Car Following Models: Explaining Older Models

[aldous]

Hello,

I'm a computer science student in desperate need for help. In the process of writing my master's thesis I've successfully developed a traffic simulation using a model I've found in the traffic research literature, the IDM, to be precise. This model seems straight-forward and is easy to grasp even for a dim mind like mine.

However, I fail to understand older car following models, such as presented by http://www-sop.inria.fr/mascotte/Philippe.Mussi/papers/esm2000.ps" and
http://arxiv.org/PS_cache/cond-mat/pdf/0002/0002177v2.pdf" ... Hopefully just because I simply don't know what some variables mean, which are apparently not introduced.

Jiménez et al. (2000) state that all car following models can be summarized by:

$$\ddot{x}_f ( t+T_r ) = \lambda * [ \dot{x}_{l}(t) - \dot{x}_{f}(t)] \quad (1)$$

$$\lambda = \frac{a_{l,m}* \dot{x}_{f}^m(t+T)}{[x_{l}(t)-x_{f}(t)]^l} \quad (2)$$

So my questions: what does $$\dot{x}_f^m$$ express? Sure, $$\dot{x}_f$$ is the velocity of vehicle $$f$$, but what is $$m$$? The vehicle's mass? Why would one want to potentiate the velocity by the mass? I'm lost! Further, I interpreted $$l$$ -- being used as an index in equation 1 -- as the leading car, $$f$$ denoting the following car. However, in equation 2, $$l$$ is used as a power? How is this to be interpreted?

Similarly, Treiber et al. (2000) state that older car following models can be reduced to that formula:
$$\dot{v}_\alpha ( t+T_r ) = \frac{-\lambda v_\alpha^m \Delta v_\alpha}{s_\alpha^l} \quad (3)$$

My question: the $$\lambda$$ in eq. 3 seems to be different to the $$\lambda$$ in eq. 2. Is it this a variable often used in physics one should just know? (It is not defined in the paper)


Thank you very much in advance for any pointers!
Alexander

 

[aldous]

Well, my questions are answered in http://www.easts.info/on-line/journal_06/1354.pdf"

$$m,l$$ are -- when used as powers -- simply parameters influencing the driving behavior, $$\lambda$$ is just any proportionality factor.


Thanks anyway,
Alexander

 

[oswaler]

Hello Alexander,

Thank you for reaching out for help with your thesis. Car following models have been a topic of research for many years and there are various models that have been proposed to explain the behavior of drivers on the road. The IDM or Intelligent Driver Model is a relatively newer model that has gained popularity due to its simplicity and ability to accurately simulate traffic flow. However, as you have mentioned, there are older models that may seem more complex and difficult to understand. I will try my best to explain some of the variables and equations you have mentioned in your post.

Firstly, in equation 1, \ddot{x}_f ( t+T_r ) represents the acceleration of the following car (f) at time t+T_r. This is dependent on the difference in velocity between the leading car (l) and the following car, as denoted by [\dot{x}_{l}(t) - \dot{x}_{f}(t)]. The value of this acceleration is then multiplied by a factor \lambda, which is defined in equation 2. In this equation, m and l are parameters that have been used to represent the mass and size of the vehicles respectively. This helps to account for the fact that larger and heavier vehicles may behave differently on the road compared to smaller ones. The power of l in the denominator is used to represent the distance between the leading and following car, as a greater distance may result in a lower acceleration for the following car.

Similarly, in equation 3, \dot{v}_\alpha ( t+T_r ) represents the change in velocity of a particular vehicle (\alpha) at time t+T_r. The value of this change is dependent on the velocity of the vehicle (v_\alpha), the difference in velocity between the following car and the leading car (\Delta v_\alpha), and the distance between the two cars (s_\alpha). The \lambda in this equation is a different variable and it is also dependent on the mass and size of the vehicles, but it is used in a different context compared to equation 2.

It is important to note that equations 1 and 3 are simplified versions of the older car following models, which may have used different variables and parameters. These equations have been proposed by Jiménez et al. (2000) and Treiber et al. (2000) as a way to summarize and compare different models. Therefore, it is not necessary to understand every single variable and