View source for Car testdrive (elliptic track)

{{Dimensions
|nd        = 1
|nx        = 7
|nu        = 3
|nw        = 1
|nc        = 1
|nri       = 7
}}

The elliptic track testdrive problem is a time optimal periodic control problem with gear shift, first introduced in <bib id="Sager2009a" />. 

== Mathematical formulation ==

The mathematical equations form a small-scale [[:Category:ODE model|ODE model]] as presented for the [[Car testdrive (lane change manoeuvre) | lane change manoeuvre]].

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 <math>w_\delta</math> the steering wheel's angular velocity. The force <math>F_\text{B}</math> controls the total braking force, while the accelerator pedal position <math>\phi</math> is translated into an accelerating force. Finally, the selected gear <math>\mu</math> influences the effective engine torque's transmission.

== Resulting MIOCP ==

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

<math>
\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), \\
 & c_\text{x}(t_0) &=& c_\text{x}(t_f), \\
 & c_\text{y}(t_0) &=& c_\text{y}(t_f), \\
 & v(t_0) &=& v(t_f), \\
 & \beta(t_0) &=& \beta(t_f) - 2\pi, \\
 & \psi(t_0) &=& \psi(t_f), \\
 & \delta(t_0) &=& \delta(t_f), \\
 & r(t,x,u) &\geq& 0, \\
 & \mu(t) &\in&  \{1, 2, 3, 4, 5\}.
\end{array} 
</math>

== Variants ==

See testdrive [[Car testdrive | overview page]].
== References ==
<biblist />

<!--List of all categories this page is part of. List characterization of solution behavior, model properties, ore presence of implementation details (e.g., AMPL for AMPL model) here -->
[[Category:MIOCP]]
[[Category:ODE model]]
[[Category:Minimum time]]
[[Category:Bang bang]]
[[Category:Transport]]
[[Category:Periodic]]