J. Cent. South Univ. (2012) 19: 974-981

DOI: 10.1007/s11771-012-1099-x

Multi-objective optimization of active steering system with force and displacement coupled control

ZHAO Wan-zhong(赵万忠)^{1, 2}, SUN Pei-kun(孙培坤)¹, LIU Shun(刘顺)¹, LIN Yi(林逸)³

1. Department of Vehicle Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China;

2. State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing 400044, China;

3. School of Mechanical and Vehicular Engineering, Beijing Institute of Technology, Beijing 100081, China

? Central South University Press and Springer-Verlag Berlin Heidelberg 2012

Abstract: A novel active steering system with force and displacement coupled control (the novel AFS system) was introduced, which has functions of both the active steering and electric power steering. Based on the model of the novel AFS system and the vehicle three-degree of freedom system, the concept and quantitative formulas of the novel AFS system steering performance were proposed. The steering road feel and steering portability were set as the optimizing targets with the steering stability and steering portability as the constraint conditions. According to the features of constrained optimization of multi-variable function, a multi-variable genetic algorithm for the system parameter optimization was designed. The simulation results show that based on parametric optimization of the multi-objective genetic algorithm, the novel AFS system can improve the steering road feel, steering portability and steering stability, thus the optimization method can provide a theoretical basis for the design and optimization of the novel AFS system.

Key words: vehicle engineering; active steering; electric power steering; multi-objective genetic algorithm

1 Introduction

The electric power steering (EPS) controls the force transfer characteristic of the steering system. It provides good steering portability and steering road feel through the assist control strategy, returnability and damping control strategy [1-3]. Because of the EPS system characteristic, it is especially suitable for the electric vehicles and new energy vehicles. Therefore, the EPS system has a very broad applicable prospect.

The active front steering (AFS) controls the displacement transfer characteristic of the steering system. It provides good handling and stability performance of the vehicle through variable transmission ratio control and active steering intervention control [4-6]. The AFS system incorporates the safety, flexibility and driving pleasure, brings the steering system’s function into full play and greatly improves the steering quality.

At present, the EPS cannot achieve variable ratio control and active steering intervention control to improve handling stability [7-9]. While still using hydraulic system to obtain steering power, the AFS system is complex and inevitably has the disadvantages of hydraulic system, and the oil volumetric flow should be adapted to the output requirements [10-11]. Compared to the EPS system, the AFS system can hardly realize the force and displacement coupled control, returnability and damping control.

Therefore, based on the technique of EPS and AFS, a novel steering system with function of both the AFS and EPS will be the main development direction of automotive steering technique in the future. A novel AFS system with force and displacement coupled control was developed in this work. Based on the dynamic model, the performance index of the portability, road feel and stability of steering was put forward and analyzed. Based on the multi-objective genetic algorithm, a multi-variable genetic algorithm for the novel AFS system was designed, and thus a theoretical basis for the design and optimization of the novel AFS system was provided.

2 Dynamic model

The structure of the novel AFS system is shown in Fig. 1. Two motors are included: one motor is assist motor, providing assistant torque to limit manual forces to a reasonable level [12-14], and the other is steering motor, providing additional steering angle to improve vehicle steering responses and avoid critical handling situations.

Fig. 1 Structure of novel AFS: 1—Torque sensor; 2—Steering motor; 3—Assist motor; 4—Rack and pinion; 5—Deceleration institution

2.1 Three-freedom model of vehicle

The three-degree of freedom dynamic differential equations for the vehicle can be expressed as

                     (1)

The parameters  and  can be described as

                                               (2)

where g is the acceleration due to gravity; u, ω_r, f, β and δ are velocity, yaw rate, roll angle, sideslip angle of the center of mass, and front-wheel steer angle of the vehicle, respectively;a, b, and h are front-wheel sideslip angle, rear-wheel sideslip angle, distance between the front axle and the center of mass, distance between the rear axle and the vehicle centre of mass, and rolling moment arm of the vehicle, respectively; m, m_s, I_x, and I_z are mass, sprung mass, moment of inertia of sprung mass about x axis, and moment of inertia of mass about z axis of the vehicle, respectively; I_xz, k₁, k₂, E₁ and E₂ are product of inertia of sprung mass about axes x and z, front-wheel cornering stiffness, rear-wheel cornering stiffness, front roll steer coefficient, and rear roll steer coefficient of the vehicle, respectively; C_f1, C_f2, D₁ and D₂ are roll angle stiffness of front suspension, roll angle stiffness of rear suspension, roll angle damping of front suspension, and rolling angle damping of rear suspension of the vehicle, respectively.

2.2 Steering input shaft model

The dynamic equation for the steering input shaft which is connected to the stator of steering motor can be expressed as

                                                  (3)

where J_s is moment of inertia of the steering input shaft and steering wheel; θ_s, B_s and T_sen1 are angle of rotation, damping coefficient, and anti-torque of the input shaft, respectively; T_h is the steering torque acting on the steering wheel.

Parameter T_sen1 can be expressed as

                                                        (4)

where K_s is the stiffness coefficient of the input shaft; θ_p is the angle of rotation of the steering motor’s stator.

2.3 Steering motor model

A permanent magnet brushless servomotor is employed as the steering motor in the novel AFS system. Based on the reference coordinate of the rotor, the voltage equations for the servomotor can be written as

                   (5)

where u_q is the voltage of the q axis; u_d is the voltage of the d axis; i_d is the current of the d axis; i_q is the current of the q axis; R_s is the resistance of the armature  winding; k is the voltage gain of the inverter forward path; k_f is the current feedback coefficient; L_s is the armature inductance; w_m-r is the relative angular velocity of the rotor to the stator; e_f is the rotational voltage; p is the pole-pairs. The electromagnetic torque caused by servomotor is obtained by

                                                                    (6)

where T_s and k_e are the electromagnetic torque and torque coefficient of the servomotor, respectively.

2.4 Steering motor stator model

Taking the transitional linkage which is connected to the stator into consideration, the dynamic equation for the stator of steering motor can be written as

                                               (7)

where J_p1 and B_p are moment of inertia and damping coefficient of the stator, respectively.

2.5 Steering motor rotor model

The dynamic equation for the of rotor steering motor may be expressed as

                                             (8)

where J_p2, T_sen2 and θ_w are the moment of inertia, anti-torque, and angle of rotation of the steering motor rotor, respectively. The T_sen2 can be expressed as

                                                        (9)

where θ_e is the rotation angle of the output shaft.

2.6 Assist motor model

A permanent magnet motor is employed as the assist motor in the novel AFS system, and the voltage equation of the assist motor is obtained as

                                                   (10)

where U, L, R, K_b, θ_m and I are voltage, inductance, resistance, back electromotive-force constant, angle of rotation, and current of the assist motor, respectively.

The electromagnetic torque caused by the assist motor can be expressed as

                                                                    (11)

where K_a is moment coefficient of the assist motor.

The dynamic equations of the mechanical part of the assist motor can be written as

                                             (12)

where J_m is moment of inertia of the assist motor and clutch; B_m, T_m and T_a are damping coefficient, electromagnetic torque, and output torque of the assist motor, respectively.

The T_a can be computed by

                                                     (13)

where K_m is the output stiffness of the assist motor and deceleration device; G is the transmission ratio from the power motor to the steering screw.

2.7 Output shaft model

The dynamic equation for output shaft is obtained as

                                 (14)

where J_e, B_e and T_r are moment of inertia, damping coefficient, and anti-torque of output shaft, respectively.

2.8 Rack and pinion model

The dynamic equation for the rack and pinion can be given by

                                                (15)

where m_r is the effective mass of the rack and pinion; b_r is the damping coefficient of the rack; x_r is the displacement of the rack; r_p is the radius of the small pinion; F_TR is the axial thrust exerted on the rack.

The F_TR can be simplified as

                                                          (16)

where k_r is the effective elasticity coefficient; F_δ is the random signal from the road. And θ_e=x_r/r_p.

3 Steering performance index

The novel AFS system should have the following performances: 1) Good road feel; 2) Good steering portability; 3) Good steering stability. In order to provide a basis for the optimization design of the novel AFS system, the quantitative formula is proposed.

3.1 Steering road feel

The steering road feel is analyzed by fastening the steering wheel. In this way, the interference information can be transferred to the driver completely. Additionally, it is easier to analyze the system, since one-degree of freedom is reduced. Assume

θ_s=0                                       (17)

When the assist motor is assumed as using the current control strategy, and the torque sensor is reduced to the torsion bar spring, the measured value of the torque sensor can be computed by

                                              (18)

where T_n is the torque received by the torque sensor.

According to the current control strategy, the current can be given by

                                                                       (19)

where K is power gain coefficient. Then, T_m can be given by

                                                 (20)

The novel AFS system can be simplified by hypothesizing the front wheel and the steering screw equivalent, and the simplified novel AFS system model contains the following information: 1) G is the transmission ratio from the power motor to steering screw; 2) n₁ is the ratio from θ_p to θ_w; 3) n₂ is the transmission ratio from the steering screw to front wheel; 4) θ_w is equivalent to θ_e. It exists the following relations:

                                                                  (21)

where δ is the angle of front wheel. According to Eqs. (3), (4), (7), (8), (12), (14), (17), (20) and (21), the steering road feel E(s), namely the transfer function from T_r to T_h, can be computed by

                               (22)

where s is the complex variable, and

 .

3.2 Steering portability

The steering portability is defined as the ratio from the yaw velocity ω_r to the angle of rotation of input shaft θ_s. According to Eqs. (4), (7), (8), (12), (14), (20) and (21), the transfer function of the steering portability can be expressed as

                                                    (23)

(24)

According to Eqs. (1)-(2),

                                                 (25)

where

3.3 Steering stability

It is essential to guarantee the steering stability while developing the novel AFS and the vehicle system. Therefore, it is indispensible to analyze how to guarantee the steering stability. So, it is necessary to analyze the requirements of steering stability.

Choose the denominator of the transfer function of steering portability (Eq. (23)) as the requirement of steering stability, then

           (26)

where

The Routh list of this function is expressed as

         (27)

In accordance with the requirements of the Routh criterion, the numbers in the first column should be all positive.

4 Parameters optimization

4.1 Design variables

Considering that some parameters of the vehicle are unchangeable, the power gain coefficient K varies with the speed, and the friction coefficients cannot be changed randomly. Therefore, n₁, G and K_s (N·m/rad) can be chosen as the design variables. Namely

                                               (28)

4.2 Objective function

In order to transfer the information from the road to the driver completely, the average frequency power of the steering road feel should be as much as possible in a certain frequency range. The objective function f is the average frequency power of the steering road feel in the (0, ω₀) frequency range, and ω₀ is designed as 40 Hz, namely,

   (29)

In order to get good steering portability for the vehicle, the average frequency power of the steering portability should be in an appropriate range. The objective function g is the average frequency power of steering portability in (0, ω₀) frequency range, and ω₀ is designed as 40 Hz, namely,

 (30)

In order to pledge both the steering road feel and steering portability, the average frequency power of the steering portability should be optimized. The objective function g_f is the sum of the average frequency power of steering road feel and steering portability with weight n in (0, ω₀) frequency range, and ω₀ is designed as 40 Hz, namely

           (31)

4.3 Constraints

The requirement of the steering portability of novel AFS system is

(32)

The steering portability g is set in a certain range  [a, b], namely,

                                                    (33)

4.4 Multi-objective optimization

Solving multi-objective problem is a decision- making problem, not only an optimization problem, since the result of multi-objective optimization is a disaggregation. When Pareto’s optimal disaggregation comes out, the best result can be chosen by decision- maker.

4.4.1 Getting normative decision-making matrix by normalizing vector

Set decision-making matrix Y={y_ij} and normalized decision-making matrix Z={z_ij} of the multi-attribute decision-making problem, then

         (34)

4.4.2 Composing weighted normalized matrix

Weight vectors are set by decision-maker, ω_w=[ω₁, ω₂, …, ω_n], then

                (35)

4.4.3 Determining ideal solution x⁺ and minus ideal solution x^-

Suppose that  is the first j property value of x⁺ and  is the first j property value of x^-, and  meet

             (36)

where j=1, 2, …, n.

4.4.4 Calculating distance from alternative point to ideal point or minus ideal point

The distance from alternative point to the ideal point or the minus ideal point can be expressed as

                 (37)

4.4.5 Calculating line effect displayed value of alternatives C_i (comprehensive evaluation)

Parameter C_i can be expressed as

                                 (38)

4.4.6 Ranking

According to the C_i value by large to small, discharge the merits of the scheme order.

4.5 Result of optimization

4.5.1 Single-objective and single-constraint

With the Routh criterion as the constraint condition, the best result is X=(10, 5, 200), when the road feel f is maximal. That is n₁=10, G=5, K_s=200 N·m/rad.

Figures 2 and 3 depict the simulation results of single-objective and single-constraint optimization. According to Figs. 2 and 3, the bandwidth increases and the phase delay decreases after optimization. Compared to 0.001 9 before optimization, the energy of the steering road feel is 0.032 4 after optimization, and it is 17 times greater than it before. Therefore, based on the multi- objective genetic algorithm optimization of this objective, the steering road feel of the novel AFS system can be optimized.

Fig. 2 Step response of steering road feel

Fig. 3 Bode reponse of steering road feel: (a) Amplitude frequency characteristics of steering road feel (I); (b) Phase frequency characteristics of steering road feel (I)

4.5.2 Single-objective and multiple-constraints

With the Routh criterion and steering portability within [1.2×10^-4, 1.8×10^-4] as the constraint conditions, the best result is X=(1.09, 5.6, 200) when the steering road feel f is maximal. That is, n₁=1.09, G=5.6, K_s=   200 N·m/rad.

Figures 4 and 5 depict the simulations results of single-objective and multiple-constrains optimization.

The energy of the steering road feel is 0.005 9 after optimization, compared to 0.001 9 before optimization, and it is 3.1 times greater than it before optimization. According to Figs. 4 and 5, the bandwidth increases and the phase delay decreases after optimization. The result shows that, steering road feel of the novel AFS system can be optimized by adopting the multi-objective genetic algorithm optimization of this objective.

4.5.3 Multiple-objectives and multiple-constraints

With the Routh criterion and steering portability within [1.2×10^-4, 1.8×10^-4] as constraint condition, the best result is X=(0.793, 5, 191), when g_f is maximal with n=100. That is, n₁=0.793, G=5, K_s=191 N·m/rad.

Fig. 4 Step response of steering road feel

Fig. 5 Bode response of steering road feel: (a) Amplitude frequency characteristics of steering road feel (II); (b) Phase frequency characteristics of steering road feel (II)

Figures 6 and 7 depict the simulations of multiple- objective and multiple-constrains optimization.

After optimization, the energy of the steering road feel is 0.004 4, and before optimization, it is 0.001 9. Additionally, the energy of the steering portability is1.799 8×10^-4 after optimization and is 1.642 5×10^-4 before optimization, so the energy of the steering portability is increased by 9% after optimization. Thereby, the energy of steering road feel is 2.31 times greater after optimization. As is shown in Figs. 6 and 7, the bandwidth increases and the phase delay decreases after optimization. Therefore, the steering road feel of the novel AFS system can be optimized by the genetic algorithm optimization of this objective.

Fig. 6 Step figure of steering road feel

Fig. 7 Bode figure of steering road feel: (a) Amplitude frequency characteristics of steering road feel (III); (b) Phase frequency characteristics of steering road feel (III)

5 Conclusions

1) The novel AFS system with force and displacement coupled control is introduced to extend the steering performance, and it has functions of both the AFS and EPS. Based on the model and the vehicle three-degree of freedom model, the steering road feel, steering portability, and steering stability are put forward and analyzed, which provides the basis for the parameter optimization of the novel AFS system.

2) According to the characteristic of constraint optimization of multivariable function, the parameters of the novel AFS system are optimized using the multi-objective genetic algorithm optimization. The simulation results show that the steering road feel and steering portability can be optimized effectively with multi-objective genetic algorithm. Thus, the optimization method can provide a theoretical basis for the design and optimization of the novel AFS system.

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(Edited by DENG Lü-xiang)

Foundation item: Project(51005115) supported by the National Natural Science Foundation of China; Project(KF11201) supported by the Science Fund of State Key Laboratory of Automotive Safety and Energy, China; Project(201105) supported by the Visiting Scholar Foundation of the State Key Laboratory of Mechanical Transmission in Chongqing University, China

Received date: 2011-02-23; Accepted date: 2011-06-27

Corresponding author: ZHAO Wan-zhong, Associate Professor, PhD; Tel: +86-25-84892200-2615; E-mail: zhaowanzhong@126.com