Improving EV Lateral Dynamics Control Using Infinity Norm Approach with Closed Form Solution

Improving EV Lateral Dynamics Control Using

Infinity Norm Approach with Closed Form Solution

A. Viehweider V. Salvucci ∗ Y. Hori T. Koseki

The University of Tokyo

ICM2013, Vicenza

Outline

1 Actuation Redundancy in EVs

2 EV Model, Control Design, and Actuator Redundancy Problem

3 Considered Approaches for Actuator Redundancy ResolutionThe 2 Norm Approach (Pseudo-inverse Matrix)The Infinity Norm Approach in Closed Form (Our Solution)

4 Simulation Description

5 Results

6 Conclusions

Outline





5 Results

6 Conclusions

Why Over-actuation in Technical Systems?

Robot Arms [Salvucci 2013] Aircraft [Harkegard 2003] EV

- Higher Cost

+ Optimization of additional criteria: minimize energy, actuator stress . . .

+ Lower sensitivity to component failure due to reconfiguration

The Electric Vehicle as an Over-actuated System

Nissan PIVO

4 Independent steering wheel

4 In-wheel motors

Our model

1 Active Front Steering (AFS)

1 Active Rear Steering (ARS)

1 yaw moment (at least 2In-Wheel motors)

Actuation Redundancy Resolution in EVs: Our Solution

Two type of approaches to resole redundancy

Expressed in closed form

Computationally easy

Often pseudoinverse based

- Input range not fully used

Based on iterative algorithm

Computationally heavy

Input range fully used

- Hard to implement in real time

Our Solution: based on infinity norm with a closed form expression

Easy to implement and full use of input range

Limitation: valid only for 3x2 allocation problem

Outline





5 Results

6 Conclusions

Bicycle Model of the EV (for the Controller Design)

Fyf

Fyr

ay

Fw

ϒ

δf

δr

lf

lr

lw

v β

δα

x = Ax+v∗ = Ax+Bu∗

[

β

γ

]

=

−(Cf +Cr )mvx

lrCr−lf Cf

mv2x

−1

lrCr−lf Cf

Jz−

l2r Cr+l2f Cf

Jzvx

[β

γ

]

+

+

[Cf

mvx

Cr

mvx0

lf Cf

Jz− lrCr

Jz1Jz

]

︸︷︷︸

B

δfδrMz

(1)

Assumptions

Three actuators are used: δf , δr , and Mz (a virtual actuator)

Steering angles (δf , δr ) are small [Pacejka 2006]

Cf, Cr roughly known [C. Sierra 2006] [B. M. Nyguyen 2011]

Yaw rate (γ) is measured

Body slip angle (β ) is accurately estimated [Nguyen 2013]

EV Lateral Dynamics Controller

v* u*

u0

constant

bounds:

Δumax

allocation

matrix:

B

Allocation Controller Electric

Vehicle

State

observer

(Body

slip

angle

observer

)

xref

x x

Robust control based on sliding mode as described in [Viehweider 2012],tracking the yaw rate γ and the body slip angle β of the vehicle.

Controller defines v∗ = [v∗1 ,v∗2 ]

T

Allocation defines u∗ = [u∗1,u∗2,u

∗3]

T

Actuator Redundancy Problem is Based on Slip Angles

[v∗1v∗2

]

︸︷︷︸

v∗

= B

u∗

︷︸︸︷

δfδrMz

= B

u∗

︷︸︸︷

αf +β (t)+ lfvx

γ(t)

αr +β (t)− lrvx

γ(t)

Mz

= B

u

︷︸︸︷

αf

αr

Mz

︸︷︷︸

v

+B

u0︷︸︸︷

β (t)+ lfvx

γ(t)

β (t)− lrvx

γ(t)

0

Actuator Redundancy Problem is v =Bu, size(v)=2, size(u)=3

Advantage of using slip angles: time invariant bounds


[v∗1v∗2

]

︸︷︷︸

v∗

= B

u∗

︷︸︸︷

δfδrMz

= B

u∗

︷︸︸︷

αf +β (t)+ lfvx

γ(t)

αr +β (t)− lrvx

γ(t)

Mz

= B

u

︷︸︸︷

αf

αr

Mz

︸︷︷︸

v

+B

u0︷︸︸︷

β (t)+ lfvx

γ(t)

β (t)− lrvx

γ(t)

0




[v∗1v∗2

]

︸︷︷︸

v∗

= B

u∗

︷︸︸︷

δfδrMz

= B

u∗

︷︸︸︷

αf +β (t)+ lfvx

γ(t)

αr +β (t)− lrvx

γ(t)

Mz

= B

u

︷︸︸︷

αf

αr

Mz

︸︷︷︸

v

+B

u0︷︸︸︷

β (t)+ lfvx

γ(t)

β (t)− lrvx

γ(t)

0




[v∗1v∗2

]

︸︷︷︸

v∗

= B

u∗

︷︸︸︷

δfδrMz

= B

u∗

︷︸︸︷

αf +β (t)+ lfvx

γ(t)

αr +β (t)− lrvx

γ(t)

Mz

= B

u

︷︸︸︷

αf

αr

Mz

︸︷︷︸

v

+B

u0︷︸︸︷

β (t)+ lfvx

γ(t)

β (t)− lrvx

γ(t)

0



Outline





5 Results

6 Conclusions

The 2 Norm Approach (Pseudo-inverse Matrix)

Moore Penrose is the simplest pseudo inverse matrix = 2 norm [Klein 1983]

2 norm optimization criteria

min

√

(αf )2

(αmf )2

+(αr )

2

(αmr )2

+(Mz )

2

(Mmz )2

s.t. v= Bu

Closed form solution

uopt =W−1BT (BW−1BT )−1v

where

W= diag(1

(αmf)2

,1

(αmr )2

,1

(Mmz )2

).

-4-3

-2-1

0 1

2 3

4 -4-3

-2-1

0 1

2 3

4-3

-2

-1

0

1

2

3

Mz

αf

αr

Mz

Our Solution: The Infinity Norm Approach in Closed-form [Salvucci 2010]

∞−norm optimization criteria

min max(|αf |αmf,|αr |αmr,|Mz |Mm

z

)

s.t. v= Bu

Closed form solution

αf =

v1b23αmf −v2b13αm

f

αmf det13+αm

r det23if case1

v1b22αmf −v2b12αm

f

αmf det12−Mm

z det23if case2

−v1(b22αmr −b23M

mz )−v2(b13M

mz −b12αm

r )Mm

z det13−αmr det12

if case3

αr =

v1b23αmr −v2b13αm

r

αmf det13+αm

r det23if case1

−v1(b21αmf +b23M

mz )+v2(b11αm

f +b13Mmz )

αmf det12−Mm

z det23if case2

−v1b21αmr +v2b11αm

r

αmr det12−Mm

z det13if case3

Mz =

−v1(b21αmf +b22αm

r )+v2(b11αmf +b12αm

r )αmf det13+αm

r det23if case1

v1b22Mmz −v2b12M

mz

αmf det12−Mm

z det23if case2

v1b21Mmz −v2b11M

mz

αmr det12−Mm

z det13if case3

-4-3

-2-1

0 1

2 3

4 -4-3

-2-1

0 1

2 3

4-3

-2

-1

0

1

2

3

Mz

αf

αr

Mz

where

det23 = b12b23−b13b22

det13 = b11b23−b13b21

det12 = b11b22−b12b21

Our Solution: The Infinity Norm Approach in Closed-form [Salvucci 2010]

Let us define 6 constant values:

kv11 = (b21αmf +b22αm

r +b23Mmz )


r −b23Mmz )

kv13 = (b21αmf −b22αm

r +b23Mmz )


r +b13Mmz )


r −b13Mmz )

kv23 = (b11αmf −b12αm

r +b13Mmz )

The 3 cases are:

case1 =(kv21v2 ≤ kv11v1 and kv22v2 ≥ kv12v1) or

(kv21v2 ≥ kv11v1 and kv22v2 ≤ kv12v1)

case2 =(kv21v2 ≤ kv11v1 and kv23v2 ≥ kv13v1) or

(kv21v2 ≥ kv11v1 and kv23v2 ≤ kv13v1)

case3 =(kv22v2 ≤ kv12v1 and kv23v2 ≤ kv13v1) or

(kv22v2 ≥ kv12v1 and kv23v2 ≥ kv13v1)

Outline





5 Results

6 Conclusions

Simulation Software: CarSim

A highly sophisticated vehicle dynamics model

Different tire slip angles at the four wheels

Load transfer

Suspension effects and non linear tyre dynamics and kinematics

Not considered: sensor noise

Simulation Parameters

Reference values is a ”sine with a dwell” steering command:

The reference for the body slip angle has been set to zero, βref = 0◦

Body slip angle (β ), yaw rate (γ), and velocity (vx ) are known

Yaw moment (Mz ) evenly distributed to the 4 wheels, Mmaxz = 2000 Nm

Maximal values for tire sleep angles are αmaxf = αmax

r =5◦

Geometric constraints: δf ,max = 17◦,δr ,max = 4.5◦

Varying parameters: δmax and velocity vx

Controller gains set once and left untouched during all simulation runs

Outline





5 Results

6 Conclusions

Speed=70km/h and maximum steering command δmax = 3◦

2 norm

(performs)

Yaw rate Steering Angle Body Slip Angle

Inf norm

(performs)

Speed=70km/h and maximum steering command δmax = 3.25◦

2 norm

(actuatorsaturate!)

Yaw rate Steering Angle Body Slip Angle

Inf norm

(performswell)


Tire slip angles αf ,αr comparison

2 norm infinity norm

2 norm leads to the violation of thebounds (5◦).

Lateral force and tire slip angles relation

⇒ EV remains in the linear region forhigher velocity (=easier to control)


2 Norm

Not stable

Click on the image or

www.youtube.com/watch?v=eDJHqf4H3VY

Infinity Norm

Stable

Click on the image or

www.youtube.com/watch?v=5jz4tgXwndQ

https://www.youtube.com/watch?v=eDJHqf4H3VY

https://www.youtube.com/watch?v=eDJHqf4H3VY

https://www.youtube.com/watch?v=5jz4tgXwndQ

https://www.youtube.com/watch?v=5jz4tgXwndQ

Maximum body slip angle β for different ”sine with a dwell”

Speed [km/h] 60 70 80 90

Norm 2 ∞ 2 ∞ 2 ∞ 2 ∞

δmax

2 0.15 0.14 0.14 0.14 0.13 0.13 0.12 0.132.25 0.16 0.16 0.14 0.15 0.14 0.14 0.15 0.142.5 0.16 0.17 0.16 0.16 0.16 0.15 0.72 0.142.75 0.17 0.18 0.18 0.17 0.73 0.16 2.45 1.093 0.18 0.19 0.18 0.18 2.35 1.11 * 1.81

3.25 0.21 0.20 1.36 0.18 * 1.90 * 2.543.5 0.20 0.21 3.13 1.57 * 2.67 * 2.953.75 0.39 0.21 * 2.36 * 3.36 * 3.174 2.37 1.11 * 3.19 * 3.53 * *

4.25 3.91 2.12 * 3.87 * 3.76 * *4.5 6.73 2.92 * 4.05 * * * *

red = actuator saturates

* = vehicle is unstable

Infinity norm is superior for higher velocities and steering angles

⇒ Actuator saturation (and instability) occurs at higher velocities/δmax

⇒ Body slip angle is quite reduced

Outline





5 Results

6 Conclusions

Conclusions

In this work we

Proposed a new algorithm based on the infinity norm optimizationcriteria, with a closed-form solution for the actuator redundancyresolution problem in EV lateral dynamics control

Compared it with the conventional 2 norm approach by simulation

Achievement

The proposed infinity norm algorithm in comparison with the 2 norm

Increased the maximum velocity at which:the EV goes in the non linear regionthe actuator saturates and the EV shows instability

Reduced the body slip angle at high velocities

Thank you for your kind attention

A. Viehweider V. Salvucci ∗ Y. Hori T. Koseki

www.hori.k.u-tokyo.ac.jp www.koseki.t.u-tokyo.ac.jp

[email protected] www.valeriosalvucci.com

http://www.hori.k.u-tokyo.ac.jp

http://koseki.t.u-tokyo.ac.jp

http://www.valeriosalvucci.com

References I

N. Ando and H. Fujimoto. Yaw-rate control for electric vehicle with active front/rearsteering and driving/braking force distribution of rear wheels. Adv. Motion Control,11th IEEE Int. Workshop on, pp. 726-731, 2010.

H. F. B. M. Nyguyen, K. Nam and Y. Hori. Proposal of cornering stiffness estimationwithout vehicle side slip angle using lateral force sensor. IIC, 2011.

A. J. C. Sierra, E. Tseng and H. Peng. Cornering stiffness estimation based on vehiclelateral dynamics. Vehicle System Dynamics: International Journal of VehicleMechanics and Mobility, pp. 24-38, 2006.

O. Harkegard. Backstepping and control allocation with applications to flight control.PhD Thesis, Dept. of Electr. Eng., Linkoping Univ., 2003.

C. A. Klein and C. H. Huang. Review of pseudoinverse control for use withkinematically redundant manipulators. IEEE Transactions on Systems, Man, andCybernetics, 13:245–250, 1983.

B. Nguyen, Y. Wang, S. Oh, H. Fujimoto, and Y. Hori. Gps based estimation ofvehicle sideslip angle using multi-rate kalman filter with prediction of course anglemeasurement residual. In Proceedings of the FISITA 2012 World AutomotiveCongress, volume 194 of Lecture Notes in Electrical Engineering, pages 597–609.Springer Berlin Heidelberg, 2013. URLhttp://dx.doi.org/10.1007/978-3-642-33829-8_56.

H. B. Pacejka. Tyre and vehicle dynamics. Elsevier, 2006.

V. Salvucci, S. Oh, and Y. Hori. Infinity norm approach for output force maximizationof manipulators driven by bi-articular actuators. In 6th Europe-Asia Congress onMechatronics (EAM), Proceedings of, 2010.

http://dx.doi.org/10.1007/978-3-642-33829-8_56

References II

V. Salvucci, Y. Kimura, S. Oh, and Y. Hori. Force maximization of biarticularlyactuated manipulators using infinity norm. IEEE/ASME Transactions onMechatronics, 18(3):1080 –1089, June 2013. ISSN 1083-4435. doi:10.1109/TMECH.2012.2193670.

A. Viehweider and Y. Hori. Electric vehicle lateral dynamics control based oninstantaneous cornering stiffness estimation and an efficient allocation scheme.MATHMOD, Conference on Mathematical Modelling, pp. 1-6, 2012.

Improving EV Lateral Dynamics Control Using Infinity Norm Approach with Closed Form Solution

Science

actuator redundancy

u3tactuator redundancy

slip anglesv1v2 v

bu f t lfvxtr tlrvxtmz

evs2 ev model

closed form expressioneasy

closed form solutiona

control design