Car testdrive (lane change manoeuvre)

Car testdrive (lane change manoeuvre)
State dimension:	1
Differential states:	7
Continuous control functions:	3
Discrete control functions:	1
Interior point equalities:	7

The mathematical equations form a small-scale ODE model.

Mathematical formulation

For $t \in [t_0, t_f]$ almost everywhere the mixed-integer optimal control problem is given by

$\begin{array}{llcl} \displaystyle \min_{x, w} & t_f \\[1.5ex] \mbox{s.t.} & \dot{x}(t) & = & f( x(t), u(t), \mu(t)), \\ & x(0) &=& x_0, \\ & \mu(t) &\in& \{1, 2, 3, 4, 5\}. \end{array}$

Parameters

These fixed values are used within the model.

$\begin{array}{llcl} Parameter & Value & Unit & Description \\ m & 1.239\cdot 10^3 & \text{kg} & Mass of the car \\ g & 9.81 & \frac{\text{m}}{\text{s}^2} & Gravity constant \\ l_\text{f} & 1.19016 & \text{m} & Front wheel distance to center of gravity \\ l_\text{r} & 1.37484 & \text{m} & Rear wheel distance to center of gravity \\ e_\text{SP} & 0.5 & \text{m} & Drag mount point distance to center of gravity \\ R & 0.302 & \text{m} & Wheel radius \\ I_\text{zz} & 1.752\cdot 10^3 & \text{kg m}^2 & Moment of inertia \\ c_\text{w} & 0.3 & -- & Air drag coefficient \\ \rho & 1.249512 & \frac{\text{kg}}{\text{m}^3} & Air density \\ A & 1.4378946874 & \text{m}^2 & Effective flow surface \\ i^1_\text{g} & 3.09 & -- & Transmission ratio of first gear \\ i^2_\text{g} & 2.002 & -- & Transmission ratio of second gear \\ i^3_\text{g} & 1.33 & -- & Transmission ratio of third gear \\ i^4_\text{g} & 1.0 & -- & Transmission ratio of fourth gear \\ i^5_\text{g} & 0.805 & -- & Transmission ratio of fifth gear \\ i_\text{t} & 3.91 & -- & Engine torque transmission ratio \\ B_\text{f} & 1.096\cdot 10^1 & -- & Pacejka coefficients (stiffness) \\ B_\text{r} & 1.267\cdot 10^1 & -- & \\ C_\text{f} & 1.3 & -- & Pacejka coefficients (shape) \\ C_\text{r} & 1.3 & -- & \\ D_\text{f} & 4.5604\cdot 10^3 & -- & Pacejka coefficients (peak) \\ D_\text{r} & 3.94781\cdot 10^3 & -- & \\ E_\text{f} & -0.5 & -- & Pacejka coefficients (curvature) \\ E_\text{r} & -0.5 & -- & \\ \end{array}$

Reference Solutions

Reference solutions for the case of a fixed time-grid are given in <bibref>Gerdts2005</bibref>. Solutions for a non-fixed time grid are given in <bibref>Gerdts2006</bibref>.

Variants

See testdrive overview page.