Steering Control

Steering Control of Autonomous Vehicles in Obstacle Avoidance Maneuvers.

Contents

Vehicle model

Bicycle model

Nonlinear model

State vector

\[ {\bf x} = \left[ \begin{array}{c} {\rm x}_1 \\ {\rm x}_2 \\ {\rm x}_3 \\ {\rm x}_4 \\ {\rm x}_5 \\ {\rm x}_6 \end{array} \right] = \left[ \begin{array}{c} x \\ y \\ \psi \\ v_{\rm T} \\ \alpha_{\rm T} \\ \dot{\psi} \end{array} \right] \]

State equations

\[ \dot{{\rm x}}_1 = {\rm x}_4 \cos \left( {\rm x}_3 + {\rm x}_5 \right) \]

\[ \dot{{\rm x}}_2 = {\rm x}_4 \sin \left( {\rm x}_3 + {\rm x}_5 \right) \]

\[ \dot{{\rm x}}_3 = {\rm x}_6 \]

\[ \dot{{\rm x}}_4 = \frac{F_{y,{\rm F}} \sin \left( {\rm x}_5 - \delta \right) + F_{y,{\rm R}} \sin {\rm x}_5}{m_{T}} \]

\[ \dot{{\rm x}}_5 = \frac{F_{y,{\rm F}} \cos \left( {\rm x}_5 - \delta \right) + F_{y,{\rm R}} \cos \alpha_{\rm T} - m_{T} {\rm x}_4 {\rm x}_6}{m_{T} {\rm x}_4} \]

\[ \dot{{\rm x}}_6 = \frac{F_{y,{\rm F}} a \cos \delta - F_{y,{\rm R}} b}{I_{T}} \]

Slip angles

\[ \alpha_{\rm F} = \arctan \left( \frac{v_{\rm T} \sin \alpha_{\rm T} + a \dot{\psi}}{ v_{\rm T} \cos \alpha_{\rm T}} \right) - \delta \]

\[ \alpha_{\rm R} = \arctan \left( \frac{v_{\rm T} \sin \alpha_{\rm T} - b \dot{\psi}}{ v_{\rm T} \cos \alpha_{\rm T}} \right) \]

Linear model

\[ \dot{x} = v_{\rm T} \]

\[ \dot{y} = v_{{\rm T},0} \left( \psi + \alpha_{{\rm T}}\right) \]

\[ \dot{\psi} = \dot{\psi} \]

\[ \dot{v}_{\rm T} = 0 \]

\[ \dot{\alpha}_{\rm T} = \frac{F_{y,{\rm F}} + F_{y,{\rm R}}}{m_{T} v_{{\rm T},0}} - \dot{\psi} \]

\[ \ddot{\psi} = \frac{a F_{y,{\rm F}} - b F_{y,{\rm R}}}{I_{T}} \]

Neglecting equations of \(x\) and \(v_T\)

\[ \left[ \begin{array}{c} \dot{y} \\ \dot{\psi} \\ \dot{\alpha}_T \\ \ddot{\psi} \end{array} \right] = \left[ \begin{array}{cccc} 0 & v_{T,0} & v_{T,0} & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & -\frac{K_F+K_R}{m_T v_{T,0}} & - \frac{m_T v_{T,0} + \frac{a K_F - b K_R}{v_{T,0}}}{m_T v_{T,0}} \\ 0 & 0 & - \frac{a K_F - b K_R}{I_T} & - \frac{a^2 K_F + b^2 K_R}{I_T v_{T,0}} \end{array} \right] \left[ \begin{array}{c} y \\ \psi \\ \alpha_T \\ \dot{\psi} \end{array} \right] + \left[ \begin{array}{c} 0 \\ 0 \\ \frac{K_F}{m_T v_{T,0}} \\ \frac{a K_F}{I_T} \end{array} \right] \delta \]

Slip angles

\[ \alpha_{{\rm F},lin} = \alpha_{{\rm T}} + \frac{a}{v_{{\rm T},0}} \dot{\psi} - \delta \]

\[ \alpha_{{\rm F},lin} = \alpha_{{\rm T}} - \frac{b}{v_{{\rm T},0}} \dot{\psi} \]

Tire model

Typical characteristic curve and slip angle definition

Pacejka

\[ F_{y} = D \sin \left[ C \arctan{B \alpha - E( B \alpha -\arctan(B \alpha))} \right] \]

Linear

\[ F_ y = K \alpha \]

Comparison of tire models

Plant model

Nonlinear vehicle + Pacejka tire

  VehicleSimpleNonlinear with properties:

        mT: 1300
        IT: 10000
         a: 1.6154
         b: 1.8846
       mF0: 700
       mR0: 600
        lT: 3.5000
        nF: 1
        nR: 1
        wT: 1.8000
       muy: 1
      tire: [1x1 VehicleDynamicsLateral.TirePacejka]
    deltaf: @ControlLaw
       Fxf: 0
       Fxr: 0

Controller design

Vehicle parameters

Linear system

A

         0   16.7000   16.7000         0
         0         0         0    1.0000
         0         0   -8.3915   -0.9324
         0         0    2.4522   -3.3606

B

         0
         0
    4.1958
   14.7147

C

     1     0     0     0

LQR design

Q

     3     0     0     0
     0     1     0     0
     0     0     1     0
     0     0     0     1

R

     1

Klqr

    1.7321    6.8987    2.5832    0.5898

Pole placement design

Kplace

    0.7936    6.6882    1.6107    0.5090

Control law

\[\delta = - {\bf K} {\bf z} + K_1 r \]

Double Lane Change Maneuver

Control - Step y

Reference - r = 2 m

\[ \delta_{max} = \pm 70 deg \]

See Also

Home