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| In [[traffic flow]] modeling, the '''intelligent driver model''' ('''IDM''') is a [[Discrete time and continuous time|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]], which lose realistic properties in the deterministic limit.
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| ==Model definition==
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| As a car-following model, the IDM describes the dynamics of the positions and velocities of single vehicles. For vehicle <math>\alpha</math>, <math>x_\alpha</math> denotes its position at time <math>t</math>, and <math>v_\alpha</math> its velocity. Furthermore, <math>l_\alpha</math> gives the length of the vehicle. To simplify notation, we define the ''net distance'' <math>s_\alpha := x_{\alpha-1} - x_\alpha - l_{\alpha-1}</math>, where <math>\alpha - 1</math> refers to the vehicle directly in front of vehicle <math>\alpha</math>, and the velocity difference, or ''approaching rate'', <math>\Delta v_\alpha := v_\alpha - v_{\alpha-1}</math>. For a simplified version of the model, the dynamics of vehicle <math>\alpha</math> are then described by the following two [[ordinary differential equation]]s:
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| :<math>\dot{x}_\alpha = \frac{\mathrm{d}x_\alpha}{\mathrm{d}t} = v_\alpha</math>
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| :<math>\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)</math>
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| :<math>\text{with }s^*(v_\alpha,\Delta v_\alpha) = s_0 + v_\alpha\,T + \frac{v_\alpha\,\Delta v_\alpha}{2\,\sqrt{a\,b}}</math>
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| <math>v_0</math>, <math>s_0</math>, <math>T</math>, <math>a</math>, and <math>b</math> are model parameters which have the following meaning:
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| * ''desired velocity'' <math>v_0</math>: the velocity the vehicle would drive at in free traffic
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| * ''minimum spacing'' <math>s_0</math>: a minimum net distance that is kept even at a complete stand-still in a traffic jam
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| * ''desired time headway'' <math>T</math>: the desired [[time headway]] to the vehicle in front
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| * ''acceleration'' <math>a</math>
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| * ''comfortable braking deceleration'' <math>b</math>
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| The exponent <math>\delta</math> is usually set to 4.
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| ==Model characteristics==
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| The acceleration of vehicle <math>\alpha</math> can be separated into a ''free road term'' and an ''interaction term'':
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| :<math>\dot{v}^\text{free}_\alpha = a\,\left( 1 - \left(\frac{v_\alpha}{v_0}\right)^\delta \right)
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| \qquad\dot{v}^\text{int}_\alpha = -a\,\left(\frac{s^*(v_\alpha,\Delta v_\alpha)}{s_\alpha}\right)^2
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| = -a\,\left(\frac{s_0 + v_\alpha\,T}{s_\alpha} + \frac{v_\alpha\,\Delta v_\alpha}{2\,\sqrt{a\,b}\,s_\alpha}\right)^2</math>
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| * ''Free road behavior:'' On a free road, the distance to the leading vehicle <math>s_\alpha</math> is large and the vehicle's acceleration is dominated by the free road term, which is approximately equal to <math>a</math> for low velocities and vanishes as <math>v_\alpha</math> approaches <math>v_0</math>. Therefore, a single vehicle on a free road will asymptotically approach its desired velocity <math>v_0</math>.
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| * ''Behavior at high approaching rates:'' For large velocity differences, the interaction term is governed by <math>-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)</math>.
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| This leads to a driving behavior that compensates velocity differences while trying not to brake much harder than the comfortable braking deceleration <math>b</math>.
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| * ''Behavior at small net distances:'' For negligible velocity differences and small net distances, the interaction term is approximately equal to <math>-a\,(s_0 + v_\alpha\,T)^2\,/\,s_\alpha^2</math>, which resembles a simple repulsive force such that small net distances are quickly enlarged towards an equilibrium net distance.
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| ==Solution Example==
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| 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:
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| :<math>\dot{x} = \frac{\mathrm{d}x}{\mathrm{d}t} = v</math>
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| :<math>\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)</math>
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| :<math>\text{with }s^*(v,\Delta v) = s_0 + v\,T + \frac{v\,\Delta v}{2\,\sqrt{a\,b}}</math>
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| For this example, the following values are given for the equation's parameters.
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| {| class="wikitable"
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| |-
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| ! Description
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| ! Value
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| |-
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| | Desired velocity
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| | 30 m/s
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| |-
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| | Safe time headway
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| | 1.5 s
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| |-
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| | Maximum acceleration
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| | 1.00 m/s<sup>2</sup>
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| |-
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| | Desired deceleration
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| | 3.00 m/s<sup>2</sup>
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| |-
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| | Acceleration exponent
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| | 4
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| |-
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| | Minimum distance
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| | 2 m
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| |-
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| | Vehicle length
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| | 5 m
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| |}
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| 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.
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| [[Image:Idm rungekutta.PNG|thumb|center|500px|Comparison of Differential Equation Solutions for intelligent driver model using RK1,3,5]]
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| 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. None-the-less, when introducing heterogeneous vehicles and both jam distance parameters, this observation could not suffice.
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| ==See also ==
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| * [[Gipps' model]]
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| * [[Newell's car-following model]]
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| * [[List of Runge–Kutta methods]]
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| * [[Traffic Simulation]]
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| ==References==
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| {{Citation
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| | last1 = Treiber | first1 = Martin
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| | last2 = Hennecke | first2 = Ansgar
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| | last3 = Helbing | first3 = Dirk
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| | title = Congested traffic states in empirical observations and microscopic simulations
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| | journal = Physical Review E
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| | volume = 62
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| | issue = 2
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| | pages = 1805–1824
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| | year = 2000
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| | doi = 10.1103/PhysRevE.62.1805 }}
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| ==External links==
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| * [http://www.traffic-simulation.de/ Interactive Java-Applet implementing the intelligent driver model]
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| * [http://www.vwi.tu-dresden.de/~treiber/MicroApplet/IDM.html Common values for modeling IDM]
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| [[Category:Road traffic management]]
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I would like to introduce myself to you, I am Andrew and my wife doesn't like it at all. What I adore doing is soccer but I don't have the time lately. For a whilst I've been in Mississippi but now I'm considering other choices. She works as a travel agent but quickly she'll be on her own.
my homepage ... love psychic readings; http://Brazil.amor-amore.com,