Fuzzy control method for a steering system consisting of a four-wheel individual steering and four-wheel individual drive electric chassis

Abstract

In this paper, we study a four-wheel, individual steering and four-wheel, individually driven, electric chassis, named the flexible chassis. A 2-FOF nonlinear steering model for the flexible chassis is established based on the “magic formula” tire model and the Ackermann steering theorem at low speed. Also, a new control method based on the fuzzy strategy is proposed for its sideslip angle control. The left front-wheel steering angle is inputted into the simulation of the vehicle, and the left-rear wheel angle is controlled by the fuzzy controller to make the sideslip angle zero. Also, the remaining steering angle of the other two wheels, which meet the Ackermann theorem, are adjusted. Comparisons between simulations of the front-wheel steering, four-wheel, proportional steering and the newly proposed steering indicate that the new control algorithm in this paper is effective.

Keywords

Electric vehicle fuzzy control four-wheel independent steering four-wheel independent driving simulation

1 Introduction

Due to environmental pollution and an energy crisis, electric vehicles (EV) have aroused great interest in research as an effective solution [1, 2]. With each wheel independently driven and independently steered by electric motors instead of an integrated driving system and the steering mechanism in a traditional automobile, four-wheel, individual drive and four-wheel, individual steering (4WID/4WIS) EV offers better driving efficiency and has control flexibility and maneuverability. Thus, it becomes a promising direction for electric vehicles [18, 23].

This paper proposes a chassis named the flexible chassis (FC), which is a 4WID/4WIS EV with in-wheel motors (IWM). Because of its Omni directional running, its main application is to work on a narrow road with a semi-closed or closed environment at low speed. This paper introduces an effective fuzzy control method for 4WID/4WIS FC, which is a nonlinear system.

The four-wheel, independent steering control system, is seen as an important way to improve the handling characteristics of the vehicle. It has been valued by domestic and foreign experts and scholars based on the state feedback control method, which is based on the linear two degree of the freedom (DOF) model [21]. Time delay control for the 4WIS vehicle using single and dual steering control strategies was proposed [22]. A linear quadratic regulator in a multi-environment was also researched. [15]. Joint control of the four-wheel steering and vehicle dynamic control was presented. [17]. A control simulation using linear predictive control proved that the steering stability has improved to some extent [20]. Side slip zeroing control for the4WS vehicle was proposed [16]. Multi-objective H^∞ optimal control for the 4WS vehicle based on yaw rate tracking was also presented [9]. Furthermore, a closed-loop, comprehensive evaluation method based on the linear control theory was proposed [14]. A variable ratio of the steering system using the optimal control method is also proposed [11]. Then, we presented an adaptive model following the control of the 4WS steering vehicle [19]. Synthesis robust control for the four-wheel steering vehicle based on yaw rare tracking was also discussed [7]. The neural network control for the 4WS based on a nonlinear model was also proposed [8]. Many control methods were demonstrated based on the sideslip angle-equaling zero without considering the Ackerman steering principle [3, 10]. Many control methods based on the Ackerman steering principle were proposed without considering nonlinear characteristics of the tires [4, 13].

From the above literature review, it can be seen that control methods based on two or three DOF, and the linear model or nonlinear model with the target of sideslip angles that equal zero, has generally been applied to controlling the four-wheel steering electric vehicles. Then various elements are selected to obtain the control strategies without considering the Ackermann steering theorem. Others take into account the Ackermann steering theorem, but the model used is built with a simple linear model using linear control strategies without taking into account the nonlinear characteristics of the tire. In this paper, a two DOF, nonlinear steering model for a four-wheel, individual steering electric chassis is established based on the “magic formula” tire model and the Ackermann steering theorem at low speed.

In this model, the input of the vehicle fuzzy control system simulation is the steering angle of left front wheel, and output is the steering angle of left rear wheel. The fuzzy controller’s goal is sideslip angle being zero. The angles of right front wheel and right rear wheel are adjusted using Ackermann steering principle, thereby four-wheel independent steering for FC can be achieved.

The rest of this paper is organized as follows: The introduction of the Flexible chassis is presented in Section 2. The FC’s kinematic model, dynamics model and tire model are presented in Section 3. The fuzzy control algorithm is provided in Section 4. The result of the experiment is compared with simulation in Section 5, followed by the conclusive remarks.

2 The introduction of the Flexible chassis

FC (as represented in Fig. 1) is made of four, independent articulations and is symmetrical. Each articulation part combines an off-centered steering (OCS) axis, a leg, an electromechanical steering lock (FBD-050 from Taiwan KAIDE) a DC, brushless motor wheel (in-wheel motor from BATTLE), and has three DOF. The technical parameters of the IWM are shown in Table 1. The OCS wheel plane is parallel to the steering axis but does not contain it. Also, the steering axis and wheel axis intersect in one point. The leg can rotate 180° around its attachment point. Once an articulation is placed at the right position, the system is designed to keep it in position by locking the electromechanical lock. A passive vertical suspension made of springs is used to connect the steerable wheels to the chassis, allowing the wheels to keep contact with the ground on uneven surfaces. FC can change its direction by changing the direction of its articulations, making the chassis Omni directional (as represented in Fig. 2), allowing it to move in tight areas (narrow environment).

The designed bearing loading of FC is not less than 200 kg. The clearance between the empty frames to the ground is 580.7 mm, and the speed for working is 3.6 km/h. Technical parameters are represented in Table.1. The walking chassis adopted the structure with high clearance, which not only ensured the farmland operation, but also optimized the function of the four-wheel, independent steering driving and control system.

3 The model of the Flexible Chassis

3.1 Kinematic model based on Ackerman steering theorem

When cars are doing the steering process at a low speed, almost no slippage of four wheels occurs. Also, the Ackerman theorem [12] must be applied. This means all the wheels will center on an instantaneous turning center scrolling, which is shown in Fig. 3.

a_i (i = 1, 2, 3, 4) refers to the steering angle of the ith off-centered wheel; F_i (i = 1, 2, 3, 4) refers to the lateral force of the ith wheel; δ_i (i = 1, 2, 3, 4) refers to the slip angle of the ith tire; a and b refer to the horizontal distance between the extended line of the instant center and front and rear axle; W refers to tread; L refers to the wheel base; O refers to the barycenter of the ideal state; O’ refers to the instant turning center; r refers to the distance between the right wheel and the instant center; u and v represent the longitudinal velocity and the transverse velocity, respectively. β refers to the vehicle’s sideslip angle.

According to Ackerman Theorem, its angular movement geometric relationship is as follows: ${\begin{matrix} α_{1} = arctan (\frac{a}{r + W}) \\ α_{2} = arctan (\frac{a}{r}) \\ α_{3} = arctan (\frac{b}{r}) \\ α_{4} = arctan (\frac{b}{r + W}) \end{matrix}$ (1) $a + b = L$ (2)

3.2 Dynamics model

In order to analyze the steering stabilization of each wheel of the four-wheel, independent steering electric and simplify the model of the system, the following reasonable assumptions are proposed:

Weight of the vehicle is evenly distributed to four wheels;

Each wheel is all the same and the tire characteristics of each tire are ignored because of the load’s changes;

Ignoring the role of the automotive suspension, the chassis’ vertical displacement movement, pitch motion about the transverse axis and rolling motion of the longitudinal axis are not considered.

Two-degree freedom, nonlinear dynamic equations [6] of the four-wheel, independent steering for the flexible chassis can be expressed as: $mu (\dot{β} + γ) = \sum_{i = 1}^{4} F_{i} cos α_{i}$ (3) $\begin{matrix} I_{z} \dot{γ} = a (F_{1} cos α_{1} + F_{2} cos α_{2}) \\ - b (F_{1} cos α_{1} + F_{2} cos α_{2}) + \frac{W}{2} \sum_{i = 1}^{4} F_{i} cos α_{i} {(- 1)}^{i + 1} \end{matrix}$ (4)

Where m is the vehicle mass, I_z is the moment of inertia.

Also, the slip angle of the ith tire δ_I (i = 1, 2, 3, 4) can be expressed as: ${\begin{matrix} δ_{1} = \frac{v + a γ}{u - c γ / - 2} - α_{1} \approx β + \frac{a γ}{u} - α_{1} \\ δ_{2} = \frac{v + a γ}{u + c γ / - 2} - α_{2} \approx β + \frac{a γ}{u} - α_{2} \\ δ_{3} = \frac{v - b γ}{u - c γ / - 2} - α_{3} \approx β - \frac{b γ}{u} - α_{3} \\ δ_{4} = \frac{v - b γ}{u + c γ / - 2} - α_{4} \approx β - \frac{b γ}{u} - α_{4} \end{matrix}$ (5)

3.3 Tire model

A traditional vehicle dynamics model is generally built using a linear tire model. This linear tire model is provided in the lower side of the angle, which is very small (less than 5°). However, when the side angle or lateral acceleration is large, the non-linear tires model is required. The Pacejka model (magic formula) is a semi-theoretical and semi-empirical model that can simulate linear and nonlinear characteristics of the tire [5]. It can also describe the steady state. The tire lateral force equation can beexpressed as: $F_{i} = D sin (C arctan (B α_{i} - E (B α_{i} - arctan (B α_{i}))))$ (6)

Where crest factor D refers to the maximum value of the curve for the tire deformation, $D = b_{1} F_{z}^{2} + b_{2} F_{z}$ and the vertical loading F_z = 1/4 mg according to the proposed assumptions;

Also, curve’s shape factor C = 1.30, due to the tire size, inflation pressure and tire tread design;

Stiffness factor B = b₃ sin(b₄ arctan((b₅F_z))/(CD));

Curve curvature factor means the shape of the curve near the maximum value, $E = b_{6} F_{z}^{2} + b_{7} F_{z} + b_{8}$ ; α_i refers to tire declination, i = 1, 2, 3, 4; b_j is fitting parameters, j = 1 -8.

4 Fuzzy control strategy

4.1 Fuzzy controller design

Based on the knowledge of human experience, the fuzzy controller can make accurate decisions in order to improve controlling accuracy and effectiveness. There are integral fuzzy sets in a fuzzy controller, and their sets are small, medium, large, etc., designed by work conditions. Fuzzy control is an intelligent control strategy that is able to mimic the way people think. Therefore, nonlinear problems can be solved in this way. In this paper, a two-dimensional fuzzy controller is applied. The error of system E and its derivative error Ec are input variables; the dynamic characteristics may reflect the input variables, as shown in Fig. 4.

Generally, no specific method could be applied to designing the fuzzy controller. However, input and output membership functions emulating human decision-making could be designed. There are three typical methods used to design a fuzzy controller:

Model of the control engineers’ knowledge.

Model based on human operators’ experience.

Mathematical model of the controlled plant.

4.2 Selection of fuzzy sets, basic domain, quantization factor and scaling factor

The research domain selection for the input of the fuzzy controller is as follows: the fuzzy domain of the sideslip angle error E is [–6, 6]; the fuzzy domain of the derivative error for the sideslip angle error Ec is [–6, 6]; and the fuzzy domain for the output of the fuzzy controller U, which is the angle ratio of the left front wheel and the left rear wheel, is [–1, 1].

According to practical experience of the four-wheel, independent steering system, the basic fuzzy domain of the sideslip angle error E and the error derivative Ec are [–0.1, 0.1] and [–0.01, 0.01], respectively; the basic fuzzy domain of the angle ratio of left front wheel and the left rear is [–1, 1].

Then, determination of the quantization factor and scaling factor is as follows:

Quantization factor of sideslip angle error: $ke = 6 / 0.1$

Quantization factor of derivative error for sideslip angle error: $kec = 6 / 0.01$

scaling factor of control variable’s output: $ku = 1 / 1 .$

Also, the fuzzy subset of E, Ec and U is {NB, N M, NS, ZO, PS, PM, PB}, in which the NB, NM, NS, ZO, PS, PM and PB represent negative big, negative medium, negative small, zero, positive small, positive medium and positive big, respectively. E and Ec stands for the error and the error rate, respectively. And the Figs. 5–7 show the membership function curves of E, Ec and U, respectively. The triangle-type membership function is adopted for E, Ec and U.

4.3 The rules of fuzzy control

According to the actual situation, the fuzzy system makes the fuzzy judgment using a simple average method. Then it does the anti-blur process, which is the transformation of the output of the fuzzy sets into the precise amount of control variables to achieve the best performance. The rules of the fuzzy controller are as follows:

If E is PB and Ec is PB, then U is NB

If E is PB and Ec is PM, then U is NB

If E is P B and Ec is P S, then U is N B

If E is P B and Ec is Z O, then U is N B

As shown in Table 2, there are a total of 49 rules in the fuzzy logical controller. Surface of the fuzzy logical controller output is demonstrated in Fig. 8.

5 Experiments and simulation analysis

In order to verify the correctness of the proposed four-wheel, independent steering nonlinear model based on the Ackermann and the effectiveness of the control strategy, comparisons were done between the proposed model, traditional, two-wheel steering (2WS) model and the steering model based on the ratio of the front and rear wheel (four-wheel steering, 4WS) [15]. A simulation was done by two different input situations:

speed of vehicle is 5 m/s, steering angle of left front wheel is 0.08 rad;

speed of vehicle is 8 m/s, steering angle of left front wheel is 0.2 rad.

Figure 9 shows the curves of the sideslip angles under Situation 1 and 2, which were simulated by the Matlab toolbox. The sideslip angle of the 4WS model, after a short fluctuation, can quickly reach a stable value, substantially, without errors, while the sideslip angle of 4WIS model also reach a stable value in a very short period of time. This indicates the effectiveness of the fuzzy control strategy. Also, due to the use of the nonlinear model, which is uncertain, the sideslip angle of the 4WIS model had a small, steady-state error within a reasonable range. The sideslip angle of the 2WS model is bigger than that of the 4WS and 4WIS models, especially in the case of the big angle. This demonstrates that the 4WS and 4WIS models could limit the changes of the sideslip angle at a mutative speed.

Figure 10 shows the curves of the yaw rate under two different situations. The yaw rate of the 2WS model with a big steering angle is very large and beyond the reasonable range. The reason is that bigger front-wheel steering angle results in larger front wheel tire’s lateral force because of tire linear model. The yaw rate of the 4WS model increases, which is due to an increase in both the front and rear lateral forces used in the linear tire model. This causes the vehicle yaw rate to also increase. Due to the nonlinear tire model, the tire lateral force in the large steering angle for the 4WIS model is basically unchanged, while the bigger angle of the tire results in the declining of the yaw rate of vehicle, as shown inEquation 2.

Figure 11 demonstrates the steering angle of 4 wheels under two different situations. Two rear wheels reached a stable state after a short period of time, and an angle of rear wheels is smaller and less angular. Meanwhile, two front wheels appear significantly different at a large steering angle. It can be seen that the fuzzy control strategy can make the wheels’ steering angle stabilized in a short time.

6 Conclusions

Traditional modeling for the four-wheel steering vehicle used in the linear model did not consider the difference of the left and right wheels’ steering angle while the vehicle was turning around. This paper proposed a nonlinear, four-wheel, independent steering model based on the Ackerman theorem and using fuzzy control. Through the Matlab simulation of the vehicle at high speed, low steering speed and low speed, as well as high steering speed, the results for the sideslip angle and the yaw rate were similar to that for the 4WS model. This proves the correctness of the model and the effectiveness of the control strategy at a small steering angle. In the big steering angle conditions, the 4WIS model can reflect the actual running state of the vehicle, as well as around the angle difference of the right and left wheels because of the applied nonlinear model. The 2WS and 4WS models are not capable of the above.

In this paper, only simulations have been done. There has been no actual testing. Therefore, for future research, an actual control system will be developed for the FC and actual road tests will be conducted to verify the feasibility of the control system.

Footnotes

Acknowledgments

This work was supported by National Natural Science Foundation of China (Grant nos. 51375401). Also, thanks to all the postgraduate students who provided their input to this study.

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