Download - 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
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
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
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
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
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
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
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
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
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 )
kv12 = (b21αmf +b22αm
r −b23Mmz )
kv13 = (b21αmf −b22αm
r +b23Mmz )
kv21 = (b11αmf +b12αm
r +b13Mmz )
kv22 = (b11αmf +b12αm
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
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
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
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
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)
Speed=70km/h and maximum steering command δmax = 3.25◦
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)
Speed=80km/h and maximum steering command δmax = 3.25◦
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
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
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
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
References I
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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.
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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.
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.