Multi－objective Optimization of Differential Steering System of Electric Vehicle with Motorized Wheels＊

Zhao Wanzhong（赵万忠），Wang Chunyan（王春燕），Duan Tingting（段婷婷） Ye Jiaji（叶嘉冀），Zhou Xie（周协）

1.Department of Vehicle 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

1 Introduction

Up to now，there are hardly any researches on the optimization of the DSS system parameters both at home and abroad，and the optimization of traditional power steering system still focuses on the genetic algorithm（GA）［3－4］.Reviewing the related literatures，the system parameters are generally optimized with steering road feel as optimization target and steering stability as constraint.Such kinds of optimizations do have some shortcomings.On the one hand，it is easy to fall into local optimal solution with GA［5－6］；on the other hand，the system is optimized without taking the steering portability into consideration［7－8］.

The multi－island genetic algorithm（MIGA）is a new optimization algorithm based on GA.It can not only keep the diversity of optimal solutions，but also increase the chances to get global optimal solutions，thus avoid getting the local optimal solutions as much as possible and restrain the precocious phenomena.In this paper，a dynamic model of the DSS system for electric vehicle with motorized wheels is built.The DSS system parameters are optimized based on MIGA，with steering road feel and steering portability as optimizing target，and steering stability as constraint.It can offer a theoretical basis for the model selection and the design for the DSS system of electric vehicle with motorized wheels.

2 Vehicle Dynamic Model

The DSS structure of the electric vehicle with motorized wheels is shown in Fig.1.Based on the traditional mechanical front－wheel－drive steering system，it has a torque sensor and a steering angle sensor in the steering shaft to measure the steering torque and the angle given by the driver，which is motivated by two in－wheel motors in the front shafts.With force and displacement coupled control of the left and right in－wheel motors，DSS of the electric vehicle with motorized wheels can coordinate the active steering safety and the steering road feel perfectly.

Fig.1 DSS of electric vehicle with motorized wheels

2.1 Three－degree－of－freedom model of vehicle

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

where gis the acceleration due to gravity；u，ωr，φ，andβare velocity，yaw rate，roll angle，sideslip angle of the vehicle，respectively；lis the distance between the two front wheels；ΔTmis the driving torque difference of the two front wheels；δis the front－wheel steer angle；α1，α2，a，b，and hare 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 center of mass，and the rolling moment arm of the vehicle，respectively；m，ms，IX，and IZare mass，sprung mass，moment of inertia of spring mass about Xaxis，moment of inertia of mass about Z axis of the vehicle，respectively；IXZ，k1，k2are product of inertia of sprung mass about Xand Zaxes，front－wheel cornering stiffness，rear－wheel cornering stiffness，respectively；Cφ1，Cφ2，D1，and D2are roll angle stiffness of front suspension，roll angle stiffness of rear suspension，roll angle damping of front suspension，roll angle damping of rear suspension of the vehicle，respectively.

2.2 Steering system model

By the mathematical analyses of the steering system，the dynamic and electromagnetic equations can be obtained as follows

where Jhis the moment of inertia of the steering input shaft and steering wheel；This the steering torque acting on the steering wheel；Bhis the damping coefficient；θhis the angle of rotation；Ksis the stiffness of the input shaft；θeis the angle of the output shaft；Ka，Je，and Beare the torque coefficient，the moment of inertia of steering output shaft，and the damping coefficient of steering output shaft，respectively；n1is the transmission ratio of the steering screw to the front wheel；Tris the anti－torque exerted on the steering output shaft；dis the offset distance of master pin of the left and right front wheels；rwis the rolling radius of the wheel；Tmis the electromagnetic torque of motor，T1the electromagnetic torque of the left motorized wheel，and T2the electromagnetic torque of the right motorized wheel；iAis the motor current.

3 Optimization Model

3.1 Design variables

The effects of some parameters on the steering operation performance are unchangeable in a real situation or just determined by experience，so some powerful and practically changeable parameters are chosen to be taken into consideration when designing variables.Those variables are Km，Ks，Je，and Bewhich stand for the torque gain coefficients of in－wheel motors，the stiffness of the input shaft，the moment of inertia of steering output shaft，and the damping coefficient of steering output shaft，respectively.

针对复杂网络的社团结构检测问题，将Newman的基于模块度函数Q的矩阵特征值和特征向量提取复杂网络中社团结构的算法和神经网络算法求解矩阵特征值特征向量的算法结合起来得到一种新的基于连续神经网络的社团结构检测算法，简称CNN算法，模型为：

3.2 Objective function

（1）Steering road feel

In order to transfer the information from the road to the driver completely，the average fre－quency power of the steering road feel should be as big as possible in a certain frequency range［9－10］.The objective function f（Km，Ks，Je，Be）is the average frequency power of the steering road feel in the（0，ω0）frequency range，andω0＝40Hz.The bigger the objective function f（Km，Ks，Je，Be）is，the better transfer characteristics of the steering road feel it has.Namely

（2）Steering portability

In order to get good steering portability for the vehicle，the average frequency power of the steering portability should be in an appropriate range［10］.The objective function f2is the average frequency power of steering portability in the（0，ω0）frequency range andω0＝40Hz，shown as

In order to guarantee the steering portability，the average frequency power of the steering portability should be in an appropriate range.The steering portability f2is set in a range［a0，b0］，namely

3.3 Constraints

The requirement of the steering stability of the electric vehicle with motorized wheels is that the numbers of the first column of the steering stability Routh table should be positive［10］，namely

4 Optimization Algorithm

4.1 Multi－island genetic algorithm

GA belongs to unclassical optimization algorithm.As an adaptive probability，it is developed by simulating the genetics and evolution proce－dure of biology in the natural environment，which includes robust searching algorithm to deal with the complicated，large－scale and multivariable non－linear inversion problems.

Based on the traditional genetic algorithm（TGA），the new characteristic of MIGA is that every individual in each population is divided into several subgroups which are called the″islands″.The whole operations of TGA such as selecting，crossing and mutation happen in every island respectively，and each island chooses some individuals to migrate to other islands at regular interval.Then，the operation repeats.

The migration process is controlled by two parameters including migration interval and migration rate.The former one stands for the generation between two adjacent migrations.The latter one means the percentage of migrated individuals in each migration.The migration operation in MIGA keeps the diversity of optimal solution and increases the chances of getting global opti－mal solutions.The optimization procedure of MIGA is as follows：（1）Optimize the initial value；（2）when it achieves preliminary convergence，a new initial value begins to operate due to mutation and migration；（3）as the operation repeats，it can avoid the local optimal solution as much as possible and restrain the precocious phenomena.

4.2 Design optimization algorithm

The issue of selecting the migration rate is complex.As the migrants are generally the optimal individuals in each subgroup，they can not only benefit the spread of excellent individuals in the whole group，but also improve the convergence speed，if the migration rate is in a high level.Meanwhile，high rate can increase the communication overhead，reduce the speed－up ratio，lead to a decline in population diversity，and even go against the feature of the algorithm that can search in multiple directions at the same time.Hence，an appropriate migration rate should be selected by experience.A small migration interval is conducive to the fusion of subgroup and allows the excellent individuals to spread throughout the subgroups timely，thus providing a favorable guide to the revolution direction of the group as well as improving the accuracy of solutions and the convergent velocity of the group.However，it will increase the communication and synchronization overhead obviously and go against the enhancement of the speed－up ratio.Additionally，the dominant position of some excellent individuals may have a negative impact on the diversity of the group and even make the revolution of the group fall into the local minimum point.On the contrary，if the migration interval is over－sized，the effect will be reversed.The migration operation of MIGA not only keeps the diversity of optimal solutions，but also improves the chance of the global optimal solution.In conclusion，the chosen parameters are as follows.

The initial values of Km，Ks，Je，Beare 10，139.82，0.002 9，2.3，respectively and the initial ranges of them are（1，50），（50，250），（0.0001，0.0100）， （1.1，10），respectively.Meanwhile，in order to ensure xj1－xj2＜0，J＞0 is set.The number of subgroups and the individ－ual number of each subgroup are set as 3and 15.The probability of replication is 0.8.The probability of crossover，which generates progeny by using the weighted average of parent，is 0.8.The mutation probability is 0.01by using the uniform method.The probability of migration，the migration interval and the maximum algebra are set as 0.3，4，and 1 000，respectively.

4.3 Optimization results

The optimization results are shown in Table 1.According to Table 1，the energy of steering road feel after optimization is 0.095 011，and it is 5.150times bigger than the one before optimization.The steering sensitivity is 0.007 320，which meets the demand.Meanwhile，all the parameters in the first column of the Routh table are bigger than zero，which meet the demand of stability.In addition，compared with GA，although the energy of steering road feel with MIGA is only increased by 0.48%，the steering sensitivity is improved by 21.01%.Therefore，the optimization result of MIGA is more advantageous when optimizing the steering system.

Table 1 Two kinds of optimization results

The bode diagram of steering road feel is shown in Fig.2.As shown in Fig.2，the bandwidth and amplitude increase，and the phase delay decreases after optimization.Additionally，compared with GA，the bandwidth and amplitude increase further with MIGA.

The bode diagram of steering sensibility is shown in Fig.3.According to Fig.3，the steering sensibility only changes a little after optimization.The differential assisted steering system optimized with MIGA can improve the steering road feel，while guaranteeing the steering sensibility and stability.Additionally，MIGA is more advantageous than GA in terms of optimization.

Fig.2 Bode diagram of steering road feel

Fig.3 Bode diagram of steering sensibility

5 Conclusions

In this paper，the DSS system with force and displacement coupled control for electric vehicle with motorized wheels and the model of the threefreedom car are built.Then，according to the features of multi－constrained optimization of multiobjective function，MIGA is designed and the parameters of system are devised with multi－objective optimization.The optimization results show that the system optimized with MIGA can improve the steering road feel，reduce the steering energy consumption efficiently while guaranteeing the operation stability and steering sensibility.Additionally，MIGA is more advantageous than GA in terms of optimization，thus getting a more satisfactory result.

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