Intelligent Driver Model
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In
traffic flow In mathematics and transportation engineering, traffic flow is the study of interactions between travellers (including pedestrians, cyclists, drivers, and their vehicles) and infrastructure (including highways, signage, and traffic control devi ...
modeling, the intelligent driver model (IDM) is a time-continuous car-following model for the simulation of freeway and urban traffic. It was developed by Treiber, Hennecke and Helbing in 2000 to improve upon results provided with other "intelligent" driver models such as
Gipps' model Gipps' model is a mathematical model for describing car-following behaviour by motorists in the United Kingdom. The model is named after Peter G. Gipps who developed it in the late-1970s under S.R.C. grants at the Transport Operations Research Gr ...
, which loses realistic properties in the deterministic limit.


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_ - x_\alpha - l_, 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_. For a simplified version of the model, the dynamics of vehicle \alpha are then described by the following two
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...
s: :\dot_\alpha = \frac = v_\alpha :\dot_\alpha = \frac = a\,\left( 1 - \left(\frac\right)^\delta - \left(\frac\right)^2 \right) :\texts^*(v_\alpha,\Delta v_\alpha) = s_0 + v_\alpha\,T + \frac 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 desired net distance. A car can't move if the distance from the car in the front is not at least s_0 * ''desired time headway'' T: the minimum possible time to the vehicle in front * ''acceleration'' a: the maximum vehicle acceleration * ''comfortable braking deceleration'' b: a positive number 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^\text_\alpha = a\,\left( 1 - \left(\frac\right)^\delta \right) \qquad\dot^\text_\alpha = -a\,\left(\frac\right)^2 = -a\,\left(\frac + \frac\right)^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\,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.


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: :\dot = \frac = v :\dot = \frac = a\,\left( 1 - \left(\frac\right)^\delta - \left(\frac\right)^2 \right) :\texts^*(v,\Delta v) = s_0 + v\,T + \frac For this example, the following values are given for the equation's parameters, in line with the original calibrated model. The two
ordinary differential equations In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...
are solved using
Runge–Kutta methods In numerical analysis, the Runge–Kutta methods ( ) are a family of implicit and explicit iterative methods, which include the Euler method, used in temporal discretization for the approximate solutions of simultaneous nonlinear equations. The ...
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 In mathematics and computational science, the Euler method (also called forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is the most basic explicit met ...
(RK1). However,
traffic wave Traffic waves, also called stop waves, ghost jams, traffic snakes or traffic shocks, are traveling disturbances in the distribution of cars on a highway. Traffic waves travel backwards relative to the cars themselves. Relative to a fixed spot on t ...
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 Gipps' model is a mathematical model for describing car-following behaviour by motorists in the United Kingdom. The model is named after Peter G. Gipps who developed it in the late-1970s under S.R.C. grants at the Transport Operations Research Gr ...
* Newell's car-following model *
Microscopic traffic flow model Microscopic traffic flow models are a class of scientific models of vehicular traffic dynamics. In contrast, to macroscopic models, microscopic traffic flow models simulate single vehicle-driver units, so the dynamic variables of the models re ...
*
List of Runge–Kutta methods Runge–Kutta methods are methods for the numerical solution of the ordinary differential equation :\frac = f(t, y). Explicit Runge–Kutta methods take the form :\begin y_ &= y_n + h \sum_^s b_i k_i \\ k_1 &= f(t_n, y_n), \\ k_2 &= f(t_n+c_2h ...
*
Traffic simulation Traffic simulation or the simulation of transportation systems is the computer simulation, mathematical modeling of transportation systems (e.g., freeway junctions, arterial routes, roundabouts, downtown grid systems, etc.) through the application o ...


References

{{Reflist


External links


Interactive JS & HTML5 implementation of the intelligent driver model showing signalized intersections


* ttp://www.traffic-simulation.de/roundabout.html 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
Road traffic management