Wang Chunyan,Zhao Wanzhong,Liu Shun,Sun Peikun

(1.College of Energy and Power Engineering,Nanjing University of Aeronautics and Astronautics,Nanjing,210016,P.R.China;2.State Key Laboratory of Mechanical Transmission,Chongqing University,Chongqing,400044,P.R.China)

Nomenclature

g/(m?s-2) Aacceleration due to gravity

u/(m?s-1) Velocity

kr/(rad?s-1) Yaw rate

O/rad Roll angle of vehicle

U/rad Sideslip angle of vehiclecentre of mass

W/rad Steer angle of front wheels

T1/rad Sideslip angle of front wheels

T2/rad Sideslip angle of front wheels

k1/(N? rad-1) Cornering stiffness coefficient of rear wheels

k2/(N? rad-1) Cornering stiffness coefficient of rear wheels

a/m Distance between vehicle centre of mass and front axle

b/m Distance between vehicle centre of mass and rear axle

m/kg Mass of vehicle

ms/kg Mass of sprung

Ix/(kg? m2) Inertia moment of sprung mass about x-axis

Iz/(kg?m2) Inertia moment of sprung mass about z-axis

Ixz/(kg?m2) Inertia product of sprung mass about x,zaxis

E1Front roll steer coefficient

E2Rear roll steer coefficient

Cφ1/(N? m? rad-1) Stiffness coefficient of roll angle of f ront suspension

Cφ2/(N? m? rad-1) Stiffness coefficient of roll angle of rear suspension

D1/(N? m? s? rad-1) Damping of roll angle of front suspension

D2/(N? m? s? rad-1) Damping of roll angleof rear suspension

Js/(kg?m2) Inertia moment of input shaft

θs/rad Rotation angle of input shaft

Bs/(N? m? s? rad-1) Damping coefficient of input shaft

Tsen1/(N?m) Anti-torque of input shaft

Th/(N?m) Torque of input shaft

Ks/(N? m? rad-1) Stiffness coefficient of input shaft

θp/rad Rotation angle of AFS motor stator

uq/V Voltage of q-axis of AFSmotor

iq/A Current of q-axis of AFSmotor

ud/V Voltage of d-axis of AFS motor

id/A Current of d-axis of AFSmotor

Rs/Ω Armature resistance of AFS motor

k/(A? N-1? m) Voltagegain coefficient of AFS motor

kfFeedback current coefficient of AFSmotor

Ls/H Armatureinductance of AFSmotor

wrf/(rad?s-1) Angular velocity of AFSmotor rotor relative to its rotor

ef/V Rotation potential of AFSmotor

p Pole pairs of AFSmotor

Ts/(N?m) Electromagnetic torque of AFSmotor

ke/(N? m? A-1) Torque coefficient of AFSmotor

Jp1/(kg?m2) Inertia moment of AFSmotor stator

Bp/(N? m? s? rad-1) Damping coefficient of AFS motor stator

Jp2/(kg?m2) Inertia moment of AFSmotor rotor

θw/rad Rotation angle of AFS motor rotor

Tsen2/(N?m) Anti-torque of AFS motor rotor

θe/rad Rotation angle of output shaft

Ka/(N? m? A-1) Moment coefficient of assist motor

Bm/(N? m? s? rad-1) Damping coefficient of assist motor

θm/rad Rotation angle of assist motor

Tm/(N?m) Electromagnetic torque of assist motor

Ta/(N?m) Output torque of assist motor

Jm/(kg?m2) Inertia moment of assist motor

Km/(N? m? rad-1) Output stiffness coefficient of assist motor

G Power transmission ratio of assist motor

U/V Terminal voltage of assist motor

L/H Inductor of assist motor

I/A Current of assist motor

R/Ω Armature resistance of assist motor

Kb/(N? m? rad-1) Back-EMF coefficient of assist motor

Je/(kg?m2) Inertia moment of output shaft

Be/(N? m? s? rad-1) Damping coefficient of output shaft

Tr/(N?m) Anti-torque of output shaf t

br/(N? m? rad-1) Damping coefficient of rack

xr/m Displacement of rack

m r/kg Effective mass of rack and pinion

rp/m Radius of pinion

FTR/N Axial thrust exerted on rack

kr/(N?m-1) Equivalent elasticity coefficient

FW/N Random signal of pavement

Mr/kg Effective mass of reduction gear,pinion and rack

Br/(N? m-1? s) Effective damping coefficient of reduction gear,pinion and rack

Kr/(N?m-1) Equivalent elasticity coefficient of tire,pinion and rack

Tn/(N?m) Torque of torque sensor

K/(A? N-1? m) Power gain coefficient

n1Transmission ratio of AFSmotor

n2Transmission ratio between steering screw and steering wheels

X Design variable vector

lb Lower bound of ideal energy rangeof steering sensibility

ub Upper bound of ideal energy range of steering sensibility

INTRODUCTION

The electric power steering(EPS)system,firstly used for mini car,has already experienced development for twenty years since 1988[1-2].It solves the problems associated with the hydraulic power steering(HPS).The motor only operates when steering assistance is needed,and hydraulic pump and piping are eliminated.In EPS,the static torque boost curves can be adjusted by modifying software in the electronic controllers without changing the torsion bars,and alternated according to vehicle velocity to improve steering feel.

Active front steering(AFS)system was introduced to improve handling stability under adverse road conditions[3-4].Compared with the conventional steering system,the mechanical linkage between the steering wheel and the front wheels of an AFS system is complemented by an ex tra angle augment motor.Therefore,a small auxiliary front wheel angle,in addition to the steering angleimposed by the driver,can be used to stabilize the vehicle besides improving vehicle steering responses and avoiding critical handling situations[5-6].

At present,the EPS system cannot achieve variable transmission ratio control and active steering control,also cannot improve the steering control stability of steering system[7-8].The existing AFS system uses the commercial HPS system,and it is complex and inevitably has the hydraulic oil leakage[9]. Additionally, compared with the EPS system to achieve power,the AFS system is difficult to achieve the returnability and damping control[10].

Therefore,on the basis of the AFS technology integrated with EPS technology,the steering system which can provide the function of both EPS and AFS systems will be the main development direction of the future automotive power steering technology.In this paper,a novel EPS system integrated with AFS function is developed.Firstly,the performance indexes of road feel,sensitivity,and operation stability of the steering are proposed.Then,the influence of parameters on the novel EPS system is analyzed.Lastly,the genetic algorithm(GA)and the coordinate rotation algorithm(CRA)are used to optimize the structure parameters of the novel EPS system,thus providing a theoretical basis for design and selection of the novel EPS system.

1 SYSTEM MODELING

The model of the novel EPSsystem is shown in Fig.1.It includes two motors:One motor is assist motor used to provide assistance torque for the EPScontrol,and the other is AFSmotor used to provide additional steering angle for the AFS control.

1.1 Three-f reedom vehicle model

The dynamic differential equation for the three-freedom vehicle can be expressed by where T1,T 2 can be described as follows

Fig.1 Model of novel EPS system

1.2 Input shaft model

Considering theviscous damping of theinput shaft and steering wheel moment of inertial,the dynamic equation for the input shaft can be expressed as

By analyzing the steering column and sensor torque,T sen1 can be expressed as

1.3 AFSmotor stator model

Considering the viscous damping of the AFS motor,the dynamic equation for the AFSmotor stator can be expressed as

1.4 AFSmotor rotor model

Considering the viscous damping of the AFS motor,the dynamic equation for the AFSmotor rotor is derived as

For the output shaft,the anti-torque T sen2 can be written as

1.5 Assist motor model

The equation for the electromagnetic torque can be given by

Based on thedynamic analysis of the mechanical parts,thedynamic equation for the assist motor can be given by

In theactual control,theequation for the assist torque can be written as

1.6 Output shaf t model

Based on the dynamic analysis of the output shaft,the dynamic equation can be expressed as

1.7 Rack and pinion model

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

where F TR is

The angle of rotation of the output shaft can be given by

According to Eqs.(1,5-7,9,11-12),the system dynamic equation can be expressed as follows

where

2 STEERING PERFORMANCE

The novel EPSsystem should have following performance:good road feel,good steering sensitivity,and good steering stability.In this paper,the concepts and quantitative expressions of the novel EPSsystem are introduced.

2.1 Road feel

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

Assume that

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

Based on the current control strategy,there is

Then Tm can be described according to Eqs.(11,17-19),shown as

When the steering wheel and the steering screw of the steering actuator are equivalent,and the rotation angles of the AFS motor rotor and the output shaft are assumed to be equal,the steering system model can be simplified as follows

With Eqs.(3-6,9-11,15-16,20-21),the transfer function from T r to T h is obtained.This transfer function is just road feel and can be expressed as

where X1,Y1,Z1can be described as follows

2.2 Steering sensitivity

The steering sensitivity is defined as the ratio between the vehicle yaw-rate and the angle of rotation of the steering wheel,its transfer function can be given by

With Eqs.(4-6,8-11,18-19,21),we have

where X 2,Y2,Z2 can be w ritten as

From Eqs.(1-2),they can be obtained as follows

2.3 Steering stability

Steering stability is stability of the novel EPS system and the vehicle system,and it should be guaranteed firstly.Therefore,it is necessary to study which circumstance can guarantee the stability of vehicle.

The denominator of the transfer function of the steering sensibility is chosen as the characteristic equation.

where Q6,Q5,Q4,Q3,Q2,Q1,Q0 can be described as follows

In accordance with the Routh criterion,it requires that all formulas in the first column of the Routh table are positive.

3 PARAMETER OPTIMIZATION

3.1 Steering stability optimization

Some parameters of vehicle cannot be changed,such as the power gain coefficient of the assist motor K,which changes with the velocity of vehicle.Additionally the viscous friction coefficient of all parts cannot be adjusted,the power transmission ratio G always tends to be restricted in the optimal design,so the transmission ratio of the AFS motor n1,the stiffness coefficient of the torque sensor K s and the moment of inertia of the assist motor J m are designed as the optimal variables.

In this paper,the optimal design variable is designed as X=(n1,Ks)for two-parameter optimization of the novel EPS system.The bound of design variable is given as Xmin=(0.1,100),X max=(32,350)respectively,and the initial design variable is taken as X0=(1,200).For three parameter optimization,the optimal design variable is designed as X=(n1,K s,J m),the bound of design variable is given as X min=(0.1,100,0.002),X max=(32,350,0.01)respectively,and the initial design variableis taken as X0=(1,200,0.008).

3.2 Objective function optimization

In order to make the information from the pavement transfer to the driver′s hand as much as possible,it requires that the mean of the frequency domain energy of the road feel within a certain range of frequency domain can be as higher as possible.The mean of thefrequency domain energy of the road feel in effective frequency range(0,k 0)of the pavement information is defined as the objective function f(X),and k 0 is set to 40 Hzin the optimal design.The objective function f(X)is described as

3.3 Constraint conditions

The steering stability conditions of the novel EPS system must be satisfied,that is the transfer function of the steering sensibility must meet the Routh criterion,so the steering stability conditions could be written as

In the same way,in order to guarantee the driver get a good steering sensitivity,it requires that the mean of the frequency domain energy of the steering sensitivity within a certain range of frequency domain remains in a reasonable area.The mean energy of the steering sensitivity in the effective frequency range(0,k 0)of the pavement information is defined as the objective function g(X),and k 0 is also set to 40 Hz in the optimal design.The objective function g(X)is described as

3.4 Optimization method

The flow chart of GA is shown in Fig.2.

Fig.2 Flow chart of GA

The optimization model of the road feel energy with GA can be described as

3.5 Optimization results

Fig.3 Fitness population with GA(three-parameter)

With GA,the population fitness values with generation change are shown in Fig.3.The optimization result is X=(6.257 29,284.798 5,0.009 82), that is n1=6.388 77,Ks=278.334 27 N? m/rad,J m=0.01 kg? m2.

With CRA,the optimization result is X=(3.644 4,350.000 0,0.003 8),that is n1=3.644 4,K s=350.000 0 N? m/rad,J m=0.003 8× 10-3kg? m2.

The road feel energy of three-parameter with different algorithms is shown in Fig.4.From Figs.3-4,with GA,the road feel energy after optimization is 0.011 287,increased 3.514 8 times than that without optimization(2.5×10-3).With CRA,the road feel energy after optimization is 8.4×10-3,increased 2.260 0 times than that without optimization(2.5×10-3).The road feel energy optimized with GA is increased by 34.37%than that optimized with CRA.It shows that the novel EPSsystem with GA of three-parameter optimization can make the road feel attain optimal result.

Fig.4 Road feel energy with different algorithms(three-parameter)

4 CONCLUSION

In this paper,the novel EPSintegrated with AFS function is proposed. The mathematical model for the novel EPS system and the three freedom steeing model are built.Then,the concepts and the quantitative expressions of road feel,sensitivity,and operation stability of the steering are introduced.These parameters are optimized by GA and CRA,and results show that the road feel of the novel EPS system is effective-ly improved.

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