Torque Allocation Strategy of 4WID In-wheel Motor Electric VehicleBased on objective Optimization*

Gang Qin1, Jianxiao Zou1, Hongbing Xu1, Xiaoshuai Xin1 and kai Li1

Abstract— To achieve a better riding stability and dynamiccharacteristic, control strategies for torque allocation are pro-posed, and then the discussions for the proposed control strat-egy has conducted. In order to solve the torque optimizationquestion, an objective optimization is proposed. Suitable charac-terizations of road condition and mechanical structure are usedin the strategy to dynamically distribute the torque demandamong four in-wheel motors. Simulation results indicate thatcontrol strategy for torque allocation with motors in normalstate and failure state can achieve the designed demands well,and the dynamic performance of the distributed driving electricvehicles has been enhanced.

I. INTRODUCTION

With the quantity of conventional vehicle growing rapidly,oil consumption also shows astonishing growth every year allover the world. However, a large number of vehicles causethe ecological environment to bear a disproportionate burden[1-5]. Therefore, as the representative of clear energy, electricvehicles(EVs) attract great attention of vehicle area. Due tothe advantages of electric vehicles, such as zero emission,high efficiency, little noise and vibration, EVs have becomea major topic in new energy vehicle research areas [6-11].

Compared with traditional electric vehicles, four wheelsindependent driving (4WID) in-wheel motor EV is able toimplement individual control of each wheel. Therefore, suchhas higher control flexibility than that of traditional electricvehicle, however there is no doubt that proper torque alloca-tion is important to maintain vehicle stability[12],[13]. 4WIDin-wheel motor EVs torque allocation strategy mainly adoptsobjective optimization and simple logic strategy, which isongoing and continues to improve the torque allocationstrategy for electric vehicles. A method of torque allocationis proposed based on economic objective function [14], butstable objective function is not consideration in paper[14].Stable objective function is employed in optimization controlmethods [15], [16], which aims to make tire-load coefficientminimal, and the concept of weight coefficient is brought out[15]. Moreover, the influence of tire slip rate on torque allo-cation is considered as well [16]. Considering the operabilityof vehicle on the road which has low adhesion coefficients,a strategy of torque allocation based on wheel differential ispresented [17]. Some methods designed for optimal torqueallocation of hybrid electric vehicle are also put forward. An

*This work is supported by National Natural Science Foundation ofChina.(No.61004048)

1Qin, Zou, Xu, Xin and Li are with the School of Automa-tion Engineering, University of Electronic Science and technology ofChina, Chengdu, 611731 China. (phone:0086-028-61831810;e-mail:qingang1234567@gmail.com).

optimal torque distribution strategy of distributing internal-combustion engine and electromotor by using fuzzy logicis proposed in literature [18]. What’s more, a method ofbraking torque distribution of 4WID hybrid electric vehiclesused in literature [19] can provide a distribution strategyof braking force, in which distribution of driving forceis neglected. Although different torque allocation strategiesare put forward rapidly, various vehicle statuses and motorfailure state are rarely considered.

The proposed torque allocation strategy is based on ob-jective optimization and is further extended this paper todiscuss different vehicle statuses and motor failure state.In order to achieve the strategy, the necessary parametersand constraints must be carried out. The required tire forcesin objective optimization are determined by motor failurecontrol strategy. The proposed constraints consist of expecteddriving moment, expected yaw moment and the proportion offront-wheel lateral force. The longitudinal forces and lateralforce coefficient are decided by the vehicle statuses. Withcombining the above parameters and constraints of 4WIDin-wheel motor electric vehicles, the mentioned strategy isput forward.

II. 4WID EV CONTROL SYSTEM

We adopt a 4WID electric vehicle structure which consistsof the driving intention recognition, yaw rate controller, 4-wheel torque allocation, and slip rate controller.The trans-ducer system is designed to gather accelerate pedal location,braking pedal location, longitudinal acceleration, transverseacceleration, yaw rate and velocity. Driving intention recog-nition influences expected Vdes vehicle velocity and steeringangle δ. The yaw rate controller is the upper layer controllerof the vehicle system, which is responsible for collecting ac-tual expected yaw moment Mz and expected vehicle drivingmoment Fdes. Moreover, on the premise of proposed Mz

and Fdes, torque allocation set as the bottom layer controlleris in charge of calculating driving moment Ti and steeringangle of four wheels δf . Specifically, for a 4WID electricvehicle, vehicle slip and balance loss must be avoided, thatis to say, we should restrict the slip rate.

Finally, steering angle, braking moment, and driving mo-ment of every in-wheel motor are sent to steering motor andin-wheel motor, respectively.

III. TORQUE ALLOCATION DESIGN BASED ONOBJECTIVE OPTIMIZATION

Motor failure control algorithm is employed to determinetire force which is required in objective optimization. As

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is depicted in Figure 2, longitudinal velocity in real timeVx, rotating speed of each wheel in real time ωi, drivingmoment of every in-wheel motor in real time Ti are totaken as input. circle of friction estimation is employed tocalculate tire adhesive ability in real time, namely circle offriction ui · Fzi. Weight coefficient calculation module is incharge of calculating longitudinal force weight factor andlateral force weight factor ci of certain objective function;objective optimization algorithm is responsible to computeoptimal tire longitudinal force and lateral force which areconverted into modified driving moment and steering anglevalues through control value computation module. Finally,the modified driving moment and steering angle values arecontrol input.

Fig. 1. 4WID EV torque allocation strategy module

A. Motor Failure Determination and Control Strategy De-sign

Generally, some damages would happen when one ormore in-wheel motors are in failure state, that is to say,there may be no torque output when the vehicle is running.Abnormal yaw rate may due to this unexpected loss oftorque. Therefore, it is necessary to consider this kind ofsituations in the torque allocation strategy to cope withemergencies.

We adopt an invalidation parameter εi to add restrictionthen objective function of objective optimization moduleswhich is expressed as:

εi =

{0 failure module1 working normally

(1)

Referring to Table I, the failure states of motors are dividedinto three modes, which are single-motor invalidation, two-motor invalidation and multi-motor invalidation. Runningstatuses of vehicle are segmented into failure condition anddeceleration condition.

Failure condition is that in-wheel motors break downwithout any effects on vehicle safety and stability, that is tosay, vehicle still keeps running status, simultaneously, alertsdriver to notice motor invalidation. Deceleration condition isa the stoppage due to safety and stability of vehicle, whichis not the research contents of this paper.

TABLE IMOTOR FAILURE CONTROL STRATEGY DESIGN

Motor failuretype

Detailedfailureconditions(invalida-tion)

Invalidation parameterεi, i=1,2,3,4 is leftfront wheel, right frontwheel, left rear wheeland right rear wheel,respectively

Vehicle stat-ues

Left-frontmotor

ε1 = 0, ε2 = ε3 =ε4 = 1

Failure

Single-motorinvalidation

Right-frontmotor

ε2 = 0, ε1 = ε3 =ε4 = 1

Failure

Left-rearmotor

ε3 = 0, ε1 = ε2 =ε4 = 1

Failure

Right-rearmotor

ε4 = 0, ε1 = ε2 =ε3 = 1

Failure

Two-frontmotor

ε1 = ε2 = 0, ε3 =ε4 = 1

Failure

Same-sidemotorinvalidation

Two-rearmotor

ε1 = ε2 = 1, ε3 =ε4 = 0

Failure

Two-leftmotor

ε1 = ε2 = ε3 =ε4 = 0

Deceleration

Two-rightmotor

ε1 = ε2 = ε3 =ε4 = 0

Deceleration

Different-side motorinvalidation

left frontmotor andright rearmotor

ε1 = ε3 = 0, ε2 =ε4 = 1

Failure

right frontmotor andleft rearmotor

ε1 = ε3 = 1, ε2 =ε4 = 0

Failure

B. Objective Optimization Function Determination

According to the force analysis, the vehicle of interest inthe proposed strategy addressed here is the front-wheel steer-ing vehicle. Since, we consider six parameters as allocationobjectives, namely front-wheel lateral forces and all-wheellongitudinal force. Furthermore, we assume the minimalproportion of all-tire forces on each circle of friction andthe maximal driving moment as optimization objective. Thatis to say, in order to establish the function of optimizationobjective, we should first ensure the utilization rate of tire-road friction coefficient reach to minimum. The function canbe taken as below:

minJ =2∑

i=1

εicxiF2xi+cyiF

2yi

(uiFzi)2

+4∑

i=3

εicxiF2xi+cyiF

2yi

min((uiFzi)2,(Tmaxiri

)2)

(2)

where εi, Fzi, ri, Tmaxi are, respectively, invalidation of

motors, tire load, tire radius, and maximum driving forceof in-wheel motors. Moreover, cxi and cyi are longitudinaltire forces weighting coefficient and lateral force weightingcoefficient of wheel i . i = 1, 2, 3, 4 represents, respectively,left front wheel, right front wheel, left rear wheel, and rightrear wheel.

C. Weighting Coefficient Estimate (Determine cxi and cyi)

As is shown in equation (2), the objective function is usedto ensure proper tire-force allocation according to proposed

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disparate values. When vehicle status get close to adhesionlimitation value, optimal utilization rate of circle of frictionmust be ensured. That is to say, reasonable allocation oflongitudinal force and lateral force is necessary. Therefore,weight coefficient in equation (2) need to be adjusted to arational status.

Vehicle statuses are separated to four kinds of control typesaccording to vehicle yaw velocity direction and expectedmetabolic yaw moment direction, as is shown in Table II.

TABLE IIFAILURE CONTROL CONDITION SORTING

Condi-tion1

Cond-ition2

Cond-ition3

Cond-ition4

Requirevehicle yawmomentdirection

Clock-wise ◦ ◦Anti-clock-wise

◦ ◦

Vehicle yawdirection

Clock-wise ◦ ◦Anti-clock-wise

◦ ◦

Although steering directions of the vehicle in condition 1and condition 4 are different, running statuses are identical.Condition 2 and 3 are similar to this. In condition 2 and 4,handling method of weight coefficient is opposite, which isalso suitable for condition 1 and 3. In this section, we takecondition 4 as an example, that expected steering direction isanticlockwise, which is the same to expected yaw moment.Understeer is engendered in this situation, such that extrayaw moment is needed to satisfy vehicle steering demand.

When the vehicle is steering anticlockwise, vertical loadtransfers from left wheel to right wheel due to the influenceof inertia. Left-wheel circle of friction reduce, simultane-ously, right-wheel circle of friction increase. In this condi-tion, calculation strategy of weight coefficient is listed asTable III according to different accelerations.

TABLE IIICALCULATION STRATEGY OF WEIGHT COEFFICIENT IN CONDITION 4

Vehicle sta-tuses Weight coefficient treatment stategy

Urgent ac-celeration

Add longitudinal force weight coefficient prop-erly,reduce lateral force weight coefficient The largerweight coefficient , the smaller force influence

Normalaccelerationor constantvelocity

Reduce tire longitudinal force weight coefficient andincrease tire lateral force weight coefficient accordingto acceleration and degree of steering

Normal de-celeration

Adjust longitudinal force weight coefficient and lateralforce weight coefficient to reasonable value accordingto deceleration and degree of steering.

Urgent de-celeration

Increase longitudinal force weight coefficient properlyand decrease lateral force weight coefficient.

In this paper, we employ longitudinal barycenter transportrate to represent degree of acceleration and deceleration and

lateral barycenter transport rate to replace degree of steering.The tire longitudinal barycenter transfer rate and lateralbarycenter transfer rate, which are expressed as bellow:

x =

lf2∑

i=1

uiFzi + lr4∑

i=3

uiFzi

4∑i=1

uiFzi

(3)

y =

B2

4∑i=1

(−1)iuiFzi

4∑i=1

uiFzi

(4)

where x and y are ranged from -0.8 to 0.8. Note that whenor is calculated beyond the proposed range, we always setthem equal to the boundary value.

−0.8 ≤ x ≤ 0.8;−0.8 ≤ y ≤ 0.8 (5)

Since, too much regulation will affect vehicle stabilityand operability if the sensitivity of steering rate is high.Therefore, we set weight coefficient cxi and cyi satisfy thefollowing equations.{

cxi + cyi = 10.1 ≤ cxi ≤ 0.5

(6)

Here we set a parameter K to represent the productof steering angle and expected yaw moment. Specifically,K >= 0 denotes condition 1 and 4; K < 0 is condition 2 and3. Then cx can be expressed as following. When K >= 0,

cxi =

cmaxx when x ≥ x

hand−acc14

cminx + 0.2( x

xhand−acc

14

+ xmax−|y|xmax )

when 0 ≤ x < xhand−acc14

cminx + 0.2( |x|

xhand−dec

14

+ xmax−|y|xmax )

when− xhand−dec14 < x < 0

cmaxx when x ≤ −x

hand−dec14

(7)Otherwise,

cxi =

cmaxx when x ≥ x

hand−acc23

cminx + 0.2( x

xhand−acc

23

+ xmax−|y|xmax )

when 0 ≤ x < xhand−acc23

cminx + 0.2( |x|

xhand−dec

23

+ xmax−|y|xmax )

when− xhand−dec23 < x < 0

cmaxx whenx ≤ −x

hand−dec23

(8)where x

hand−acc14 and x

hand−dec14 denote urgent accelera-

tion and deceleration threshold in condition 1 and 4. Herewe set them equal to 0.7 and 0.5, respectively. Similarly,xhand−acc23 and x

hand−dec23 which are qual to 0.5 and 0.7 are

urgent acceleration and deceleration threshold in condition 2and 3.

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D. Circle of friction Estimate (Determine ui · Fzi)

Circle of friction can be calculated according to equation(9). Constant K is determined by a series of membershipparameters: static road friction coefficient ui−static, staticroad tire load Fzi−static, slip rate within the scope of tirelinearity range λi−linear and longitudinal force within thescope of tire linearity range Fxi−linear. If the above tireinformation is known, K can be calculated [20].

uFz = K · cxiK = ui−static · Fzi−static ·

λi−linear

Fxi−linear

(9)

E. Objective Optimization Constraints Determination

In order to satisfy the expected velocity, longitudinal forcemust satisfy the output of velocity control module, namelylongitudinal driving force by each tire as is shown in equation(10). In equation (11), the longitudinal force and lateral forcesupplied by each tire must satisfy yaw moment. Tire netforce is the sum of all-tire longitudinal force, which ensuresvehicle driving capability and is influenced by acceleratorpedal input.

Fx−des =4∑

i=1

εi · Fxi (10)

Mz = lf (Fy1+Fy2)+B

2(

4∑i=1

(−1)i ·εi ·Fxi) (11)

where Fx−des denotes expected driving force; lf is thedistance of front wheel and centroid; B is the distancebetween the two front wheels. Note that B = Bf = Br here;Moreover, the yaw moment is represented by Mz which isneeded to ensure stability of vehicle.

Since front-wheel steering electric vehicles are focusedin this paper, the steering angles of the two front tires areconsistent. Thus, the relationship between the extra tire forceof left front wheel and right front wheel is given by equation(12).

Fy1

u1Fz1=

Fy2

u2Fz2(12)

Equation (10), (11), (12) express three constraints ofsix independent variables in objective function J . Thesethree constraints restraint vehicles longitudinal motion, yawmotion and lateral force, respectively.

F. Solution Algorithm of Objective Optimization

As for the objective function and constrains are various atdifferent time, the solution of equation must be in real time.

With analysis of equation (3), the proposed six unknownparameters are equivalent. We take normal condition ε1 =ε2 = ε3 = ε4 = 1 here for example. Other conditions aresimilar.

In equation (1), Fxi becomes zero when in-wheel motorsbreak down. We set left-front-wheel lateral force and twofront-wheel longitudinal forces as unknown parameters. Thefollowing equations are obtained by integrating equations(10), equation (11) and equation (12) together.

Fx3 = −Fx1 +lf (u1Fz1+u2Fz2)

Bu1Fz1Fy1

+Fx−des

2 − Mz

B

(13)

Fx4 = −Fx2 − lf (u1Fz1+u2Fz2)Bu1Fz1

Fy1

+Fx−des

2 + Mz

B

(14)

Fy2 =u2Fz2

u1Fz1Fy1 (15)

In order to solve actual extremal problem of objectivefunction, we substitute equation (13), equation (14), equation(15) into equation (3) as desired conditions. Partial deriva-tives of Fx1, Fx2 and Fy1 are calculated and equal to zero,which is depicted as

∂J∂Fx1

= 2( cx1

(u1Fz1)2+ cx3

min((u3Fz3)2,(Tmax3r3

)2))Fx1

−2lf (u1Fz1+u2Fz2

Bu1Fz1

cx3

min((u3Fz3)2,(Tmax3r3

)2)Fy1

−2(Fx−des

2 − Mz

B ) cx3

min((u3Fz3)2,(Tmax3r3

)2)

= 0

(16)

∂J∂Fy1

= −2lf (u1Fz1+u2Fz2)

Bru1Fz1· cx3

min((u3Fz3)2,(Tmax3 /r3)2)

· Fx1

+2lf (u1Fz1+u2Fz2)

Bru1Fz1· cx4

min((u4Fz4)2,(Tmax4 /r4)2)

· Fx2

+2

(cy1

(u1Fz1)2+(

u2Fz2

u1Fz1

)2

· cy2

(u2Fz2)2

+(

lf (u1Fz1+u2Fz2)Bru1Fz1

)2

·(

cx3

min((u3Fz3)2,(Tmax3 /r3)2)

+ cx4

min((u4Fz4)2,(Tmax4 /r4)2)

))· Fy1

+2lf (u1Fz1+u2Fz2)

Bru1Fz1·((

Fx des

2 − Mz

Br

)· cx3

min((u3Fz3)2,(Tmax3 /r3)2)

−(

Fx des

2 + Mz

Br

)· cx4

min((u4Fz4)2,(Tmax4 /r4)2)

)= 0;

(17)

∂J∂Fx2

= 2( cx2

(u2Fz2)2+ cx4

min((u4Fz4)2,(Tmax4r4

)2))Fx2

+2lf (u1Fz1+u2Fz2

Bu1Fz1

cx4

min((u4Fz4)2,(Tmax4r4

)2)Fy1

−2(Fx−des

2 + Mz

B ) cx4

min((u4Fz4)2,(Tmax4r4

)2)

= 0;

(18)

According to (16), (17), (18), a three-variable simple equa-tion is expressed as follows, where, Fx1, Fx2, Fy1are definedas three independent unknown parameters in proposed equa-tion.

Fx1 = c5(b2c3c+c2e+c4a

2e+c3b2e)

G

Fx2 = b2(−c1c3c−c5c3c+ec5c1)G + c

Fy1 = b(−c1c3c−c5c3c+ec5c1)G

G = a2c1c4 + a2c4c5 + b2c3c+ b2c3c5+b2c1c5 + c1c2 + c2c5

(19)

where these are represented by friction coefficient, verticaltire load, expected velocity, needed yaw moment, maximum

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in-wheel motor torque, tire radius, wheel tread and wheel-base.

Then according to equation (13), equation (14) and equa-tion (15), we can get Fx3, Fx4, Fy2.

Fx3 = e− c5(b2c3c+c2e+c4a

2e+c3b2e)

G

− b2(−c1c3c−c3c5c+c1c5e)G

Fx4 = 0

Fy2 = ab(−c1c3c−c5c3c+ec1c5)G

(20)

where c1 = cx1

(u1Fz1)2; c2 =

cy1

(u1Fz1)2; c3 = cx2

(u2Fz2)2;

c4 =cy2

(u2Fz2)2; c5 = cx3

min((u3Fz3)2,(Tmax3r3

)2); c6 =

cx4

min((u4Fz4)2;(Tmax4r4

)2); a = u2Fz2

u1Fz1; b = − lf (u1Fz1+u2Fz2)

Bu1Fz1;

e =Fx−des

2 − Mz

B , c =Fx−des

2 + Mz

B .Steering angle is confirmed by tire longitudinal force

module in equation (21). In-wheel motor torque is resolvedby equation (22).

δ =Fy

ca(21)

Ti = Fxiri (22)

where δ is modified steering angle; Fy is lateral force of leftfront tire; ca is stiffness of tire lateral force, which is intrinsicparameter of tire. Ti is driving force of in-wheel motor i.

IV. SIMULATION AND INTERPRETATION OFRESULTS

To investigate the effectiveness of the torque allocationstrategy, 4WID electric vehicle simulation model of DY-NAware is employed part of simulation parameters are listedin Table IV.

TABLE IVMAIN SIMULATION PARAMETERS OF VEHICLE

Parameters ValueVehicle quality m 1296 kg

Tire radius radius r 0.1651 mDistance of centroid and rear axle lr 1.32 mDistance of centroid and front axle lf 1.25 m

Distance between two front(rear) wheels B 1.405 mVertical height of vehicle centroid to ground h 0.415 mRotational inertia of vehicle spin on Z axle Iz 1750 kg ∗m2

Simulation results of hunting condition (turning the wheelurgently), el motor failure etc are conducted.

A. Hunting Condition

With the purpose of imitating the hunting condition, weset friction coefficient u = 0.7.(Note that u = 0.7 is normalfriction coefficient of dry asphalt road)In this case, velocityof 4WID electric vehicle increases from 0 to 70km/h. Andthe vehicle bypasses 7 barriers within 8 to 25 seconds.Simulation curve is expressed as below.

According to the results shown in the Fig.2, Fig.3, Fig.4,Fig.5, Fig.6 and Fig.7, overshoot of the vehicle whould bereduced efficiently based on torque allocation strategy when

Fig. 2. Steering Angle Input Fig. 3. Corrected Steering Angle

Fig. 4. Front Wheel Torque ofTorque Allocation Strategy

Fig. 5. Rear Wheel Torque ofTorque Allocation Strategy

vehicle takes a sudden turn to avoid obstacles. Fig.6 andFig.7 show the comparison results between torque allocationstrategy and equal allocation arithmetic. It is observed thatyaw rate of torque allocation arithmetic is closed to expectedyaw rate, and the vehicle bypasses obstacles successfully asits running track shows. But equal allocation arithmetic haspoor capacity of yaw rate tracking, and vehicle loses stabilityaround 20th second.

B. Single Motor Invalidation

Right rear motor invalidation is taken as example here torepresent single motor invalidation. In this condition, velocityof 4WID electric vehicle increases from 0 to 80km/h. Andwhen right rear motor breaks down at 5th second, outputtorque becomes zero. In this condition, friction coefficientu = 0.8. Simulation curve is shown as follows. According toFig.8 and Fig.9, right rear motor breaks down at 5th second.At the same time, running track of equal allocation controlarithmetic deviates from expected running track. But if we

Fig. 6. Yaw Rate of Vehicle Fig. 7. Running Track of Vehicle

Fig. 8. Yaw Rate of Vehicle Fig. 9. Running Track of Vehicl

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employ torque allocation arithmetic, it has better controleffect when motors break down. And the basic running trackis closed to normal running track.

C. Double Motors Invalidation

Another condition is adopted as example to representdiagonal motors invalidation which is left front wheel andright rear wheel both fail. In this condition, velocity of4WID electric vehicle increases from 0 to 80km/h. And whenleft front wheel and right rear motor fail at 5th second,output torque can only be zero. In this condition, frictioncoefficient u = 0.8, and this friction coefficient is normalfriction coefficient of dry asphalt road. Simulation curve isshown as follows. According to Fig.10 and Fig.11, three

Fig. 10. Yaw Rate of Vehicle Fig. 11. Running Track of Vehicl

running tracks are well overlap in straight driving. However,equal allocation curve deviates far from normal track. Withcomparing invalidation track curves of torque allocation andequal allocation, torque allocation has better control effectand ensures running track comes closer to normal track.

V. CONCLUSION

With consideration of different running statuses and motorfailure conditions, the torque allocation strategy of the 4WIDin-wheel motor electric vehicle based on objective optimiza-tion is proposed which aims for greater stability. Accordingto simulation results, we get the following conclusions.

• Torque allocation strategy of 4WID EV can implementstability control.

• Torque allocation strategy of 4WID EV with equalallocation arithmetic, which has better capacity of sta-bility control in urgent turning, slow turning, urgentacceleration conditions and other conditions.

• In motor failure conditions, torque allocation strategycan implement stability control.

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