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

$\begin{array}{llcl} \min_{x, w} & t_{f} \\ s.t. & \dot{x} (t) & = & f (x (t), u (t), μ (t)), \\ x (0) & = & x_{0}, \\ μ (t) & \in & {1, 2, 3, 4, 5} . \end{array}$

Parameters

These fixed values are used within the model.

$\begin{array}{llcl} P a r a m e t e r & V a l u e & U n i t & D e s c r i p t i o n \\ m & 1.239 \cdot 1 0^{3} & kg & M a s s o f t h e c a r \\ g & 9.81 & \frac{m}{s^{2}} & G r a v i t y c o n s t a n t \\ l_{f} & 1.19016 & m & F r o n t w h e e l d i s t a n c e t o c e n t e r o f g r a v i t y \\ l_{r} & 1.37484 & m & R e a r w h e e l d i s t a n c e t o c e n t e r o f g r a v i t y \\ e_{SP} & 0.5 & m & D r a g m o u n t p o i n t d i s t a n c e t o c e n t e r o f g r a v i t y \\ R & 0.302 & m & W h e e l r a d i u s \\ I_{zz} & 1.752 \cdot 1 0^{3} & {kg m}^{2} & M o m e n t o f i n e r t i a \\ c_{w} & 0.3 & - - & A i r d r a g c o e f f i c i e n t \\ ρ & 1.249512 & \frac{kg}{m^{3}} & A i r d e n s i t y \\ A & 1.4378946874 & m^{2} & E f f e c t i v e f l o w s u r f a c e \\ i_{g}^{1} & 3.09 & - - & T r a n s m i s s i o n r a t i o o f f i r s t g e a r \\ i_{g}^{2} & 2.002 & - - & T r a n s m i s s i o n r a t i o o f s e c o n d g e a r \\ i_{g}^{3} & 1.33 & - - & T r a n s m i s s i o n r a t i o o f t h i r d g e a r \\ i_{g}^{4} & 1.0 & - - & T r a n s m i s s i o n r a t i o o f f o u r t h g e a r \\ i_{g}^{5} & 0.805 & - - & T r a n s m i s s i o n r a t i o o f f i f t h g e a r \\ i_{t} & 3.91 & - - & E n g i n e t o r q u e t r a n s m i s s i o n r a t i o \\ B_{f} & 1.096 \cdot 1 0^{1} & - - & P a c e j k a c o e f f i c i e n t s (s t i f f n e s s) \\ B_{r} & 1.267 \cdot 1 0^{1} & - - \\ C_{f} & 1.3 & - - & P a c e j k a c o e f f i c i e n t s (s h a p e) \\ C_{r} & 1.3 & - - \\ D_{f} & 4.5604 \cdot 1 0^{3} & - - & P a c e j k a c o e f f i c i e n t s (p e a k) \\ D_{r} & 3.94781 \cdot 1 0^{3} & - - \\ E_{f} & - 0.5 & - - & P a c e j k a c o e f f i c i e n t s (c u r v a t u r e) \\ E_{r} & - 0.5 & - - \end{array}$

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>.

Source Code

C

Variants

See testdrive overview page.