optimum traction force distribution for stability
TRANSCRIPT
Optimum Traction Force Distribution for StabilityImprovement of 4WD EV
In Critical Driving ConditionPeng He
The University of TokyoDepartment of Electrical Engineering
4-6-1 Komaba, Meguro, Tokyo 153-8505, [email protected]
Yoichi HoriThe University of Tokyo
Institute of Industrial Science4-6-1 Komaba, Meguro, Tokyo 153-8505, Japan
Abstract— This paper is concerned with integrated yaw sta-bility control strategy for 4WD EV driven by in-wheel motors.The strategy consists of double control loops. One is the uppercontrol loop, which uses2−DOF control method to specifycontrol effectors which are required for keeping EV yaw stability.The other is the lower control loop, which is used to determinecontrol inputs for four driving motors by optimum traction forcedistribution method. This distribution method is implemented byredundancy control.
The simulation results indicate that handling stability of this4WD EV is improved by proposed strategy, especially when theEV drives in critical conditions.
I. I NTRODUCTION
In order to improve maneuverability, stability and obtainmuch more driving traction force, many electric vehicles (EVs)are equipped with more than two actuating motors.
Further, the power transfer construction of EV is changeddays after days. The latest EV is driven by four motorizedwheels, which mean driving motors are fitted into wheels andcan be controlled independently[1][2].
For example, ”UOT March II”, which is shown in Fig.1,has four in-wheel driving motors (4WD). Control strategies,such as ABS, TCS and VSC, can be implemented for that EVeasily by advanced motion control of in-wheel electric motors.
Fig. 1. ”UOT March II” and in-wheel motors.
For planar motion control of EV, yaw rate and side slipangle are often chosen to be controlled. However, the controlmethods, such as in the paper[3], have contributed in improvingvehicle performance and guaranting safety for 2WD vehicles,which means that vehicle has two driving wheels.
However, in 4WD case, due to four driving motors, controlsystem has four permitted independent control inputs[4]. Andthose control inputs are all avaiable to control yaw rate andside slip angle. Because the number of indepent control inputsmuch than the number of controlled variables, therefore, thereis an actuator redundancy problem[5].
If we design control laws for 4WD EV, one issue whichmust be faced is how to deal with such actuator redundancy,which means how to distribute traction force for four drivingmotors.
Further, in order to avoid slip or lock and keep stability,especially when EV drives in critical conditions, distributingtraction force should consider the conditions of tire friction.
When 4WD EV drives in critical or dangerous conditions,even if one driving motor failures, the redundancy of controlinputs can be used to deal with those things and avoidinstability.
In this paper, we discuss a method optimally and dy-namically to distribute traction force by actuator redundancycontrol, which permits us to consider constraints of actuatingmotors (limitation of tire friction force). The ”optimally”means the best combination of redundant set of actuators. The”dynamically” means the resolution depends not only on thecurrent conditions but also on the previous sampling instant.
By using that optimum traction force distribution, we alsopropose an integrated yaw stability control algorithm for 4WDEV. As Fig.3 shows, the strategy consists of double controlloops. The upper control loop composes optimal feedback con-trol and yaw moment feedforward control. The lower controlloop is dynamic traction force distribution control. This loopis used to calculating control inputs for four driving motors.In this loop, an actuator redundancy problem is resolved.
The remainder of this paper is organized as follows: insection 2, 4WD EV modeling is mentioned; in section 3,integrated yaw stability control algorithm for 4WD EV is
discussed; in section 4, application of proposed algorithm forimproving stability of 4WD EV in critical driving is discussed;in last section, preliminary conclusions and future works arementioned.
II. 4WD EV M ODELING
In order to design control law for 4WD EV (”UOT MarchII”), a linear model is used, which is abstracted from the EVshown in Fig.1. The model is described in Fig.2. We assumethat only front wheels can be steered and steering angles offront wheels are equal to each other. In Fig.2,δf is the steeringangle.V is the velocity vector of center of gravity (COG) ofEV. All Fij mean traction forces.β represents sideslip angleof COG.γ represents yaw rate of COG.Mz is yaw moment.β andγ are controlled output variables.Mz andδf are controlinputs. The state equation is given by Eq.1[6][7].
X
Y
Fx2Fx1
Fx3 Fx4
Fy1
Fy3Fy4
Fy2
dr
df
βδf δfV
Mz
γ
lf
lr
Fig. 2. Linear vehicle model for dynamic analysis
(β̇γ̇
)=
[a11 a12
a21 a22
]
︸ ︷︷ ︸A
(βγ
)+
[b11 b12
b21 b22
]
︸ ︷︷ ︸B
(δf
Mz
)(1)
where
A =
[− 2(Cf+Cr)
MV −1− 2(lf Cf−lrCr)MV 2
− 2(lf Cf−lrCr)Iz
− 2(l2f Cf+l2rCr)
IzV
]
B =[ 2Cf
MV 02lf Cf
Iz
1Iz
]
M is the vehicle mass.Iz is the inertia moment.Cf (Cr) is thecornering stiffness of front (rear) tire.lf (lr) is the distrancefrom the front (rear) axial to the COG.
The transfer functions from steering angleδf and yawmomentMz to yaw rateγ are
γ =Gγ(0)(1 + Tγs)1 + 2ζ
ωns + 1
ω2ns2
δf +GM (0)(1 + TMs)1 + 2ζ
ωns + 1
ω2ns2
Mz (2)
= G(s)δf + H(s)Mz
where, Gγ(0) and GM (0) are steady gains.Gγ(0) =Vl
11+KV 2 and GM (0) = V (Cf+Cr)
2l2Cf Cr(1+KV 2) . Tγ and TM are
time constants.Tγ = Mlf V2lCr
and TM = MV2(Cf+Cr) . l is the
length of chassis (l = lf + lr). K is the stability factor of EV,K = −M
2l2lf Cf−lrCr
Cf Cr.
ζ is the damping coefficient.ωn is the natural frequency ofcontrol system.
ωn =2l
V
√CfCr
MIz
√1 + KV 2 (3)
ζ =M(l2fCf + l2rCr) + Iz(Cf + Cr)
2l√
MIzCfCr(1 + KV 2)(4)
III. I NTEGRATED YAW STABILITY CONTROL ALGORITHM
The concept of the proposed control algorithm is shownin Fig.3. The reference input commands areX1, ..., Xp. Thecontrol inputs for motors areU1, ..., Un and the controlledoutput variables of EV areY1, ..., Yp. The upper control loopcomposes optimal states feedback control and yaw momentfeedforward control. This control loop is used for calculating”virtual” control effectorsV1, ..., Vm, which is required forrealizing steady and dynamic characteristics of control system.However, the ”virtual” control effectors could not directly becontrol inputs(U1, ..., Un) for motors. Therefore, in the lowercontrol loop, we map the ”virtual” control effectors to physicalcontrol inputs for motors by optimum dynamic traction forcedistribution. Then we control EV to achieve the ”virtual”control effectors by traction force control of in-wheel motors.The traction force distribution is implemented by redundancycontrol.
}controlled
output variables
control inputs
feedback
reference
input commands
dynamic
distribution
control
virtual
control effectors
Vm
m < n
...
V1
Un
...
U1
motors
...
Body
EV
}
...
optimal
control
-+
Yp
Y1
Xp
X1
+
feed
forward
}lower control loopUpper control loop
Fig. 3. Block diagram of control strategy for EV
For planar motion control of 4WD EV, steering angleδf
and accerelation pedal angleθ are given by driver. And then,a reference model is used for calculating reference inputcommandsγ∗ andβ∗ in accordance with driver’s commands.The controlled output variables are yaw rateγ and slip angleβ. The ”virtual” control effectors are yaw momentMz andactive steering angles. The physical control inputs are tractionforce commands for motors.
In this study,θ is assumed to be zero and reference inputcommand is onlyγ∗. The ”virtual” control effector is yawmomentMz.
The upper control loop includes reference model control,feedforward control which is designed as aP− controller,feedback control which is designed by using aLQG regulator.Therefore, robust ”2-DOF” control algorithm is used for thedesign of upper control loop.
The lower control loop consists of dynamic optimal tractionforce distribution, which is used to determine control inputs formotors according to ”virtual” control effector and tire frictionconditions.
The control objective is to make the EV to follow the desiredyaw rate.
A. Reference model
According to Eq.2, yaw rate response from steering angleof ”UOT March II” is shown in Fig.4.
10-2
10-1
100
101
102
-90
-45
0
-20
-10
0
10
20
30
Ph
ase
(rad
)G
ain
(d
B)
Frequency (rad/sec)
40 km/h70 km/h
100 km/h120 km/h
Fig. 4. Yaw response of ”UOT March II”
In order to improve yaw rate phase delay and thus makelateral response quick, the reference model is defined as follow
γ∗ = G(s)refδf =Gγ(0)(1 + Tγs)1 + 2ζ
ω′ns + 1
ω′n2 s2
δf (5)
whereγ∗ is the target yaw rate. We make reference modelto have quick response, which meansω′n > ωn. In this study,we putω′n = 1.5ωn. The response is shown in Fig.5.
B. Feedforward control
According to Eqs.2 and 5, in order to make the real yawrate of EV fit the target value, we calculate the required yawmomentMff as one part of ”virtual” control effectorMz
[6].
γ∗ − γ = G(s)refδf −G(s)δf −H(s)Mff = 0 (6)
Mff =G(s)ref −G(s)
H(s)δf (7)
Based on Eq.7, feedforward control law is designed as aP−controller
P =Gγ(0)Tγ
GM (0)TM
ω′n2 − ω2
n
ω2n
= 2lfCf (ω′n
2
ω2n
− 1) (8)
-10
0
10
20
30
10-1
-90
-45
0
Ph
ase
(rad
)
Frequency (rad/sec)
w/o control
w/o control
reference
reference
100
101
102
Gai
n (
dB
)
-20
Fig. 5. Yaw response of reference model(V = 60km/h)
C. Feedback control
However, due to inaccuracy of EV model and effect ofdisturbance, Eq.8 is incomplete to satisfy Eq.7. For example,Cf , which is the cornering power of front tire, changes in realtime. It is difficult for us to know its true value. Therefore,we compensate the ”P-” controllor by using aLQG regulator.The controlled variables areγ and β. The LQG regulator isdesigned as follows
min J =∫ ∞
0
(ET QE + MTfbRMfb)dt (9)
whereQ andR are weighting matrices and decided accordingto experiments.E = (γ∗− γ, β∗− β)T . Yaw momentMfb isanother part of ”virtual” control effectorMz.
We can calculateMfb by
Mbf = −KE (10)
K = R−1BT S (11)
whereS is the solution of Algebraic Ricatti Equation
AT S + SA + Q− SBR−1BT S = 0 (12)
D. Optimum traction force distribution
In 4WD case, four driving motors can be used for motioncontrol. As Fig.6 shows, the inputs of lower control loop areyaw momentMz and driving forceFac (Fac is assumed toequal to total friction force). As mentioned above, the virtualcontrol effector is yaw momentMz, Mz = Mff + Mfb.
According to virtual control effectorMz, dynamic optimaltraction force distribution is used to determine control inputcommands for four motors to• keep the tire from slipping or locking.• obtain the maximum tire load ratio.• achieve the control effectors.• improve yaw rate stability.
According to Fig.2, if four tires are all in adhesion states, yawmomentMr can be obtained by
Mr = dfFx1 + drFx3 − dfFx2 − drFx4 (13)
wheredf anddr are front and rear tread.Fx1, Fx2, Fx3, Fx4
are traction forces.Optimum traction force distribution control is used to con-
vert ”virtual” control effectorMz to physical control inputs foravailable sets of motors. According to Eq.13, the distributedtraction forces can realize yaw momentMr in real time. Byoptimum traction force distribution control, the error betweenthe ”virtual” control effectorMz and the achieved real yawmoment Mr is to be minimized. Therefore, the requiredcontrol objective will be obtained due toMr ≈ Mz. And atthe same time, tire driving conditions are considered to keeptire from spinning or slipping.
Distribution control is generally used for solving this kindof problem. It is often used for flight control or marinevessel control[8][9]. It was also previously used for automo-tive control[10][11]. However, it was a static inverse map-ping method[12]. Some resolutions mainly focus on how toeasily implement or how to efficiently calculate for controlsystem[12].
That means,
min ‖W1F‖2 (14)
subject toMz = Bv · F.WhereW1 is weighting matrix.F = (F̃x1, F̃x2, F̃x3, F̃x4)T
are control input commands for motors. Virtual control vectorBv = (df ,−df , dr,−dr).
Control inputsF are obtained by
F = W−11 (BvW−1
1 )†Mz (15)
where† means pseudo-inverse of matrix.In order to consider dynamics of tire friction, we try to
extend the above static resolution to a dynamic optimum one.The ”dynamic” means redundancy resolution which dependsnot only on the current conditions but also on the resolutionof previous sampling instant[13].
According to Eq.13, Eq.14 is extended to
min ‖W1F(t)‖2 + ‖W2(F(t)− F(t− T ))‖2 (16)
subject toMz = Bv · F. WhereW2 is weighting matrix too.Control inputsF are obtained by
F(t) = W−11 (BvW−1
1 )†Mz + PF(t− T ) (17)
whereP = (I − W−1(BvW−1)†Bv)W 22
W 2 , † means pseudo-inverse of matrix,W 2 = W 2
1 + W 22 .
Eq.17 is just like a digital first order low pass filter, whichmeans dynamic redundancy resolutionF(t) can be obtainedby filtering a desired or reference static value. It is this pointthat the change of frequency and extent of control inputs canbe dynamically controlled quite well[13]. It is easy to verifythat the eigenvalues of matrixP are bounded by 1, whichmeans Eq.17 is asymptotically stable[13].
IV. SIMULATION
The actual control diagram is shown in Fig.6. In accordancewith what mentioned above, the upper control loop uses ”2-DOF” algorithm by integrating with yaw moment feedforwardand states feedback control. The lower control loop is dynamicoptimum traction force distribution control. The control objec-tive is to make the EV to follow the desired yaw rate.
driver
reference model
feedforward controller
feedback controller
optimum traction force distribution
+ -
+
γ∗ γ
MfbMffδf
FacEq.(17)
motor sensors
EV Body
motor
motor
motor sensors
control
inputs
+
δf
Eq.(5)
Eq.(8) Eq.(9)
Fig. 6. Integrated yaw stability control algorithm for 4WD EV
The specifications of ”UOT March II”, which are used forsimulations, are shown in TABLE I.
TABLE I
THE SPECIFICATIONS OF”UOT M ARCH II”
Dimensions (L×W×H) 3695×1660×1525 (mm)Weight 1100kgInertia moment 3.76×103 kgm2
Tire Radius 0.26mMax. output 36kWMax. torque 77Nm
Motor Max. speed 8700rpm(/unit) reduction gear ratio 5.0
Type Panasonic EC-EV1238Battery Capacity 38Ah
Voltage 12V × 19 modules
A. Verification
In order to verify the effectiveness and efficiency of thecalculated virtual control effector, we performed two kinds oftests for EV (” UOT March II”). In this case, optimum tractionforce distribution control is not used.
0 5 10 15
-6
-3
0
3
6
10
13
Time (sec)
Ste
erin
g a
ng
le (
deg
ree)
-10
-13
Fig. 7. Input signal for Lane change test of ”UOT March II”
First, driver input singal and simulation result are shown inFigs.7 and 8, which show that the yaw rate can be controlledto well fit the reference value.
-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0.02
Yaw
rat
e (r
ad/s
ec)
0 5 10 15Time (sec)
FFFF+FB
ref
Fig. 8. Yaw response of ”UOT March II” in lane change test(V = 60km/h)
The second test is ”J turn” test. Driver input singal andsimulation result are shown in Figs.9 and 10, which also showthe effectiveness of control algorithm.
0
10
Ste
erin
g a
ngle
(deg
ree)
5
0 5 10 15Time (sec)
Fig. 9. Input signal for J-turn test of ”UOT March II”
Therefore, it can be deduced that the upper control loopcan give the quite right virtual control effector to satisfy thecontrol requirement.
B. Comparison
Simulation is performed to compare the proposed controlmethod with the one which does not use traction force distri-bution method. We use ”Lane change” test for that comparison.In this case, the friction forces of left and right tire are differentto each other. The friction coefficient of left side is 0.8, butright one is 0.3. It means that EV drives ”Lane change” on asplit µ road.
We suppose that the traction force distribution methodwhich we want to compare is
Mr = drFx3 − drFx4 (18)
As Fig.11 shows, control method which does not use opti-mum traction force distribution, can not make the controlled
0
0.005
0.01
0.015
0.02
0 5 10 15Time (sec)
Yaw
rat
e (r
ad/s
ec)
FFFF+FB
ref
Fig. 10. Yaw response of ”UOT March II” in J-turn test(V = 60km/h)
yaw rate to follow the reference value quickly and accurately.It cannot keep EV handling from instability.
On the contrary, as Fig.12 shows, when we use optimumtraction force distribution control, which is realized by takingadvantage of redundancy of control inputs, the controlledyaw rate can follow the reference value quite well. It meansstability of EV can be improved.
0 2 4 6 8 10 12-0.4
Yaw
rat
e (r
ad/s
ec)
Time (sec)
-0.2
0
0.2
0.4
0.6
0.8
1
w/o optimum distributionref
Fig. 11. Yaw rate response of EV on splitµ road without optimum tractionforce distribution
As Fig.13 shows, when it does not take full advantage ofthe redundancy of control inputs for traction force distritution,the rear slip. It causes EV being instability, which is shownin Fig.15. For this case, the EV can not make ”Lane change”under the test condition.
When redundancy of control inputs is considered and wellused for optimum traction force distribution, the coorespond-ing tire slip ratio is kept in a safty domain, which is shown inFig.14. For this case, the EV can realize stable ”Lane change”under the same test condition.
V. CONCLUSIONS ANDFUTURE WORKS
Simulation results show that the proposed control strategycan improve the stability of 4WD EV, especially EV drives incritical conditions. Optimum traction force distribution controlcan be an effective method when take full advantage of redun-
Time (sec)0 5 10 15
Yaw
rat
e (r
ad/s
ec)
-0.4
-0.3
-0.2
-0.1
0
0.4
0.3
0.2
0.1
w/ optimal distributionref
Fig. 12. Yaw rate response of EV on splitµ road with optimum tractionforce distribution
0 2 4 6 8 10 12Time (sec)
0
0.2
0.4
0.6
0.8
1
-0.2
-0.4
-0.6
-0.8
-1
Sli
p r
atio
rear left
rear right
Fig. 13. Slip ratio of rear tires when optimum traction force control is notused
dancy of control inputs. That method bring advantages overcontrol strategies which do not use control inputs redundancy.
Futher, using redundancy of control inputs will have moremetrits. For example, tire friction constraints can be taken intoaccount, control reconfiguration can be performed when motorfailure, and algorithm calculating time can be saved becauseof the two control loops design.
For future work, this control strategy need to be refined andexamined by more experiments. The singularity problem ofoptimal dynamic redundancy resolution need to be discussed.BecauseLQG regulator is a full state feedback controller,more research should be focused on some observers for theestimation of slip angleβ or cornering power, which aredifficult to know in real time[14].
REFERENCES
[1] Yoichi Hori, ”Future Vehicle driven by Electricity and Control -Researchon 4 Wheel Motored ’UOT March II’,”IEEE Transactions on IndustrialElectronics,Vol. 51, No. 5, pp. 954–962, 2004.10.
[2] J. Marshall, M. Kazerani, ”Design of an Efficient Fuel Cell Vehicle Driv-etrain, Featuring a Novel Boost Converter,”The 31st Annual Conferenceof the IEEE Industrial Electronics Society,Raleigh, North Carolina, USA,2005. 11.
[3] Y. Furukawa and M. Abe, ”Direct yaw moment control with estimatingside-slip angle using on-board-tire-model,” inProc. AVEC’98,Nagoya,Japan, 1998.
0.15
Time (sec)0 5 10 15
Sli
p r
atio
0.1
0.05
0
-0.05
-0.1
-0.15
-0.2
rear left
rear right
Fig. 14. Slip ratio of rear tires when optimum traction force control is used
w/o optimum traction force distribution
w/ optimum traction force distribution
Fig. 15. ”Lane change” trajectory of EV controlled by differentmethods(V = 60km/h)
[4] Shin-ichiro Sakai, Hideo Sado and Yoichi Hori, ”Motion control in anelectric vehicle with 4 independently driven in-wheel motors,”IEEETrans. on Mechatronics,Vol. 4, No. 1, pp. 9–16, 1999.
[5] Peng He, Yoichi Hori, ”Resolving Actuator Redundancy for 4WD ElectricVehicle by Sequential Quadratic Optimum Method,”The 19th JapanIndustry Applications Society Conference,Fukui, Japan, 2005.8.
[6] Peng He, Yoichi Hori, Makoto Kamachi, Kevin Walters and HiroakiYoshida, ”Future Motion Control to be Realized by In-wheel MotoredElectric Vehicle,” The 31st Annual Conference of the IEEE IndustrialElectronics Society,Raleigh, North Carolina, USA, 2005. 11.
[7] Hiroshi Fujimoto, Akio Tsumasaka and Toshihiko Noguchi, ”Direct Yaw-moment Control of Electric Vehicle Based on Cornering Stiffness Esti-mation,” The 31st Annual Conference of the IEEE Industrial ElectronicsSociety,Raleigh, North Carolina, USA, 2005. 11.
[8] Buffington, J. M., ”Tailless aircraft control allocation,”AIAA Guidance,Navigation, and Control Conference and Exhibit,New Orleans, LA, 1997.
[9] Omerdic, E. and G.N. Roberts, ”Thruster fault diagnosis and accommo-dation for open-frame underwater vehicles,”Special feature GCUV 2003,Control Engineering Practice,2004.
[10] Y.Hattori, K.Koibuchi, and T. Yokoyama, ”Force and moment controlwith nonlinear optimum distribution for vehicle dynamics,”In Proc. ofthe 6th International symposium on advanced vehicle control,Hiroshima,2002.
[11] John Harris Plumlee, ”Multi-Input Ground Vehicle Control UsingQuadratic Programming Based Control Allocation Techniques,”Masterthesis,The Graduate Faculty of Auburn University, Auburn University,2004.8.
[12] John A. M. Petersen, Marc Bodson, ”Constrained Quadratic Program-ming Techniques for Control Allocation,”Proceedings of the42nd IEEEConference on Decision and Control,Maui, Hawaii, USA, 2003.12.
[13] Ola Harkegard, ”Backstepping and Control Allocation with Applicationsto Flight Control,” Ph.D. Thesis,Submitted to the Department of Elec-trical Engineering, Linkoping University, Linkoping, Sweden, 2003.
[14] Yoshifumi Aoki, Toshiyuki Uchida, Yoichi Hori, ”Experimental Demon-stration of Body Slip Angle Control based on a Novel Linear Observerfor Electric Vehicle,”The 31st Annual Conference of the IEEE IndustrialElectronics Society,Raleigh, North Carolina, USA, 2005. 11.