Proposal of a Range Extension Control Systemwith Arbitrary Steering for In-Wheel Motor Electric

Vehicle with Four Wheel SteeringToshihiro Yone and Hiroshi Fujimoto

The University of Tokyo5-1-5, Kashiwanoha, Kashiwa, Chiba, 277-8561 Japan

Phone: +81-4-7136-3881Fax: +81-4-7136-3881

Email: yone@hflab.k.u-tokyo.ac.jp, fujimoto@k.u-tokyo.ac.jp

Abstract— In this paper, a range extension control system withvehicle dynamics control is proposed. Front and rear activesteering and torque difference between left and right in-wheelmotors enable yaw rate and side-slip angle control with higherenergy efficiency. The effectiveness of the proposed method wasverified by both simulations and experiments.

I. INTRODUCTION

Electric vehicles (EVs) are expected to be a drastic solutionto the current environmental and energy problems by deprivinginternal combustion engine vehicles (ICEVs) of its position.Furthermore, EVs have advantages in motion control com-pared with ICEVs [1].

Due to limited cruising range per charge, EVs have beenprevented from wide spreading. In order to solve this prob-lem, efficiency of motors [2] and regenerative torque control[3] were studied. From the view point of motor efficiencyimprovement control, research of torque and angular velocitypattern that maximise efficiency during acceleration and de-celeration was carried out [4]. To decrease EV’s energy con-sumption, torque distribution to traction motors with differentcharasteristics was studied [5].

On the other hand, the authors’ research group proposedrange extension control systems (RECSs) [6], [7], [8]. Thesesystems do not involve changes of vehicle structure such asadditional clutch [5] and motor type. RECS extends cruisingrange by motion control of vehicle. In article [8] consumedenergy during cornering is decreased by making differencebetween torques of left and right motors. However, the systemis only effective when the vehicle is traveling at a constantvelocity and a constant radius of curvature. That is, the systemcould not be applied in case yaw rate is not constant.

In this paper, to apply RECS to the normal steering situa-tions, RECS that can be applied even when steering commandchanges arbitrarily is proposed. This proposed RECS not onlyminimises consumed energy, but also controls yaw rate andside slip angle to enable vehicle to follow a target trajectoryby utilising four wheel steering and a moment that is createdby torque differece between left and rear in-wheel motors.

(a) FPEV2-Kanon (b) In-wheel motor

Fig. 1. Experimental vehicle

II. EXPERIMENTAL VEHICLE

In this research, an original electric vehicle “FPEV–2Kanon”, manufactured by the authors’ research group, isused. This vehicle has four outer-rotor type in-wheel motors.Therefore, driving and braking force distribution to each motorcan be used to produce torque around the z-axis of thevehicle. Front and rear electric power steerings are equipped,which enable active control of both front and rear steeringmechanisms. Power train of the vehicle consists of battery,chopper, inverters and motors. Input voltage (Vdc) and currents(Idc) of inverters are measurable.

III. D YNAMICS EQUATIONS AND TRAVELING RESISTANCE

In this section, traveling resistance is calculated from dy-namics equations.

Fx = Fxfl + Fxfr + Fxrl + Fxrr (1)

Fy = Fyfl + Fyfr + Fyrl + Fyrr (2)

Fy = MV (β + γ) (3)

Mz = Iγ

= Fyfllf + Fyfrlf − Fyrllr − Fyrrlr +Nz (4)

Nz :=df2(−Fxfl + Fyfr) +

dr2(−Fxrl + Fyrr) (5)

Fx is sum of driving or braking force,Fy is total lat-eral force, Mz is yaw moment applied to the vehicle,Fxfl, Fxfr, Fxrl, Fxrr are driving or braking forces of eachtyre, Fyfl, Fyfr, Fyrl, Fyrr are lateral force of each tyre,Mis mass of the vehicle,V is vehicle velocity,β is side slip

angle of the vehicle,γ is yaw-rate,I is inertia around z-axis, lf , lr are the distances to front and rear axles from thecentre of gravity(CG),df , dr are front and rear treadwidth.Nz represents moment by differencial torque of right andleft motors and shown by Eq.(5). Tyre slip angle is an anglebetween surface of the tyre and direction of movement of thevehicle and written asαfl, αfr, αrl, αrr. When front and rearsteering (written asδf and δr) are small,approximations canbe made as below.

αfl = αfr = αf ≃ β +lfγ

V− δf (6)

αrl = αrr = αr ≃ β − lrγ

V− δr (7)

Lateral forces are proportional to tyre slip angle and areapproximated as below.

Fyfl = Fyfr = Fyf ≃ −Cfαf (8)

Fyfr = Fyrr = Fyr ≃ −Crαr (9)

Therefore, dynamics equations of lateral and yaw motions areestablished as below.

x = Ax+Bu (10)

A =

[a11 a12a21 a22

]=

[−2(Cf+Cr)

MV−2(lfCf−lrCr)

MV 2 − 1−2(lfCf−lrCr

I

−2(l2fCf+l2rCr)

IV

](11)

B =

[b11 b12 b13b21 b22 b23

]=

[2Cf

MV2Cr

MV 02lfCf

I2lrCr

I1I

](12)

x =

[βγ

],u =

δfδrNz

For the vertical direction of the vehicle, when the vehicletravels at a constant speed, driving forceFx equals to travelingfriction Fr. Traveling friction consists of cornering drag forceFcd, rolling friction and air resistance, which is shown asbelow. µ0 is the coefficient of the rolling friction, normalforces of front and rear tyres areNf and Nr, and otherdisturbances areFdis.

Fr = Fcd + µ0Nf cos δf + µ0Nr cos δr + Fdis (13)

Cornering drag forceFcdis the sum of cornering drag forcesFcr work on each tyres as Fig. 3. When, tyre slipping anglesα are small, cornering drag forces are calculated as below.

Fcr = Fy sinα (14)

Fcd = 2Fcrf cosβ + 2Fcrr cosβ (15)

Thus, from the fact that tyre side slip angles and steeringangles are small andNf +Nr = Mg, by substituting Eq. (6),Eq.(7) and Eq.(15) into Eq.(13), the following approximationsis derived.

Fr ≃ 2Cfα2f + 2Crα

2r + µ0Mg (16)

Fig. 2. Bicycle model of Vehicle Dynamics

Fig. 3. Cornering Drag on each tyres

IV. D ISTRIBUTION LAW FOR RECS

A. Consumed electrical power

In this paper, only front in-wheel motors are utilised. Con-sumed electrical powerP is treated as an evaluation functionand front and rear steering anglesδf , δr, as well as differentialtorqueNz are calculated to minimise the consumed electricalpower. Consumed electrical power consists of motor outputPm, copper lossPc and iron lossPi.

P = Pm + Pc + Pi (17)

At first, motor outputPm is examined. Motor output isexpressed by wheel angular velocityωfl, ωfr and wheel torqueTfl, Tfr.

Pm = ωflTfl + ωfrTfr (18)

On the assumption that slip does not occur between tyres androad, wheel angular velocity is calcurated as below.

ωfl =1

r

(V − dr

2γ

)(19)

ωfr =1

r

(V +

dr2γ

)(20)

r is radius of the tyres. When each wheel angular accelerationis small, torqueTi is proportional to driving force.

Ti = rFxi (21)

Driving force and differential torque are distributed to left andright motors evenly.[

Fxfl

Fxfr

]=

[12 − 1

df12

1df

][Fr

Nz

](22)

Motor output Pm is able to be expressed as follows bytraveling resistanceFr, longitudinal vehicle velocityVx, yawrate γ and differential torqueNz. By substituting Eq.(20) toEq.(22) into Eq.(18), motor outputPm is calculated as follows[8].

Pm = VxFr + γNz (23)

Secondary, copper loss is discussed as following. copperloss is proportional to square of current and under d-axiscurrent id = 0 control, torqueTi are proportional to q-axiscurrentsiqrl, iqrr.

Pc = Ra

(i2qfl + i2qfr

)(24)

iqi =Ti

Pnϕa=

Ti

Kt(25)

Here, Coeffients of right and left motors are equal.Ra isresistance value of armature windings,Pn is number ofmagnetic pole pairs of a stator,ϕa is interlinkage flux,Kt

is coefficient of torque. Thus, copper loss is expressed fromEq.(22) as

Pc =Rar

2

K2t

(1

2F 2r +

2

d2fN2

z

)(26)

For the last, iron lossPi is ignored. Because the velocity ofthe vehicle in this paper is low, the iron loss is small enoughto be ignored. By substituting Eqs.(16), (23) and (26) intoEq.(17), cosumed electrical powerP is calculated as follows.For αf and αr, the terms that have order larger than 2 areignored.

P = 2Cf

(Rar

2

K2t

µ0Mg + Vx

)α2f

+2Cr

(Rar

2

K2t

µ0Mg + Vx

)α2r

+Nzγ +2Rar

2

K2t d

2r

N2z + (µ0MG)

2 (27)

B. Distribution law to minimise the consumed electrical power

Dynamic equations of lateral and yaw-motions are ex-pressed as[

−2Cf −2Cr 0−2Cf lf 2Crlr 1

] αf

αr

Nz

=

[Fy

Mz

](28)

X is defined as the coefficient matrix in the left side of theequation,z is defined as a column vector of tyre slip anglesand differential torque[αf αr Nz]

T and column vector on the

right side of the equation is defined asb. WhenW is definedas weighting matrix, solution of the method of weighted leastsquares iszopt and expressed as

P = zTWz (29)

zopt = W−1XT(XW−1XT)−1b (30)

W :=

w11 0−Cf

MV

0 w22−Cr

MV−Cf

MV−Cr

MV2Rar

2

K2t d

2r

(31)

w11 := 2Cf

(Rar

2

K2t

µ0Mg + Vx

)w22 := 2Cr

(Rar

2

K2t

µ0Mg + Vx

)Treating lateral force and yaw-moment as commands, tyre sideslip angles and differential torque that minimise the consumedelectrical power are calculated from Eq.(30).

V. CONTROL SYSTEM DESIGN

When the velocity of the vehicle is constant, the dynamicsof the vehicle is expressed by side slip angle and yaw-rate.In this article, side slip angle and yaw rate are controlledby total yaw-momentMz and lateral forceFy. Side slipangle is difficult to be measured directly, therefore, differentkinds of algorithms were studied [10]. Here, side slip angleis observed by a side slip angle observer (SAO) [11] and yawrate is acquired by a gyro sensor that placed at the centreof gravity. Lateral force observer (LFO) and yaw-momentobserver (YMO) are designed to control side slip angle andyaw rate [12]. Observers are designed to nominal the plantincluding the destribution law, which decouple controls of sideslip angle and yaw rate, and strengthen the controls againstdisturbance and parameter errors. In addition, two degree offreedom control is constructed to control side slip angle andyaw-rate. The block diagram is shown in Fig. 4.

A. Side-slip angle observer

Not only yaw rateγ, δf andδr, but also lateral accelerationay is utilised to construct a robust and linear observer. FromEqs.(3) and (13), lateral accelerationay is expressed as

ay = V (a11β + a12γ + b11δf + b12δr + γ) (32)

Herea11 ∼ a22, b11 ∼ b23 are the components of the matrixesA, B. x, u are utilised to express the output equation asbelow.

y = Cx+Du (33)

C =

[0 1

V (a11) V (a12 + 1)

](34)

D =

[0 0 0

V b11 V b12 0

], y =

[γay

],

By utilising these output equations, an observer is designed asbelow to observe side slip angleβ for control.

˙x = Ax+Bu−K(y − y) (35)

y = Cx+Du (36)

Fig. 4. Block Diagram of proposal method

B. Yaw-moment Observer

Considering the control inputs and disturbances to thevehicle, dynamics equation of yaw motion is described asbelow.

Idγ

dt= Mz +Ntd (37)

Where Ntd is disturbance yaw-moment. By constructing adisturbance observer, disturbance yaw-moment is suppressedand Eq.(37) is nominalised as following.

γ =1

InsMz = Pγ(s)Mz (38)

C. Lateral Force Observer

Lateral dynamic equations with disturbanceYd included isgiven as

MV

(dβ

dt+ γ

)= Fy + Ytd (39)

Eq.(39) is transformed into

MV β = Fy + Ytd (40)

Ytd := −MV γ + Yd (41)

Eq. (41) is nominalised by disturbance observer into

β =1

MnVnsFy = Pβ(s)Fy (42)

D. 2 degree of freedom control

To control the nominalised plant by YMO and LFO, 2degree of freedom control is utilised. Cut off frequenciesof low-pass filtersGβ(s) and Gγ(s) to proper the inversesystem in feedfoward control are10 rad/s. When commandsare compared with output in feedback control, commands arefiltered by the same low pass filter as the feedfoward control.The yaw rate is controlled by P control and the side slip angleis controlled by PI control.

VI. SIMULATION

For vehicle dynamics evaluation, sinusoidal input to steeringis a effective and widely used methodology [13]. In thisresearch, although this proposal RECS can be applied to any

trajectory, to compare proposed method with conventionalmethod, a sinusoidal input in utilised as follows.

u∗ =

δ∗fδ∗rN∗

z

=

A sinωt00

(43)

x∗ = Ax∗ +Bu∗ (44)

Target trajectory is designed as a trajectory that the vehicletravel by steering front steer in sinusoidal input, without rearsteering and differential torque as is shown in Eq.(43). Sideslip angle and yaw rate calculated by the bicycle model thatshown in Eq.(44), are treated as command.

A travel that steered only by front steering is defined asconventional method and compared with proposal method. Thefront steering in conventional method are settled as amplitudeA is 0.1 rad and angular velocityω is 1.0 rad/s. The constantvelocity of the vehicle is15 km/h. Cut off frequency of SAOis 30 rad/s and that of LFO and YMO are10 rad/s. Feedback gains of yaw-rateγ and side slip angleβ are determinedby the pole placement method and the poles are placed at−6 rad/s.

The results of the simulation for4π seconds are shown inFig. 5. Only output of and loss at the motors are considered sothat efficiency of chopper and inverters are100 %. Fig. 5(a),Fig. 5(b) show that the vehicle is following the yaw rate andside slip angle commands. The steering angles of conventionaland proposal methods are shown in Fig. 5(d) and Fig. 5(e) andthe differential torque in Fig. 5(c). Fig. 5(f) shows consumedenergy in4π seconds. This consumed energy is integration ofconsumed electrical power. By the proposal method, consumedenergy decreased for9.1 %. The consumed energy convertedinto traveling distance per1 kwh of energy is shown on TableI. Table I shows that traveling distance increased for0.8 kmper 1 kwh of energy stored in batteries.

VII. E XPERIMENTS

Experiments were conducted in the same conditions as thesimulations. The experiments were held in experimental areain the university and the vehicle expressed in section 2 wasutilised. Vehicle velocity was controlled by PI controller andits feedback gains were designed by pole assignment methodto place the pole at−1 rad/s. On the experiment, consumedenergy was calculated from products of input voltages andcurrents of inverters. Thus, inverter loss and iron loss wereincluded. Experiments were performed for 4 times each ofconvention and proposal method. The average of consumedenergy and standard deviation as errorbar are shown in Fig.6(f).

Fig. 6(a), Fig. 6(b), Fig. 6(d), Fig. 6(e) shows that sideslip angle and yaw rate are corresponding. The front and rearsteering angles and differential torque are shown in Fig. 6(c),Fig. 6(g), Fig. 6(h). Fig. 6(f) shows that consumed energy isdecreasing for13.4 %. On tableII traveling distance per1 kWhis shown.800 m increase of traveling distance is able to beconfirmed.

TABLE I

TRAVELING DISTANCE PER1 KWH(SIMULATION )

Battery Capacity Without RECS With RECS

1 kWh 5.14 km 5.65 km5 kWh 25.7 km 28.3 km

16 kWh 82.2 km 90.42 km

TABLE II

TRAVELING DISTANCE PER1 KWH(EXPERIMENTS)

Battery Capacity Without RECS With RECS

1 kWh 5.18 km 5.98 km5 kWh 25.9 km 29.9 km

16 kWh 82.9 km 95.7 km

VIII. C ONCLUSION

In this article, a range extension control system is proposedand its effect is confirmed by both simulations and exper-iments. Future works include consistent of range extensionand controllability especially considering safety. In addition,yaw rate and side slip angle can be changed to achieve higherefficiency.

ACKNOWLEDGEMENT

This research was partly supported by Industrial Tech-nology Research Grant Program from New Energy and In-dustrial Technology Development Organization(NEDO) ofJapan (number 05A48701d), and by the Ministry of Educa-tion, Culture, Sports, Science and Technology grant (number22246057).

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0 2 4 6 8 10 12−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

Time [s]

Sid

e sl

ip a

ngle

[rad

]

conventionalproposed

(a) Side-slip angle

0 2 4 6 8 10 12

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Time [s]

Yaw

rate

[rad

/s]

conventionalproposed

(b) Yaw Rate

0 2 4 6 8 10 12−80

−60

−40

−20

0

20

40

60

80

Time [s]

Yaw

mom

ent [

Nm

]

conventionalproposed

(c) Differencial Torque

0 2 4 6 8 10 12

−0.1

−0.05

0

0.05

0.1

Time [s]

Ste

erin

g an

gle

[rad

]

FrontRear

(d) Steering angle (conventional)

0 2 4 6 8 10 12

−0.1

−0.05

0

0.05

0.1

Time [s]

Ste

erin

g an

gle

[rad

]

FrontRear

(e) Steering angle (proposed) (f) Electricity Consumed

Fig. 5. Simulation

0 2 4 6 8 10 12−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

Time [s]

Sid

e sl

ip a

ngle

[rad

]

conventionalreference

(a) Side slip angle (conventional)

0 2 4 6 8 10 12−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

Time [s]

Sid

e sl

ip a

ngle

[rad

]

proposedreference

(b) Side slip angle (proposed)

0 2 4 6 8 10 12−40

−20

0

20

40

Time [s]

Yaw

mom

ent [

Nm

]

conventionalproposed

(c) Differencial Torque

0 2 4 6 8 10 12

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Time [s]

Yaw

rate

[rad

/s]

proposedreference

(d) Yaw Rate (conventional)

0 2 4 6 8 10 12

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Time [s]

Yaw

rate

[rad

/s]

conventionalreference

(e) Yaw Rate (proposed)

conventional proposed0

2

4

6

8

10

12

Ele

ctric

ity C

onsu

mpt

ion

[kW

s]

(f) Electricity Consumption

0 2 4 6 8 10 12−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Time [s]

Ste

erin

g an

gle

[rad

]

FrontRear

(g) Steering angle (conventional)

0 2 4 6 8 10 12−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Time [s]

Ste

erin

g an

gle

[rad

]

FrontRear

(h) Steering angle (proposed)

Fig. 6. Experimental Results