# Car testdrive (lane change manoeuvre)

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

The testdrive control problem is a time optimal double lane change maneouvre with gear shift. It has been introduced as a benchmark problem for mixed-integer optimal control by [Gerdts2005]Author: M. Gerdts
Journal: Optimal Control Applications and Methods
Pages: 1--18
Title: Solving mixed-integer optimal control problems by Branch\&Bound: A case study from automobile test-driving with gear shift
Volume: 26
Year: 2005 .

## Mathematical formulation

The mathematical equations form a small-scale ODE model.

The vehicle dynamics are based on a single-track model, derived under the simplifying assumption that rolling and pitching of the car body can be neglected. Consequentially, only a single front and rear wheel is modeled, located in the virtual center of the original two wheels. Motion of the car body is considered on the horizontal plane only.

Four controls represent the driver's choice on steering and velocity. We denote with $w_\delta$ the steering wheel's angular velocity. The force $F_\text{B}$ controls the total braking force, while the accelerator pedal position $\phi$ is translated into an accelerating force. Finally, the selected gear $\mu$ influences the effective engine torque's transmission.

## Resulting MIOCP

For $t \in [t_0, t_f]$ almost everywhere the mixed-integer optimal control problem is given by $\begin{array}{llcl} \displaystyle \min_{x(\cdot), u(\cdot), \mu(\cdot)} & t_\text{f} \\[1.5ex] \mbox{s.t.} & \dot{x} & = & f(t, x, u, \mu), \\ & x(t_0) &=& x_0, \\ & r(t,x,u) &\geq& 0, \\ & \mu(t) &\in& \{1, 2, 3, 4, 5\}. \end{array}$

## Parameters

These fixed values are used within the model.

Symbol Value Unit Description $m$ 1.239e+3 kg Mass of the car $g$ 9.81 m/s^2 Gravity constant $l_\text{f}$ 1.19016 m Front wheel distance to center of gravity $l_\text{r}$ 1.37484 m Rear wheel distance to center of gravity $e_\text{SP}$ 0.5 m Drag mount point distance to center of gravity $R$ 0.302 m Wheel radius $I_\text{zz}$ 1.752e+3 kg m^2 Moment of inertia $c_\text{w}$ 0.3 - Air drag coefficient $\varrho$ 1.249512 kg/m^3 Air density $A$ 1.4378946874 m^2 Effective flow surface $i_\text{g}$ 3.09, 2.002, 1.33, 1.0, 0.805 - Transmission ratios for the five gears $i_\text{t}$ 3.91 - Engine transmission ratio $B_\text{f}$ 1.096e+1 - Pacejka coefficients (stiffness) $B_\text{r}$ 1.267e+1 - $C_\text{f}$ 1.3 - Pacejka coefficients (shape) $C_\text{r}$ 1.3 - $D_\text{f}$ 4.5604e+3 - Pacejka coefficients (peak) $D_\text{r}$ 3.94781e+3 - $E_\text{f}$ -0.5 - Pacejka coefficients (curvature) $E_\text{r}$ -0.5 -

## Test course

The double-lane change manoeuvre presented in [Gerdts2005]Author: M. Gerdts
Journal: Optimal Control Applications and Methods
Pages: 1--18
Title: Solving mixed-integer optimal control problems by Branch\&Bound: A case study from automobile test-driving with gear shift
Volume: 26
Year: 2005 is realized by constraining the car's position onto a prescribed track at any time $t\in[t_0,t_\text{f}]$. Starting in the left position with an initial prescribed velocity, the driver is asked to manage a change of lanes modeled by an offset of 3.5 meters in the track. Afterwards he is asked to return to the starting lane. This manoeuvre can be regarded as an overtaking move or as an evasive action taken to avoid hitting an obstacle suddenly appearing on the starting lane.

From a mathematical point of view, the test track is described by setting up piecewise cubic spline functions $P_\text{l}(x)$ and $P_\text{r}(x)$ modeling the top and bottom track boundary, given a horizontal position $x$. \begin{align} P_\text{l}(x) &:=& \left\{ \begin{array}{llrcl} 0 & \text{if } & & x & \leq 44, \\ 4\; h_2\; (x-44)^3 & \text{if } & 44 < & x & \leq 44.5, \\ 4\; h_2\; (x-45)^3 + h_2 & \text{if } & 44.5 < & x & \leq 45, \\ h_2 & \text{if } & 45 < & x & \leq 70, \\ 4\; h_2\; (70-x)^3 + h_2 & \text{if } & 70 < & x & \leq 70.5, \\ 4\; h_2\; (71-x)^3 & \text{if } & 70.5 < & x & \leq 71, \\ 0 & \text{if } & 71 < & x. & \\ \end{array} \right. \\ P_\text{u}(x) &:=& \left\{ \begin{array}{llrcl} h_1 & \text{if } & & x & \leq 15, \\ 4\; (h_3-h_1)\; (x-15)^3 + h_1 & \text{if } & 15 < & x & \leq 15.5, \\ 4\; (h_3-h_1)\; (x-16)^3 + h_3 & \text{if } & 15.5 < & x & \leq 16, \\ h_3 & \text{if } & 16 < & x & \leq 94, \\ 4\; (h_3-h_4)\; (94-x)^3 + h_3 & \text{if } & 94 < & x & \leq 94.5, \\ 4\; (h_3-h_4)\; (95-x)^3 + h_4 & \text{if } & 94.5 < & x & \leq 95, \\ h_4 & \text{if } & 95 < & x. & \\ \end{array} \right. \end{align}

where $B=1.5\;\text{m}$ is the car's width and $h_1 := 1.1\; B + 0.25, \quad h_2 := 3.5, \quad h_3 := 1.2\; B + 3.75,\quad h_4 := 1.3\; B + 0.25.$

## Reference Solutions

Reference solutions for the case of a fixed time-grid are given in [Gerdts2005]Author: M. Gerdts
Journal: Optimal Control Applications and Methods
Pages: 1--18
Title: Solving mixed-integer optimal control problems by Branch\&Bound: A case study from automobile test-driving with gear shift
Volume: 26
Year: 2005 . Solutions for a non-fixed time grid are given in [Gerdts2006]Author: M. Gerdts
Journal: Optimal Control Applications and Methods
Number: 3
Pages: 169--182
Title: A variable time transformation method for mixed-integer optimal control problems
Volume: 27
Year: 2006 .

## Source Code

Model descriptions are available in

## Variants

See testdrive overview page.