Your SlideShare is downloading. ×
0
2010 IEEE American Control Conference
2010 IEEE American Control Conference
2010 IEEE American Control Conference
2010 IEEE American Control Conference
2010 IEEE American Control Conference
2010 IEEE American Control Conference
2010 IEEE American Control Conference
2010 IEEE American Control Conference
2010 IEEE American Control Conference
2010 IEEE American Control Conference
2010 IEEE American Control Conference
2010 IEEE American Control Conference
2010 IEEE American Control Conference
2010 IEEE American Control Conference
2010 IEEE American Control Conference
2010 IEEE American Control Conference
Upcoming SlideShare
Loading in...5
×

Thanks for flagging this SlideShare!

Oops! An error has occurred.

×
Saving this for later? Get the SlideShare app to save on your phone or tablet. Read anywhere, anytime – even offline.
Text the download link to your phone
Standard text messaging rates apply

2010 IEEE American Control Conference

317

Published on

Published in: Economy & Finance, Technology
0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total Views
317
On Slideshare
0
From Embeds
0
Number of Embeds
0
Actions
Shares
0
Downloads
5
Comments
0
Likes
0
Embeds 0
No embeds

Report content
Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
No notes for slide

Transcript

  • 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.eduVirginia Tech ACC 2010 - s1
  • 2. Outline  Motivations  Game Theory  System Model GAME THEORY  Control Derivation  Simulation and Results  ConclusionsVirginia Tech ACC 2010 - s2
  • 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.auVirginia Tech ACC 2010 - s3
  • 4. Interaction Model  Driver / VSC interaction model: Driver’s Driver’s Processing Unit Action Unit Vehicle System VSC Processing & Action UnitVirginia Tech ACC 2010 - s4
  • 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.Virginia Tech ACC 2010 - s5
  • 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 ) 2Virginia Tech ACC 2010 - s6
  • 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 FzFLVirginia Tech ACC 2010 - s7
  • 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 ]   TVirginia Tech ACC 2010 - s8
  • 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 . Virginia Tech ACC 2010 - s9
  • 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 }Virginia TechACC 2010 - s10
  • 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.Virginia TechACC 2010 - s11
  • 12. Simulation  Vehicle: 2-axle Van  Maneuver: standard “Moose” test at 60 kphVirginia TechACC 2010 - s12
  • 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 =Virginia TechACC 2010 - s13
  • 14. Results unit strategy Nash LQR Driver 97,363 162,060 Controller 9,734,700 16,204,000Virginia TechACC 2010 - s14
  • 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.Virginia TechACC 2010 - s15
  • 16. Thank You ! GAME THEORYVirginia TechACC 2010 - s16

×