Car testdrive (elliptic track)
From Mintoc
| Car testdrive (elliptic track) | |
|---|---|
| State dimension: | 1 |
| Differential states: | 7 |
| Continuous control functions: | 3 |
| Discrete control functions: | 1 |
| Interior point inequalities: | 7 |
Contents |
Mathematical formulation
The mathematical equations form a small-scale ODE model as presented for the 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 wδ the steering wheel's angular velocity. The force FB controls the total braking force, while the accelerator pedal position φ is translated into an accelerating force. Finally, the selected gear μ influences the effective engine torque's transmission.
Resulting MIOCP
For ![t \in [t_0, t_f]](/images/math/5/5/8/55823791d9100bcb5461801aff4f6edd.png)
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}(t) & = & f(t, x(t), u(t), \mu(t)), \\
& 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(t),u(t)) &\geq& 0, \\
& \mu(t) &\in& \{1, 2, 3, 4, 5\}.
\end{array}](/images/math/e/5/9/e59c3fd245fe4ea4c28a631bc39744dc.png)
Variants
See testdrive overview page.
