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General Motors Car Following Model

The basic philosophy of car following model is from Newtonian mechanics, where the acceleration may be regarded as the response of a matter to the stimulus it receives in the form of the force it receives from the interaction with other particles in the system. Hence, the basic philosophy of car-following theories can be summarized by the following equation

begin{displaymath}[{mathrm{Response}}]_n alpha  [{mathrm Stimulus}]_n
            end{displaymath} (1)



for the nth vehicle (n=1, 2, ...). Each driver can respond to the surrounding traffic conditions only by accelerating or decelerating the vehicle. As mentioned earlier, different theories on car-following have arisen because of the difference in views regarding the nature of the stimulus. The stimulus may be composed of the speed of the vehicle, relative speeds, distance headway etc, and hence, it is not a single variable, but a function and can be represented as,

            {a_n^t}= f_{sti}{(v_n,{Delta{x_n}},{Delta{v_n}})}
            end{displaymath} (2)



where $f_{sti}$ is the stimulus function that depends on the speed of the current vehicle, relative position and speed with the front vehicle.


Follow-the-leader model

The car following model proposed by General motors is based on follow-the leader concept. This is based on two assumptions; (a) higher the speed of the vehicle, higher will be the spacing between the vehicles and (b) to avoid collision, driver must maintain a safe distance with the vehicle ahead.

Let $Delta x^{t}_{n+1}$ is the gap available for $(n+1)^{th} $ vehicle, and let $Delta x_{safe}$ is the safe distance, $v^t_{n+1}$ and $v^t_{n}$ are the velocities, the gap required is given by,

            Delta x^{t}_{n+1}= Delta{x}_{safe} + tau v^{t}_{n+1}
            end{displaymath} (3)



where $tau$ is a sensitivity coefficient. The above equation can be written as

            x_n - x^{t}_{n+1} = Delta{x}_{safe} + tau v^t_{n+1}
            end{displaymath} (4)



Differentiating the above equation with respect to time, we get


v_n^{t}-v_{n+1}^t = tau.a_{n+1}^t
a^t_{n+1} = frac{1}{tau}[v^t_{n}-v^t_{n+1}]



General Motors has proposed various forms of sensitivity coefficient term resulting in five generations of models. The most general model has the form,

            a^t_{n+1} =
            end{displaymath} (5)



where $l$ is a distance headway exponent and can take values from +4 to -1, $m$ is a speed exponent and can take values from -2 to +2, and $alpha$ is a sensitivity coefficient. These parameters are to be calibrated using field data. This equation is the core of traffic simulation models.

In computer, implementation of the simulation models, three things need to be remembered:

  1. A driver will react to the change in speed of the front vehicle after a time gap called the reaction time during which the follower perceives the change in speed and react to it.
  2. The vehicle position, speed and acceleration will be updated at certain time intervals depending on the accuracy required. Lower the time interval, higher the accuracy.
  3. Vehicle position and speed is governed by Newton's laws of motion, and the acceleration is governed by the car following model.

Therefore, the governing equations of a traffic flow can be developed as below. Let $Delta{T}$ is the reaction time, and $Delta{t}$ is the updation time, the governing equations can be written as,

$displaystyle v_n^t$ $textstyle =$ $displaystyle v_n^{t-Delta t}+a_n^{t-Delta t}times Delta t$ (6)
$displaystyle x_n^t$ $textstyle =$ $displaystyle x_n^{t-Delta t}+v_n^{t-Delta t}times Delta
            t+frac{1}{2}a_n^{t-Delta t}Delta t^2$ (7)
$displaystyle a^t_{n+1}$ $textstyle =$ $displaystyle left[frac{alpha_{l,m}{(v^{t}_{n+1}})^m}{{(x^{t-Delta
            T}_n-x^{t-Delta T}_{n+1})^l}}right](v_n^{t-Delta T}-v^{t-Delta T}_{n+1})$ (8)



The equation (6)  is a simulation version of the Newton's simple law of motion $v = u + at$ and equation (8) is the simulation version of the Newton's another equation $s = ut+frac{1}{2}a{t^2}$. The acceleration of the follower vehicle depends upon the relative velocity of the leader and the follower vehicle, sensitivity coefficient and the gap between the vehicles.

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