Difference between revisions of "Car testdrive (elliptic track)"
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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. | 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}(t) & = & f(t, x(t), u(t), \mu(t)), \\ | ||
| + | & x(t_0) &=& x_0, \\ | ||
| + | & r(t,x(t),u(t)) &\geq& 0, \\ | ||
| + | & \mu(t) &\in& \{1, 2, 3, 4, 5\}. | ||
| + | \end{array} | ||
| + | </math> | ||
| + | |||
| + | == Source Code == | ||
| + | |||
| + | |||
| + | == Variants == | ||
| + | |||
| + | See testdrive [[Car testdrive | overview page]]. | ||
== References == | == References == | ||
<bibreferences/> | <bibreferences/> | ||
Revision as of 15:00, 25 November 2008
| Car testdrive (elliptic track) | |
|---|---|
| State dimension: | 1 |
| Differential states: | 7 |
| Continuous control functions: | 3 |
| Discrete control functions: | 1 |
| Interior point inequalities: | 7 |
The elliptic track testdrive problem is a time optimal periodic control problem with gear shift, first introduced in <bibref>Sager2009a</bibref>.
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 
the steering wheel's angular velocity. The force 
controls the total braking force, while the accelerator pedal position 
is translated into an accelerating force. Finally, the selected gear 
influences the effective engine torque's transmission.
Resulting MIOCP
For ![t \in [t_0, t_f]](https://mintoc.de/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)), \\
& x(t_0) &=& x_0, \\
& r(t,x(t),u(t)) &\geq& 0, \\
& \mu(t) &\in& \{1, 2, 3, 4, 5\}.
\end{array}](https://mintoc.de/images/math/0/0/3/00382d7408dddcf936f8d70afb496034.png)
Source Code
Variants
See testdrive overview page.
References
<bibreferences/>
