2010 IEEE American Control Conference

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2010 IEEE American Control Conference

  1. 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. 2. Outline  Motivations  Game Theory  System Model GAME THEORY  Control Derivation  Simulation and Results  ConclusionsVirginia Tech ACC 2010 - s2
  3. 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. 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. 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. 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. 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. 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. 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. 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. 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. 12. Simulation  Vehicle: 2-axle Van  Maneuver: standard “Moose” test at 60 kphVirginia TechACC 2010 - s12
  13. 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. 14. Results unit strategy Nash LQR Driver 97,363 162,060 Controller 9,734,700 16,204,000Virginia TechACC 2010 - s14
  15. 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. 16. Thank You ! GAME THEORYVirginia TechACC 2010 - s16

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