# The Car-Following Model and Its Applications in the V2X Environment: A Historical Review
## 3 Research on Car-Following Behavior in the V2X Environment
### Modelling the Car-Following Behavior
#### 3.2.4. Electronic Throttle Open Angle Information
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Research** & **Information** & **Expression** \\ \hline
[53] & position & \(\frac{\partial^{2}x_{n}(t)}{dt^{2}}=\alpha[\left\{V_{F}(x_{n+1}(t)-x_{n}(t))+V _{B}(x_{n}(t)-x_{n-1}(t))\right\}-v_{n}(t)]\) \\ \hline
[54] & position & \(\frac{d^{2}x_{n}(t)}{dt^{2}}=\alpha[V(\Delta x_{n+1},(t),\Delta x_{n+1}(t), \Delta x_{n}(t),\Delta x_{n-1}(t),\ldots,\Delta x_{n-k}.(t))-v_{n}(t)]\) \\ \hline
[55] & position & \(\frac{dx_{j}(t+\tau)}{dt^{2}}=pV_{F}\big{(}\Delta x_{j}(t),\Delta x_{j+1}(t), \ldots,\Delta x_{j+n-1}(t)\big{)}+(1-p)H(h_{c}-\Delta x_{j-1}(t))V_{B}\big{(} \Delta x_{j-1}(t)\big{)}\) \\ \hline
[56] & velocity & \(\frac{d^{2}x_{n}(t)}{dt^{2}}=\kappa[V(\Delta x_{n}(t))-v_{n}(t)]+\kappa\left( \sum\limits_{l=1}^{n}a_{l}\Delta x_{n+l-1}(t)+\sum\limits_{l=1}^{n}\beta_{l} \Delta v_{n-l}(t)\right)\) \\ \hline
[57] & position, velocity & \(\frac{d^{2}x_{n}(t)}{dt^{2}}=\alpha[V(\Delta x_{n}(t),\Delta x_{n+1}(t), \ldots,\Delta x_{n+m}(t);\Delta x_{n-1}(t))-v_{n}(t)]\) \\ \hline
[58] & position & \(\frac{d^{2}x_{n}(t)}{dt^{2}}=\alpha[pV_{F}(\Delta x_{n}(t))+(1-p)V_{B}(\Delta x _{n-1}(t))-v_{n}(t)]+\kappa\Delta v_{n}(t)\) \\ \hline
[61] & position, velocity & \(\frac{d^{2}x_{n}(t)}{dt^{2}}=\kappa_{l}[V(\Delta x_{n-1,n-2}(t),\Delta x_{n-1, n-3}(t))-v_{n-1}(t)]\) \\  & & \(+\kappa_{c}[V(\Delta x_{n\alpha-1}(t),\Delta x_{n,\alpha-2}(t))-v_{n}(t)]\) \\  & & \(+\kappa_{f}[V(\Delta x_{n+1,n}(t),\Delta x_{n+1,n-1}(t))-v_{n+1}(t)]\) \\ \hline
[59] & position, velocity & \(\frac{d^{2}x_{n}(t)}{dt^{2}}=\alpha[V(\Delta x_{n}(t),\Delta x_{n+1}(t), \ldots,\Delta x_{n+m}(t);\Delta x_{n-1}(t))-v_{n}(t)]\) \\  & & \(+\kappa G(\Delta v_{n}(t),\Delta v_{n+1}(t),\ldots,\Delta v_{n+m}(t);\Delta x _{n-1}(t))\) \\ \hline
[60] & position, velocity & \(\frac{d^{2}x_{n}(t)}{dt^{2}}=\alpha[(1-p^{x})V(\Delta x_{n}(t))+p^{x}(-V( \Delta x_{n-1}(t)))-v_{n}(t)]\) \\  & & \(+r(1-p^{x})\Delta v_{n}(t)+rp^{v}(-\Delta v_{n-1}(t))\) \\ \hline
[62] & position, velocity & \(\frac{d^{2}x_{n}(t)}{dt^{2}}=\alpha[pV_{F}(\Delta x_{n}(t))+(1-p)(V_{B}( \Delta x_{n-1}(t)))-v_{n}(t)]\) \\  & & \(+\lambda[q\Delta v_{n}(t)+(1-q)\Delta v_{n+1}(t)]\) \\ \hline
[63] & velocity, acceleration & \(\frac{d^{2}x_{n}(t+t_{d})}{dt^{2}}=\alpha_{n}^{0}\bigg{[}1-\left(\frac{v_{n} (t)}{dt^{2}}\right)^{4}\bigg{]}-\left[\gamma_{f}\sum\limits_{l=1}^{Q_{f}} \xi^{s*}\Big{(}v_{n-l,+1}(t),\Delta v_{n-l,+1}(t)\Big{)}\times\lambda_{l}g^{0 }_{n-l,+1}\bigg{(}\frac{s^{*}_{n-l,l+1}}{s^{*}_{n-l,+1}}\bigg{)}^{2}\right.\) \\  & & \(+\tau_{f}\sum\limits_{l=1}^{Q_{f}}\xi^{s*}(v_{n+l,-1}(t),\Delta v_{n+l,-1}(t) )\times\lambda_{l}g^{0}_{n+l,-1}\bigg{(}\frac{s^{*}_{n+l,-1}}{s^{*}_{n+l,-1}} \bigg{)}^{2}\bigg{]}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Incorporated information in the above models and their expression. where \(a_{n}(t)\) is the acceleration of the vehicle, \(v_{0}\) is the velocity in steady state, \(b\) and \(c\) are parameters depending on \(v_{0}\), \(\widetilde{\theta}_{n}\) is the ETOA, and \(d_{n}\) is an adjustment item. In the traditional traffic environment without the V2X environment, the driver can only obtain the information about ETOA of his/her own vehicle. However, like other kinds of information that are difficult to obtain or successively obtain, the ETOA information can be collected and then exchanged with other units in the V2X environment. According to this, some researchers modeled the car-following behavior with consideration of the ETOA information based on [PERSON]'s model. [PERSON] et al. incorporated the ETOA information and proposed an extended FVD model, i.e., the throttle-based FVD (T-FVD) model [65]. Soon after, based on the T-FVD model, [PERSON] et al. further incorporated the headway and relative velocity of two preceding vehicles and the influence exerted by the preceding vehicles not driving on the center of the lane, and proposed a new model, i.e., the Non-Lane-Based Car-following (NLBCF) model [66]. [PERSON] et al.