Electronic Differential Control With Vehicle State Observer Based on Extended Kalman Filter

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Electronic Differential Control with Vehicle State Observer Based on Extended

Kalman Filter

Li Jun1, Duan Jie-li1, and Tian Shen2

1College of EngineeringSouth China Agricultural University

Guangzhou, Chinae-mail: [email protected]

2School of Civil Engineering and TransportationSouth China University of Technology

Guangzhou, Chinae-mail: [email protected]

AbstractThe growing demands for high level of energy

efficiency have given much new impetus to the development of

electric vehicle in Agricultural Machines. To improve the

performance of the state observer for an 8-DOF four-wheel

vehicle, a state observer based on the Extended Kalman Filter

is proposed. With the observer, the tire-road forces and the

vehicle roll velocity are estimated accurately according to the

simulation results. With the consideration of the nonlinearcharacters of tire and electric vehicle, the electronic

differential controller is proposed to select the wheel slip ratios

as the control tracking variables and distribute the torques to

ensure the steering stability by using modified exponent-

approaching sliding mode control method. The proposed

electronic differential controller is robust to improve the

handling performance and control response of electric vehicle

with the numerical verification.

Keywords- electrical differential control; Extended Kalman

Filter; sliding mode control; Magic Formula tyre model

I. INTRODUCTION

Presently, continuing interests of the green agriculturalequipments have further stimulated the development ofelectric off-road vehicles. The in-wheel motor multi-drivesystems for electric off-road vehicles have advantages overthe classical construction with one central machine. Wheelmotor drive systems provide better handling stability withthe electronic differential control during steering manoeuvres.

However, it is difficult for the steering control system toobtain directly some parameters including the sideslip angle,yaw rate and lateral velocity of vehicle. Therefore, severalvirtual observers had been employed to estimate the states ofnonlinear dynamic vehicle. The sliding mode observers were

proposed to replace the expensive sensors used for themeasurement of tires forces, side slip angle and velocity of

vehicle [1, 2]. The H observer-based controller was provedto have considerable improvements in the vehicle handlingwhen the sideslip angle was unavailable for measurement [3].To address vehicle nonlinearities and unmodeled dynamics,the unscented Kalman filtering technique was applied for theobserver of lateral tire forces and sideslip angle [4, 5].

Different from the mechanical differential, the forces androtational speeds of the electrical differential motors cannotautomatically coordinate while cornering for the requiredchanges of steering angle and torques. Based on Ackermanturning geometry, many approaches taken account of the

speed difference between the drive wheels when steeringwere proposed [6, 7]. Other steering controllers using thevehicle speed as input were designed to guarantee theelectric vehicle dynamics and stability [8, 9].

In this paper, an eight-DOF nonlinear model of vehicle isdeveloped with some general assumptions to obtain steeringdynamics. Consequently, a vehicle state observer based on

the Extended Kalman Filter (EKF) is proposed. To improvethe performance of electronic differential controller, themodified exponent-approaching sliding mode control schemeis implemented with the selection of wheel torques as controlvariables. The proposed controller is robust to improve thehandling performance and control response of vehicle.

II. VEHICLE STEERING MODEL

To describe vehicle dynamic movements including rolland yaw, a vehicle model with eight degrees of freedom isdeveloped in this section, which comprising the longitudinalmotion in x and y direction, rolling motion around x-axis,yawing motion around z-axis and wheel rotational motion offour tires.

Assume that the front steering angle and side slipangle at vehicle center of gravityare small, thenvxV vyV sin cos1

WhereVis the total velocity at vehicle center of gravity, vxand vy are the longitudinal and lateral velocity at vehiclecenter of gravity.

Ignore the aerodynamic forces and rolling resistance ofthe tires, thus, the eight-DOF nonlinear model of vehicle can

be obtained [10]

(1)While mis the vehicle mass, mSis the mass of the sprungmass, FxfandFxrare the longitudinal tire force on front andrear tires,FyfandFyrare the lateral tire force on front and reartires,ris the yaw rate of vehicle,is the vehicle roll angle,hsis the distance between the sprung mass center of gravityto the x-axis, lfand lr are the longitudinal distance fromvehicle center of gravity to front and rear tires, C is the

( )( ) ( )

( )

1 2 1 2 1 2

1 2 1 2 1 2

1 2 1 2 1 2

( )

2

x r xf xf xr xr yf yf

x r s s xf xf yf yf yr yr

z z r xz f xf f xf f yf f yf r yr r yr

f f

X mv F F F F F F

Y mv m h F F F F F F

M I I l F l F l F l F l F l F

dl K

= + = + + + +

= + = + + + + +

= = + + +

+

( ) ( )

2f r

r r

x x xz r s s x r s s

dl K

d d

M I I m h v K m h g C

= + = +

2011 International Conference on Computer Distributed Control and Intelligent Environmental Monitoring

978-0-7695-4350-5/11 $26.00 2011 IEEE

DOI 10.1109/CDCIEM.2011.376

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equivalent damping coefficient of the rolling motion, KfandKr are the front and rear wheel tire camber thrustcoefficients, df/dand dr/dare the front and rear wheelcamber angles per unit roll angle, Izis the total yaw momentof inertia around the vertical axis passing through the vehiclecenter of gravity, and Ixis the rolling moment of inertia ofthe sprung mass around the x-axis.

Using (1), the vehicle longitudinal acceleration axandlateral accelerationayare

[ ]( )[ ]

1 2 1 2 1 2

1 2 1 2 1 2

( )x xf xf xr xr yf yf

y xf xf yf yf yr yr

a F F F F F F m

a F F F F F F m

= + + + += + + + + +

(2)

The longitudinal slip ratioof the front or rear wheel isdefined as

( ), , , ,

, =1, 2fi ri fi ri fi ri fi ri

r u u i = (3)

Wherefi,riis the rotational speed of front or rear wheel,risthe wheel effective radius. Variable u fi,ri is the velocity offront or rear wheel, and it can be calculated from

1 2

1 2

2 2

2 2

f f

wf r f wf r f

r r

wr r r wr r r

b bu V l u V l

b bu V l u V l

= = + +

= + = +

(4)

Wherebf,brare the front and rear track of vehicle.The slip angles at front and rear tires are [11]

1 2

1 2

arctan arctan2 2

arctan arctan2 2

x f r x f r

f f

x f r x f r

x r r x r r

r r

x r r x r r

v l v l

v b v b

v l v l

v b v b

+ += =

+

= = +

(5)With the definition of the height of center of gravity of

vehicle as constant h, the normal loadFzfi,zri of front and reartires can be written as

( )

( )

( 1)2 2

, =1, 2

( 1)2 2

i y xr

zfi

f r f f r

if y x

zri

f r r f r

ma h ma hl mgF

l l b l l i

l ma h ma hmgF

l l b l l

= + + +

= + ++ +

(6)

The Magic Formula tire model is used to describe tirebehavior as follows:

( ){ }[ ]sin arctan arctanv

h

y D C Bx E Bx Bx

Y y S

x X S

= = +=

(7)

Where Y are the output variables including longitudinalforce Fx, lateral force Fyor aligning moment Mz, Xare theinput variables including slip angle or slip ratio ,Shis thehorizontal shift,Svis the vertical shift.

With the selection of the constants a1, a2, , a8, thestiffness factor B, shape factor C, peak value D andcurvature factor E can be expressed as functions of the

normal loadFzas follows

( )( ) ( )3 4 5

2 2

1 2 6 7 8

sin arctan lateral force

,z

z z z z

BCD a a a F

D a F a F E a F a F a

=

= = + + (8)

To describe the lateral transient behavior of tires, therelaxation lengthis introduced, so that,

( ), , ,

, =1, 2x

yfi yri yfi yrii yfi yri

vF F F i

= (9)

Where,yfi yri

F is the lateral force in the Magic Formula tyre

model frame.

III. STATE OBSERVER DESIGN

Due to the EKF is an optimal stochastic observer in theleast-square sense for the estimation of nonlinear systems,the proposed vehicle state observer is designed by using theEKF technique to estimate the tire-force and dynamics ofvehicle especially including the roll angular velocity.

Defining:State vector:

( ) ( )[ ]1 2 1 2 1 2 1 2

, , , , , , , ,r yf yf yr yr xf xf xr xr

x F F F F F F F F = + +

Input vector: [ ]1 2 1 2 1 2 1 2, , , , , , , ,f f r r fz fz rz rzu F F F F =

Output vector: [ ], , ,r x y x

y a a v=

For this application, the nonlinear eight-DOF dynamicstate-space equations of vehicle are expressed as

( ) ( )( ) ( ) ( ) ( )( ) ( ) ( )( ) ( )

0 00 ~ , , ( ) ~ (0, ), ( ) ~ (0, )

x t A x t x t Bu t Gw t

y t Cx t v t

x x P w t Q v t R

= + += +

(10)

Where w(t) and v(t) are the zero-mean Gaussian processnoise and measurement noise vectors with covariancematricesQandRrespectively.

Assuming a sampling time of Ts, the corresponding

discrete time model can be obtained as( )

1

~ (0, ), ~ (0, )

k d k k d k k

k d k k

k K k K

x A x x B u w

y C x v

w Q v R

+ = + += + (11)

WhereAdI+ATs , BdBTs ,Cd=C, RK=R/Ts,

0

( ) ,s

s

TA T T

k K sw e Gw d Q GQG T

= The prediction of the state covariance requires the

online computation of Jacobian matrix, defined as

( )[ ]1

1k

k d k k x xA A x x

x +

+ =

=

(12)

The recursive algorithm of the Extended Kalman Filter

can be derived as below.Step 1: State prediction (time update)

( )1 1 1 1

,T

k d k k d k k k k k K x A x x B u P A P A Q

+ + + += + = + (13)

Step 2: Innovation (measurement update)

( )1 1 1 1 1 1 1 1 1

,k k k k d k k k k d k

x x K y C x P P K C P+ + + + + + + + +

= + = (14)

The Extended Kalman Filter gain matrixKis calculated by

( )1

1 1 1

T T

k k d d k d K K P C C P C R

+ + += + (15)

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IV. ELECTRONICDIFFERENTIALCONTROLLER DESIGN

To avoid skidding, the speed of the inner wheel has tobe different from that of the outer wheel while cornering.Assume that the front and rear tracks of the vehicle areequal and the drive wheels are turning without slip. So that,the conventional electronic differential controller of electric

vehicle is designed using the Ackerman turning geometry.In the geometry, the difference between the angular speedsof the wheel drives is expressed by the relation

,0

,

2

2

f r

i f r

R b

R b

+=

(16)

Wherei andoare the angular velocities of the inner and

outer drive wheels,Ris the radius of rotation.However, the longitudinal slip of tires and nonlinear

dynamic characters of vehicle cannot be neglected for bettersteering control. Therefore, an electronic differential controlscheme based on sliding mode control and EKF stateobserver is proposed. The proposed controller chooses the

tire longitudinal slip ratios as control variables not the wheelspeed, and distributes the torque value to track the referenceslip ratio of each drive wheel by using the sliding modecontrol. The EKF state observer is employed to estimate thevehicle state parameters and feedback them to the steeringcontroller.

The dynamics of dual rear wheel drives are given by

, =1, 2w ri ri xri

J T F r i = (17)

WhereJwis the wheel moment of inertia, Triis the torqueof rear wheel.

Note that the variables and both are the functions of the

slip ratios from (3), (6) and (7), the speed u ri and normal

load Fzri of the rear tires can be considered as constants

compared to the slip ratio when the control system is stableand if the sampling period is short enough. So that, (17) can

be rewritten as

, =1, 2ri ri ri

A BT i = + (18)

Further, the sliding surface of the electronic differential

control scheme is given as

( )*ri ri ri

S c

= (19)

Whereri*is the reference longitudinal slip ratio which is relative to the adhesion coefficient.

To guarantee rapid responses and depress system ripple,

(19) can be derived as follow

Figure 1. The block diagram of the proposed controller

ri

ri ri

ri

SS S

S

= +

(20)

Wherec,,andare positive control constants.Since the control performance of a discrete system is

much influenced by the sampling periodTsand constant, itis necessary to modify the exponential approximation law.

The discrete modified form of (20) is selected as follow

( ) ( )

( ) ( )( )

ri

ri ri ri

ri

S kS k S k S k

S k

= +

(21)

Then, (21) can be rewritten in the form

( ) ( )

1 ( ) ( ) ( )

( )

ri

ri ri s ri s ri

ri

S kS k S k T S k T S k

S k

+ =

+(22)

When the sampling period Tsis sufficiently short, the

existence of sliding surface for a discrete controller can be

guaranteed if the following equation is satisfied

{[ ( 1) ( )]sign[ ( )] 0[ ( 1) ( )]sign[ ( )] 0S k S k S k

S k S k S k

+

(23)

Substituting (22) into (23), it is easy to prove that themodified discrete sliding mode control system is globalasymptotical stable when the values of the control constants

c,and all are within their bound.

V. SIMULATION RESULTS

To confirm the effectiveness of the proposed nonlinearvehicle state observer and electronic differential controller,some numerical simulations are implemented with the 8DOFmodel and Magic Formula tire model. The specifications ofvehicle is listed in Table I. The vehicle is undergoing anextreme maneuver with a small steering angle = 2 (positiveindicates a left turn), and a constant speed of 20m/s.

TABLE I. SPECIFICATIONS OFVEHICLE WITHDUAL WHEEL DRIVES

Vehicle mass,m 1252kg

Distance from vehicle C.G. to front tire,lf 1.2 m

longitudinal distance from vehicle C.G. to rear tire,lr 1.37m

Height of vehicle C.G., h 0.53m

Wheel effective radius,r 0.32m

Yaw moment of inertia around vertical axis,Iz2027kgm2

Wheel moment of inertia,J 2.4 kgm2

Rolling moment of inertia around x-axis,Ix 381kgm2

Sprung mass,ms 1020kg

Distance between sprung mass C.G. to x-axis,hs 0.42m

Vehicle track,bf= b r 1.4m

SMC Controller EKF Observer

*

r

1rT

2rT

1xrF

2xrF

Magic Formula Tire Model

zrF

2r2f

1r

2r

zfF

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(a)

(b)

(c)

Figure 2. Simulation results of the proposed EKF observer

(a)

(b)

Figure 3. Responses of rear torques, wheel velocities and slip ratios with

the proposed sliding model controller

Fig. 2 (a)-(c) show the responses of side slip angle, rollangular velocity, yaw rate and lateral forces with slidingmode steering control. The results demonstrate that the

proposed vehicle state observer based on EKF is a suitableestimator and matches the actual parameters excellently.These torque values, velocities and slip ratios of the rear leftand right wheel are chosen to ensure differential controlfeasibility as seen in Fig. 3 (a)-(b). With a stepwise left turn,the modified exponent-approaching sliding mode controller

can adaptively adjust the inner and outer wheel drive torqueto track the reference slip ratios rapidly.

VI. CONLUSIONS

This paper discusses the electronic differential control foran electric vehicle with independently wheel drives. Aneight-DOF four-wheel model for state observer was derived.

With the proposed EKF observer, the important parametersaffecting vehicle stability and the risk of skipping can beestimate, especially including the tire-road forces and thevehicle roll angular velocity. Different to the conventionalelectronic differential controller, the control variables of the

proposed controller based on modified exponent approachingsliding mode control method are the slip ratios of the wheeldrives not the rotational speeds. The proposed observer andcontroller are validated by simulation results.

ACKNOWLEDGMENT

The author would like to acknowledge the referees, Prof.Hong Tian-shen for helpful discussions on this topic and thefinancial support of Department of Education of Guangdong

Province, China.

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Electronic Differential Control With Vehicle State Observer Based on Extended Kalman Filter