqrt{x_{2}}} \end{matrix}$ (8)

Next, the least square method is introduced to fit the parameters (x₁, x₂) in Formula (7), and then the optimal slip ratio and the maximum friction coefficient under the condition of driving road surface are obtained from Formula (8) [11].

2.2.2 Nonlinear least square parameter fitting

The least square method was proposed by Gauss in 1829, which is a mathematical optimization method for finding unknown data by minimizing the square of error. The parameter fitting is an important application of least square method. The formula of least square method is defined as follows. Its mathematical description is to find the optimal solution to minimize the target function F (x) in a limited domain [10]. $F (x) = \frac{1}{2} \sum_{i = 1}^{m} {(f_{i} (x))}^{2}$ (9) Where, f_i : Rⁿ → R, i = 1, 2, 3, ⋯ , m represents the known function, m ⩾ n.

The parameter fitting is to design a model M (X, t) with parameter x for a given set of data sampling points { (t_i, y_i)}, so that x^* exists. Thus: $y_{i} = M (x^{*}, t_{i}) + ɛ_{i}$ (10)

In the formula, ɛ_i denotes the measurement error of model.

For any x, the function f_i is defined as: $f_{i} (x) = y_{i} - M (x, t_{i}), i = 1, 2, 3, \dots, m$ (11)

The principle of least square method in parameter fitting is to find a parameter x^*, so as to minimize the sum of squares of f_i (x).

The least square method is to find the optimal solution to minimize the objective function F (x) in a limited domain. According to different definition domains, the optimal solution is divided into the local optimal solution and the global optimal solution. The global optimal solution is that the objective function reach the minimum independent variable in the whole definition domain. The global optimal solution is defined as follows: $For F : R^{n} \to R, x^{*} = arg min_{x} {F (x)} exists$ (12)

When using the least square method to solve the practical problems, we need to get the local optimal solution of objective function, and the independent variable that makes the objective function reach the minimum in local definition domain is the local optimal solution. Therefore, the process of parameter fitting by the least square method is also the local optimal solution of objective function. Its expression is as follows: $For F : R^{n} \to R, x - x^{*} < δ, δ > 0$ (13)

If the target function F is continuous and differentiable, the following Taylor expansion exists. $F (x + h) = F (x) + \frac{1}{2} h^{T} Hh + O ({∥ h ∥}^{3})$ (14)

Let g be the gradient matrix of target function F, then: $g = F^{'} (x) = {[\begin{matrix} \frac{\partial F (x)}{\partial x_{1}} & \dots & \frac{\partial F (x)}{\partial x_{n}} \end{matrix}]}^{T}$ (15)

Let H be Hessian Matrix, thus: $H = F^{'} (x) = [\frac{\partial^{2} F}{\partial x_{i} \partial x_{j}}]$ (16)

The fitting result of nonlinear least square parameter is: $ζ = F (x) \circ \int \int \int ⨀ f_{i} (x) \cdot y_{i} / g \cdot H$ (17)

2.2.3 Estimation fitting of two-parameter optimal slip rate

According to the above fitting results of nonlinear least square parameters, the estimation fitting process of setting the two-parameter optimal slip rate is shown in Fig. 3.

Fig. 3

Estimation flow of optimal slip rate.

In Fig. 3, according to the functional relationship between the slip rate and the friction coefficient, the two-parameter least square estimation algorithm is used to calculate the optimal slip rate under the condition of driving road [17].

The damping method is used to solve the two-parameter least square method and thus to integrate the slip rate and the adhesion coefficient, the function expression is obtained. $μ (x_{1}, x_{2}, λ) = \frac{30 λ}{1 + 1 + x_{1} λ + x_{2} λ^{2}}$ (18)

During the calculation of optimal slip rate, the search range of parameters (x₁, x₂) which need to be estimated is constrained at first, so as to avoid the excessive divergence of the fitting results caused by the adverse sampling results. When limiting the search fields, we should consider the following aspects:

Firstly, it is necessary to judge whether the initial search location x₀ belongs to the search domain before the first update iteration. If not, we must adjust x₀ to the search domain closest, as shown in Fig. 4. In order to ensure the stability of algorithm, a parameter ɛ is introduced to represent the thickness tolerance of search boundary, which is used to adjust the distance between the initial location x₀ and the search boundary [18, 19]. The value of ɛ is 0.01 [6].

Fig. 4

Judgment and adjustment of initial position of optimization.

Secondly, it is necessary to judge whether the result x + h generated by iteration is beyond the scope of search area. If yes, it is necessary to adjust the search vector h so that the iteration result x_k+1 will not be outside the search area [8]. See Fig. 5:

Fig. 5

Adjustment from inside to boundary.

Through the above process, the optimal slip ratio is shown as follows: $λ_{p} = \frac{\sum_{i = 1}^{k} μ (x_{1}, x_{2}, λ)}{ζ}$ (19)

In the above process, the estimation for the optimal slip rate was completed, which prepares for the final control of wheel slip rate [4].

2.3 Design of slip rate controller

Based on the estimated optimal slip rate, the slip rate controller is designed by the multi-agent system. The specific design process is as follows.

In this article, a multi-agent system composed of five intelligence agents is used to design the slip rate controller, where No. 0 is the leader agent, No.1, No.2, No.3 and No.4 are the follower agent. The topology of multi-agent system is shown in Fig. 6.

Fig. 6

Topology of multi-agent system.

In multi-agent system, if the master node changes continuously and dynamically, the purpose of control is to enable the slave node to track the motion state of master node quickly, so as to achieve the state consistency between master node and slave node. The dynamic equations of master node and slave node are as follows: ${\begin{matrix} x_{0} = v_{0}, x_{0} \in R^{m} \\ v_{0} = - k_{0} x_{0} - k_{1} v_{0} + u_{0}, v_{0} \in R^{m} \\ x_{t} = v_{t}, x_{t} \in R^{w} \\ v_{t} = - k_{0} x_{t} - k_{1} v_{t} + u_{t} + δ_{t}, v_{t} \in R^{w} \end{matrix}$ (20)

In formula, x₀ denotes the position of leader agent; v₀ denotes the speed of leader agent; x_t denotes the position of follower agent; v_t denotes the speed of follower agent; k₀ denotes the feedback gain of position; k₁ denotes the feedback gain of speed; u_t denotes the control input of multi-agent system; δ_t denotes the bounded disturbance of multi-agent system.

Thus, the error equation is: ${\begin{matrix} ɛ_{1}^{'} = ɛ_{2} \\ ɛ_{2} = x_{0} + v_{0} + {Bk}_{0} ɛ_{1} - k_{1} ɛ_{2} \end{matrix}$ (21)

The threshold value of slip rate controller is: $S = ɛ_{1}^{'} + β ɛ_{2}^{α}$ (22) In the formula, β = diag [β₁, β₂, β₃, ⋯ , β_n].

2.4 Parameter optimization of slip rate controller

Based on the above threshold value of slip rate controller, the fuzzy algorithm is used to optimize the parameters of slip rate controller, so that the optimal slip rate remains, and the control for wheel slip rate can be achieved.

The fuzzy algorithm belongs to the intelligent algorithm. When we don’t understand the system model in depth, or objective reasons limit the research on the control model of system, the intelligent algorithm often plays an inessential role. At this time, the fuzzy algorithm is adopted. The common fuzzy algorithms include mean fuzzy and Gaussian fuzzy.

The schematic design of fuzzy algorithm is shown in Fig. 7.

Fig. 7

Theory of fuzzy algorithm.

According to the fuzzy algorithm shown in Fig. 7, the parameters of the slip rate controller are optimized. The specific process of parameter optimization is shown in Fig. 8.

Fig. 8

Flow chart of parameter optimization of slip rate controller based on fuzzy algorithm.

Through the above process, the wheel slip rate can be controlled, which provides more effective support for the safe driving.

3 Results

In order to prove the control performance of the proposed method, a simulation experiment is designed. The specific experimental process is as follows.

3.1 Establishment of simulation environment

In order to ensure the accuracy of experimental data, the simulation model of automobile brake system is built in Fig. 9.

Fig. 9

Simulation model of automobile brake system.

According to the least square method in Simulink simulation software, the estimation model of optimal slip rate is established, as shown in Fig. 10. The input is the vehicle driving speed, vehicle deceleration and wheel angular speed. The optimal slip rate in the current road condition is obtained by the least square algorithm.

Fig. 10

Model of optimal slip rate estimation algorithm.

In Simulink simulation software, the fuzzy algorithm is used to build the sliding mode controller of optimal slip rate. The input contains the driving speed, the estimated optimal slip rate and the driving deceleration. The output is the total braking force that is needed to realize the ideal emergency braking. According to S function in MATLAB simulation software, the regenerative braking system model established in AMESim software is connected with the braking system model. According to the specific requirements of experiment, we need to set the simulation experiment parameters. See Table 2:

Table 2

Parameter setting of simulation experiment

Parameter	Parameter definition	Parameter value
M	Automobile quality	1359 kg
Lf	Distance from center of mass to front Axle	1.050 m
Lr	Distance from center of mass to rear Axle	1.569 m
H	Distance from center of mass to ground	0.67 m
Cf	Front wheel side deflection stiffness	87320 N/Deg
Cr	Rear wheel side deflection stiffness	125900 N/Deg
Iz	Moment of Inertia about Z Axis	1763.8 kg.M2

3.2 Analysis of simulation results

According to the experimental environment and the experimental parameters, the feedback control method of wheel slip rate state, the wheel slip rate sliding mode control method, the wheel slip rate control method based on the back wheel state monitoring and the proposed method were adopted for the comparison experiment. The comparison between the automobile speed and the slip rate is shown in Fig. 11.

Fig. 11

Comparison of vehicle speeds and slip rates.

According to the analysis of Fig. 11(a), compared with the traditional method, the method in this paper takes the shortest time at the same speed, which shows that the method has the shortest braking time. According to the analysis of Fig. 11(b), the optimal slip ratio is kept between 0.20–0.50, the slip ratio of the method in this paper is kept between 0.18–0.45, the slip ratio of the sliding control method of automobile wheel slip ratio is controlled between 0.05–0.55, the slip ratio of the state feedback control method of automobile wheel slip ratio is kept between 0.14–0.53, and the slip ratio control method of automobile wheel based on the state monitoring of automobile rear wheel is kept between 0.14–0.53 The slip ratio is kept between 0.10–0.60. By analyzing the above data, it can be seen that the slip ratio of the three traditional methods is in a large range, and there are many differences with the optimal slip ratio. Compared with the optimal slip ratio, the slip ratio of this method is closest to the optimal slip ratio, which shows that the control effect of the proposed method is better.

4 Discussion

The slip ratio is an important parameter in emergency braking. If the slip ratio is too small, the braking time of will be too long. If the slip ratio is too large, the wheels will be locked. When the front wheel is locked, the car will have serious side slip under the small lateral force. When the rear wheel is locked, the steering will be impossible. When the front wheels and rear wheels are locked at the same time, the car will slide straight. In order to ensure the brake safety, it is necessary to make the slip ratio be close to the optimal value. This will prevent the car from locking. At the same time, the friction between car and road is the largest, so that the car has good brake performance. For different road conditions, the optimal slip ratio between vehicle and road is different, so the research on the control of slip ratio of vehicle emergency braking is helpful to improve the control ability of ABS and ensure the brake safety.

Simulation results show that compared with the current control methods of wheel slip rate, the proposed method greatly improves the control effect. Therefore, the proposed control method of wheel slip rate has better performance.

5 Conclusions

The proposed control method greatly improves the control effect, so the method of controlling the wheel slip rate has better performance. This method provides more effective support for the safe driving. However, the proposed method does not take into account the time-varying parameters and abrupt changes in road conditions, and there is still room for improvement in the control effect. Further optimization research is needed for the method. In the future, the above two points will be the research focus to further improve the method in this paper.

Footnotes

Acknowledgments

The research was supported by Research Project of Hubei Provincial Department of Education (No. B2019057).

References

Minematsu

, Hanaoka

, Takeshita

, et al., Long-term wheel-running can prevent deterioration of bone properties in diabetes mellitus model rats, Journal of Musculoskeletal & Neuronal Interactions17(1) (2017), 433–443.

Yang

, Yang

, Liu

, et al., Research on 3d positioning of handheld terminal based on particle swarm optimization, Journal of Internet Technology20(2) (2019), 563–572.

Fang

, Mu

, Deng

, et al., Fast detection of the fuzzy communities based on leader-driven algorithm, International Journal of Modern Physics B64(23) (2017), 1850058.

Wang

, Shi

, Man

, et al., Continuous fast nonsingular terminal sliding mode control of automotive electronic throttle systems using finite-time exact observe, IEEE Transactions on Industrial Electronics12(99) (2018), 1–1.

Cui

, Hu

and Li

, Establishment and applications of thermal forming limit margin field graph for automobile components, Zhongguo Jixie Gongcheng/China Mechanical Engineering28(3) (2017), 358–365.

Fan

and Wang

, A two-phase fuzzy clustering algorithm based on neurodynamic optimization with its application for PolSAR image segmentation, IEEE Transactions on Fuzzy Systems26(1) (2018), 72–83.

, Jiang

, Song

, et al., Analysis and validation of transient thermal model for automobile cabin, Applied Thermal Engineering74(10) (2017), 122.

Cuevas-Martinez

J.C.

, Yuste-Delgado

A.J.

and Trivino-Cabrera

, Cluster head enhanced election type-2 fuzzy algorithm for wireless sensor networks, IEEE Communications Letters17(99) (2017), 1–1.

Wang

J.Y.

, Strength optimization design and simulation of automobile shape shell structure, Computer Simulation34(03) (2017), 144–147.

10.

Prasad

K.V.

, Hanumesh

and Vajravelu

, MHD mixed convection heat transfer over a non-linear slender elastic sheet with variable fluid properties, Applied Mathematics and Nonlinear Sciences2(2) (2017), 351–366.

11.

Verde

, Harby

and Corberan

J.M.

, Optimization of thermal design and geometrical parameters of a flat tube-fin adsorbent bed for automobile air-conditioning, Applied Thermal Engineering111(1359) (2017), 489–502.

12.

Ammar

M.K.

, Amin

M.R.

and Hassan

M.H.M.

, Visibility intervals between two artificial satellites under the action of Earth oblateness, Applied Mathematics and Nonlinear Sciences3(2) (2018), 353–374.

13.

Chevalier

M.L.

, Philippe

H.L.

, Replumaz

, et al., Temporally constant slip rate along the Ganzi fault, NW Xianshuihe fault system, eastern Tibet, Geological Society of America Bulletin130(3) (2017), 104–105.

14.

Yang

M.S.

and Nataliani

, A feature-reduction fuzzy clustering algorithm based on feature-weighted entropy, IEEE Transactions on Fuzzy Systems52(99) (2017), 1–1.

15.

Ali

M.Z.

, Awad

N.H.

, Reynolds

R.G.

, et al., A balanced fuzzy cultural algorithm with a modified levy flight search for real parameter optimization, Information Sciences12(7) (2018), 447.

16.

Gao

, Farahani

M.R.

, Aslam

, et al., Distance learning techniques for ontology similarity measuring and ontology mapping, Cluster Computing-the Journal of Networks Software Tools and Applications20(2) (2017), 959–968.

17.

Liu

X.W.

, Yuan

D.Y.

and Su

, Late pleistocene slip rate of the northern Qilian Shan frontal thrust, western Hexi Corridor, China, Terra Nova29(6) (2017), 238–244.

18.

Qin

, Lou

, Zhao

and Zhang

, Research on relationship between tourism income and economic growth based on meta-analysis, Applied Mathematics & Nonlinear Sciences3 (2018), 105–114.

19.

Sagna

, Multidimensional BSDE with Poisson jumps of Osgood type, Applied Mathematics and Nonlinear Sciences4(2) (2019), 387–394.