Intelligent Driver Model

In traffic flow modeling, the Intelligent Driver Model (IDM) is a time-continuous car-following model for the simulation of freeway and urban traffic.

Model definition

As a car-following model, the IDM describes the dynamics of the positions and velocities of single vehicles. For vehicle $alpha$, $x\_alpha$ denotes its position at time $t$, and $v\_alpha$ its velocity. Furthermore, $l\_alpha$ gives the length of the vehicle. To simplify notation, we define the "net distance" $s\_alpha\; :=\; x\_\{alpha-1\}\; -\; x\_alpha\; -\; l\_\{alpha-1\}$, where $alpha\; -\; 1$ refers to the vehicle directly in front of vehicle $alpha$, and the velocity difference, or "approaching rate", $Delta\; v\_alpha\; :=\; v\_alpha\; -\; v\_\{alpha-1\}$. The dynamics of vehicle $alpha$ are then described by the following two ordinary differential equations:

:$dot\{x\}\_alpha\; =\; frac\{mathrm\{d\}x\_alpha\}\{mathrm\{d\}t\}\; =\; v\_alpha$
:$dot\{v\}\_alpha\; =\; frac\{mathrm\{d\}v\_alpha\}\{mathrm\{d\}t\}\; =\; a,left(\; 1\; -\; left(frac\{v\_alpha\}\{v\_0\}\; ight)^delta\; -\; left(frac\{s^*(v\_alpha,Delta\; v\_alpha)\}\{s\_alpha\}\; ight)^2\; ight)$
:$ext\{with\; \}s^*(v\_alpha,Delta\; v\_alpha)\; =\; s\_0\; +\; v\_alpha,T\; +\; frac\{v\_alpha,Delta\; v\_alpha\}\{2,sqrt\{a,b$

$v\_0$, $s\_0$, $T$, $a$, and $b$ are model parameters which have the following meaning:
* "desired velocity" $v\_0$: the velocity the vehicle would drive at in free traffic
* "minimum spacing" $s\_0$: a minimum net distance that is kept even at a complete stand-still in a traffic jam
* "desired time headway" $T$: the desired time headway to the vehicle in front
* "acceleration" $a$
* "comfortable braking deceleration" $b$The exponent $delta$ is usually set to 4.

Model characteristics

The acceleration of vehicle $alpha$ can be separated into a "free road term" and an "interaction term":

:$dot\{v\}^\; ext\{free\}\_alpha\; =\; a,left(\; 1\; -\; left(frac\{v\_alpha\}\{v\_0\}\; ight)^delta\; ight)qquaddot\{v\}^\; ext\{int\}\_alpha\; =\; -a,left(frac\{s^*(v\_alpha,Delta\; v\_alpha)\}\{s\_alpha\}\; ight)^2=\; -a,left(frac\{s\_0\; +\; v\_alpha,T\}\{s\_alpha\}\; +\; frac\{v\_alpha,Delta\; v\_alpha\}\{2,sqrt\{a,b\},s\_alpha\}\; ight)^2$

"Free road behavior:" On a free road, the distance to the leading vehicle $s\_alpha$ is large and the vehicle's acceleration is dominated by the free road term, which is approximately equal to $a$ for low velocities and vanishes as $v\_alpha$ approaches $v\_0$. Therefore, a single vehicle on a free road will asymptotically approach its desired velocity $v\_0$.

"Behavior at high approaching rates:" For large velocity differences, the interaction term is governed by $-a,(v\_alpha,Delta\; v\_alpha)^2,/,(2,sqrt\{a,b\},s\_alpha)^2\; =\; -(v\_alpha,Delta\; v\_alpha)^2,/,(4,b,s\_alpha^2)$. This leads to a driving behavior that compensates velocity differences while trying not to brake much harder than the comfortable braking deceleration $b$.

"Behavior at small net distances:" For negligible velocity differences and small net distances, the interaction term is approximately equal to $-a,(s\_0\; +\; v\_alpha,T)^2,/,s\_alpha^2$, which resembles a simple repulsive force such that small net distances are quickly enlarged towards an equilibrium net distance.

References

Citation
last1 = Treiber | first1 = Martin
last2 = Hennecke | first2 = Ansgar
last3 = Helbing | first3 = Dirk
title = Congested traffic states in empirical observations and microscopic simulations
journal = Physical Review E
volume = 62
issue = 2
pages = 1805–1824
year = 2000

External links

[http://www.traffic-simulation.de/ Interactive Java-Applet implementing the Intelligent Driver Model]