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.

Motivation

Mathematical formulation

We consider 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 $w_{δ}$ the steering wheel's angular velocity. The force $F_{B}$ 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 $t \in [t_{0}, t_{f}]$ 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.

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

Test course

The double-lane change manoeuvre presented in <bibref>Gerdts2005</bibref> is realized by constraining the car's position onto a prescribed track at any time $t \in [t_{0}, t_{f}]$ . 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 $P_{l} (x)$ and $P_{r} (x)$ modeling the top and bottom track boundary, given a horizontal position $x$ .

$\begin{aligned} P_{l} (x) & : = & {\begin{array}{llrcl} 0 & if & x & \leq 44, \\ 4 h_{2} (x - 44)^{3} & if & 44 < & x & \leq 44.5, \\ 4 h_{2} (x - 45)^{3} + h_{2} & if & 44.5 < & x & \leq 45, \\ h_{2} & if & 45 < & x & \leq 70, \\ 4 h_{2} (70 - x)^{3} + h_{2} & if & 70 < & x & \leq 70.5, \\ 4 h_{2} (71 - x)^{3} & if & 70.5 < & x & \leq 71, \\ 0 & if & 71 < & x . \end{array} \\ P_{u} (x) & : = & {\begin{array}{llrcl} h_{1} & if & x & \leq 15, \\ 4 (h_{3} - h_{1}) (x - 15)^{3} + h_{1} & if & 15 < & x & \leq 15.5, \\ 4 (h_{3} - h_{1}) (x - 16)^{3} + h_{3} & if & 15.5 < & x & \leq 16, \\ h_{3} & if & 16 < & x & \leq 94, \\ 4 (h_{3} - h_{4}) (94 - x)^{3} + h_{3} & if & 94 < & x & \leq 94.5, \\ 4 (h_{3} - h_{4}) (95 - x)^{3} + h_{4} & if & 94.5 < & x & \leq 95, \\ h_{4} & if & 95 < & x . \end{array} \end{aligned}$

where $B = 1.5 m$ is the car's width and

$h_{1} : = 1.1 B + 0.25, h_{2} : = 3.5, h_{3} : = 1.2 B + 3.75, h_{4} : = 1.3 B + 0.25 .$

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.

References