1. Cooperative DYC System Design for
Optimal Vehicle Handling Enhancement
Virginia Tech
C N E F RV H LE
E T R O E IC
S S E S& S F T
Y TM A EY
S.H. Tamaddoni *, S. Taheri, M. Ahmadian
Center for Vehicle Systems and Safety (CVeSS)
Department of Mechanical Engineering
Virginia Tech, USA
* email: tamaddoni@vt.edu
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2. Outline
 Motivations
 Game Theory
 System Model GAME
THEORY
 Control Derivation
 Simulation and Results
 Conclusions
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3. Motivations
 Vehicle Stability Control (VSC) improves vehicle
stability and handling performance.
 Ferguson (2007) has shown that VSC can reduce
• single-vehicle crashes by 30-50% in cars and SUVs,
• fatal rollover crashes by 70-90% regardless of vehicle type.
© www.racq.com.au
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4. Interaction Model
 Driver / VSC interaction model:
Driver’s Driver’s
Processing Unit Action Unit
Vehicle System
VSC Processing
& Action Unit
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5. Game Theory
 The systems are governed by several controllers, i.e., decision
makers or players, where each controller aims to minimize its
own cost function.
 No player can improve his/her payoff
by deviating unilaterally from his/her
Nash strategy once the equilibrium is
attained.
 For a game with a sufficiently small
planning horizon, there is a unique
linear feedback Nash equilibrium that
© Andrew Gelman
can be computed by solving a set of
so-called Nash Riccati differential equations.
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6. Primary Objectives
 Driver:
• Steer the vehicle through the maneuver
 Controller:
• Guarantee vehicle handling stabiltity where the desired
value of yaw rate is obtained from Wong (2001):
vx
ψ desired =
 δF
(lF + lB )(1 + K us v x )
2
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7. Evaluation Model
 The evaluation vehicle model includes
• longitudinal & lateral dynamics
• yaw, roll, pitch motions
• combined-slip Pacejka tire model
• steering system model Y
φ sR
X sL
• 4-wheel ABS system Z
ψ FyBL
vy FxBL
vx FzBL lB
FyFR
FyFL lF
FxFR
FxFL
α FR FzFR
δF
α FL δF
FzFL
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8. Control Model
 2-DOF bicycle model CG ψ
• y: absolute lateral position Y
• ψ: absolute yaw angle
X
 δ
x =Ax + B1u1 + B 2 u2 , u1 = F , u2 =M zc
0 1 vx 0   0 
 0
C + Cα B lF Cα F − lB Cα B   C 
0
0 − α F 0 −vx −   αF 
 mv x mv x   m   
= 
A =  , B1 = ,B 0
0 0 0 1 0  2  
   
 lF Cα F − lB Cα B lF Cα F + lB Cα B 
2 2
 lF Cα F  1
0 0 −   Iz   Iz 
 
 I z vx I z vx   
x(t0 ) = [ y0 y0 ψ 0 ψ 0 ]
 
T
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9. Theorem 1: Certain system
Let the strategies (δ , M ) be such that there exist
*
f
*
zc
solutions ( P1 , P2 ) to the differential equations
∂H i * * ∂H i * *
*
∂γ
Pi = i ) −
( ( x , δ f , M zc , Pi ) . j ,
d
− x , δ f , M zc , P
* *
dt ∂x ∂ui ∂x
in which,
H i ( x, δ f , M zc , Pi )= xT Qi x + ri1δ f2 + ri 2 M zc + PiT ( Ax + B1δ f + B 2 M zc ) ,
2
such that,
∂H i * * *
∂ui
( x , δ f , M zc , Pi ) = 0,
and x* satisfies
 x* (t ) = Ax* (t ) + B1δ * + B 2 M zc ,
 f
*
 *
 x (t0 ) = x0 .

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10. Theorem 1: Certain system
Then,
(δ *
f , M zc )
is a Nash equilibrium with respect to the
*
memoryless perfect state information structure, and
the following equalities hold:
K i (t ) x (t )
−
ui* = − Rii 1BT Pi (t ),
i
i ∈ {δ , M } , u ∈ {δ f , M zc }
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11. Theorem 2: linear feedback
Suppose ( K1 , K 2 ) satisfy the coupled Riccati equations

K1 =− K1A − Q1 + K1S1K1 + K1S 2 K 2 + K 2S 2 K1 − K 2S1 K22 ,
− AT K 1

K 2 = − K 2 A − Q 2 + K 2S 2 K 2 + K 2S1K1 + K1S1K 2 − K1S 2 K ,
− AT K 2 11
where
= Bi R ii1BT , Sij B j R −1R ij R −1BTj .
Si = −
i jj jj
Then the pair of strategies
(δ *
f , M zc ) =(t ) x, − R 22 BT K 2 (t ) x )
*
( −R111B1T K1
− −1
2
is a linear feedback Nash equilibrium.
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12. Simulation
 Vehicle: 2-axle Van
 Maneuver: standard “Moose” test at 60 kph
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13. Simulation
 Selected Q & R matrices:
1 0 0 0  0 0 0 0
0 0.1 0 0  0 0.1 0 0
= =
Qδ  , QM  ,
0 0 0 . 0 1  0 0 0 0
   
 0 0 0 0  0
. 1 0 0 0 1

= 10, R δ M 0,
R δδ =
= 10−5 , R M δ 103
R MM =
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14. Results
unit strategy Nash LQR
Driver 97,363 162,060
Controller 9,734,700 16,204,000
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15. Conclusions
 A novel cooperative optimal control strategy for
driver/VSC interactions is introduced:
• The driver’s steering input and the controller’s compensated
yaw moment are defined as two dynamic players of the
game “vehicle stability”
• GT-based VSC is optimally more involved in stabilizing the
vehicle compared to the common LQR controllers.
• GT-based VSC improves vehicle handling stability more
than the common LQR controllers can do with the same
driver and controller cost matrices.
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16. Thank You !
GAME
THEORY
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