• Images
• Videos
• Documents
• Books
• Articles

Model definition

As a car-following model, the IDM describes the dynamics of the positions and velocities of single vehicles.

The influencing factors of the IDM are the speed of the vehicle, the bumper-to-bumper gap to the leading vehicle, and the relative speed of the two vehicles. The model output is the acceleration chosen by the driver for that situation. The model parameters describe the driving style.¹

The IDM equation, for the dynamics of vehicle α {\displaystyle \alpha } , reads as follows²³:

v ˙ α = a ( 1 − ( v α v 0 ) δ − ( s ∗ ( v α , Δ v α ) s α ) 2 ) {\displaystyle {\dot {v}}_{\alpha }=a\left(1-\left({\frac {v_{\alpha }}{v_{0}}}\right)^{\delta }-\left({\frac {s^{*}(v_{\alpha },\Delta v_{\alpha })}{s_{\alpha }}}\right)^{2}\right)} with  s ∗ ( v α , Δ v α ) = s 0 + v α T + v α Δ v α 2 a b {\displaystyle {\text{with }}s^{*}(v_{\alpha },\Delta v_{\alpha })=s_{0}+v_{\alpha }\,T+{\frac {v_{\alpha }\,\Delta v_{\alpha }}{2\,{\sqrt {a\,b}}}}}

where:

• v {\displaystyle v} is the velocity of the vehicle;
• s {\displaystyle s} is the bumper-to-bumper distance to the next vehicle;
• Δ v {\displaystyle \Delta v} is velocity difference (approaching rate) to the next vehicle;
• v 0 {\displaystyle v_{0}} is desired velocity of the vehicle — the speed it would drive at in free traffic;

• s 0 {\displaystyle s_{0}} is the minimum gap to the next vehicle — kept when traffic is at a standstill;
• T {\displaystyle T} is the minimum possible time to the next vehicle;
• a {\displaystyle a} is the maximum vehicle acceleration;
• b {\displaystyle b} is the target deceleration rate (i.e. comfortable braking rate), a positive number.

The exponent δ {\displaystyle \delta } is usually set to 4.

Model characteristics

The acceleration of vehicle α {\displaystyle \alpha } can be separated into a free road term and an interaction term:

v ˙ α free = a ( 1 − ( v α v 0 ) δ ) {\displaystyle \qquad {\dot {v}}_{\alpha }^{\text{free}}=a\,\left(1-\left({\frac {v_{\alpha }}{v_{0}}}\right)^{\delta }\right)}

v ˙ α int = − a ( s ∗ ( v α , Δ v α ) s α ) 2 = − a ( s 0 + v α T s α + v α Δ v α 2 a b s α ) 2 {\displaystyle \qquad {\dot {v}}_{\alpha }^{\text{int}}=-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 {\displaystyle s_{\alpha }} is large and the vehicle's acceleration is dominated by the free road term, which is approximately equal to a {\displaystyle a} for low velocities and vanishes as v α {\displaystyle v_{\alpha }} approaches v 0 {\displaystyle v_{0}} . Therefore, a single vehicle on a free road will asymptotically approach its desired velocity v 0 {\displaystyle v_{0}} .

• Behavior at high approaching rates: For large velocity differences, the interaction term is governed by − a ( v α Δ v α ) 2 / ( 2 a b s α ) 2 = − ( v α Δ v α ) 2 / ( 4 b s α 2 ) {\displaystyle -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 {\displaystyle 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 α T ) 2 / s α 2 {\displaystyle -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.

Solution example

Let's assume a ring road with 50 vehicles. Then, vehicle 1 will follow vehicle 50. Initial speeds are given and since all vehicles are considered equal, vector ODEs are further simplified to:

x ˙ = d x d t = v {\displaystyle {\dot {x}}={\frac {\mathrm {d} x}{\mathrm {d} t}}=v} v ˙ = d v d t = a ( 1 − ( v v 0 ) δ − ( s ∗ ( v , Δ v ) s ) 2 ) {\displaystyle {\dot {v}}={\frac {\mathrm {d} v}{\mathrm {d} t}}=a\,\left(1-\left({\frac {v}{v_{0}}}\right)^{\delta }-\left({\frac {s^{*}(v,\Delta v)}{s}}\right)^{2}\right)} with  s ∗ ( v , Δ v ) = s 0 + v T + v Δ v 2 a b {\displaystyle {\text{with }}s^{*}(v,\Delta v)=s_{0}+v\,T+{\frac {v\,\Delta v}{2\,{\sqrt {a\,b}}}}}

For this example, the following values are given for the equation's parameters, in line with the original calibrated model.

Variable	Description	Value
v 0 {\displaystyle v_{0}}	Desired velocity	30 m/s
T {\displaystyle T}	Safe time headway	1.5 s
a {\displaystyle a}	Maximum acceleration	0.73 m/s2
b {\displaystyle b}	Comfortable Deceleration	1.67 m/s2
δ {\displaystyle \delta }	Acceleration exponent	4
s 0 {\displaystyle s_{0}}	Minimum distance	2 m
-	Vehicle length	5 m

The two ordinary differential equations are solved using Runge–Kutta methods of orders 1, 3, and 5 with the same time step, to show the effects of computational accuracy in the results.

This comparison shows that the IDM does not show extremely irrealistic properties such as negative velocities or vehicles sharing the same space even for from a low order method such as with the Euler's method (RK1). However, traffic wave propagation is not as accurately represented as in the higher order methods, RK3 and RK 5. These last two methods show no significant differences, which lead to conclude that a solution for IDM reaches acceptable results from RK3 upwards and no additional computational requirements would be needed. Nonetheless, when introducing heterogeneous vehicles and both jam distance parameters, this observation could not suffice.

See also

• Gipps' model
• Newell's car-following model
• Microscopic traffic flow model
• List of Runge–Kutta methods
• Traffic simulation

External links

• Interactive JS & HTML5 implementation of the intelligent driver model showing signalized intersections
• Interactive JS & HTML5 implementation showing stop & Go waves on a ring road. Links to other scenarios can be found there
• Interactive JS & HTML5 implementation showing the effect of different traffic rules at roundabouts
• Common values of the IDM parameters and hints for the simulation
• Textbook on traffic flow dynamics with a free downloadable chapter on the IDM

References

1. traffic-simulation.de https://traffic-simulation.de/info/info_IDM.html. Retrieved 2025-06-20. {{cite web}}: Missing or empty |title= (help) https://traffic-simulation.de/info/info_IDM.html ↩

2. Treiber, Martin; Hennecke, Ansgar; Helbing, Dirk (2000-08-01). "Congested traffic states in empirical observations and microscopic simulations". Physical Review E. 62 (2): 1805–1824. arXiv:cond-mat/0002177. Bibcode:2000PhRvE..62.1805T. doi:10.1103/PhysRevE.62.1805. ISSN 1063-651X. PMID 11088643. S2CID 1100293. https://link.aps.org/doi/10.1103/PhysRevE.62.1805 ↩

3. traffic-simulation.de https://traffic-simulation.de/info/info_IDM.html. Retrieved 2025-06-20. {{cite web}}: Missing or empty |title= (help) https://traffic-simulation.de/info/info_IDM.html ↩