Intelligent Driver Model

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

1 Model definition
2 Model characteristics
3 References
4 External links

[edit] Model definition

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

$\dot{x}_\alpha = \frac{\mathrm{d}x_\alpha}{\mathrm{d}t} = v_\alpha$

$\dot{v}_\alpha = \frac{\mathrm{d}v_\alpha}{\mathrm{d}t} = a\,\left( 1 - \left(\frac{v_\alpha}{v_0}\right)^\delta - \left(\frac{s^*(v_\alpha,\Delta v_\alpha)}{s_\alpha}\right)^2 \right)$

$\text{with }s^*(v_\alpha,\Delta v_\alpha) = s_0 + v_\alpha\,T + \frac{v_\alpha\,\Delta v_\alpha}{2\,\sqrt{a\,b}}$

$v 0$ , $s 0$ , $T$ , $a$ , 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 $δ$ is usually set to 4.

[edit] Model characteristics

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

$\dot{v}^\text{free}_\alpha = a\,\left( 1 - \left(\frac{v_\alpha}{v_0}\right)^\delta \right) \qquad\dot{v}^\text{int}_\alpha = -a\,\left(\frac{s^*(v_\alpha,\Delta v_\alpha)}{s_\alpha}\right)^2 = -a\,\left(\frac{s_0 + v_\alpha\,T}{s_\alpha} + \frac{v_\alpha\,\Delta v_\alpha}{2\,\sqrt{a\,b}\,s_\alpha}\right)^2$

Free road behavior: On a free road, the distance to the leading vehicle $s α$ 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 $v α$ 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.

[edit] References

Treiber, Martin; Hennecke, Ansgar & Helbing, Dirk (2000), “Congested traffic states in empirical observations and microscopic simulations”, Physical Review E 62 (2): 1805–1824