# Game Theory-Based Interactive Control for Human-Machine Cooperative Driving
## 4 Vehicle Interactive Control Strategy
### Driving Cost Function
#### 4.3.1 Driving Style Cost Function of SVs  
This article considers two important indicators reflecting the personalized driving styles of surrounding vehicles: driving safety and travel efficiency. The driving style cost function of surrounding vehicles \(V_{i}\) is expressed as follows:  
\[J^{V_{i}}=(1-k^{V_{i}})J_{s}^{V_{i}}+k^{V_{i}}J_{e}^{V_{i}} \tag{20}\]  
where \(J_{s}^{V_{i}}\), \(J_{e}^{V_{i}}\) represent the cost functions of driving safety and travel efficiency. It is worth noting that \(k^{V_{i}}\) is a weight coefficient used to adjust the degree of aggressiveness of driving style, \(k^{V_{i}}\in(0,1)\). The larger value of \(k^{V_{i}}\) represents a driving style that pursues higher travel efficiency and lower driving safety, which means a higher level of aggressiveness. On the contrary, the smaller the value of \(k^{V_{i}}\), the lower the level of aggressiveness. The corresponding relationship between the weight coefficient and the driving aggressiveness level is defined in Table 1.  
The cost function of driving safety \(J_{s}^{V_{i}}\) consists of three parts, namely, vehicle longitudinal threat, lateral threat, and lane change threat. It is defined as follows:  
\[J_{s}^{V_{i}}=J_{s-x}^{V_{i}}+J_{s-y}^{V_{i}}+J_{s-lc}^{V_{i}} \tag{21}\]  
where \(J_{s-x}^{V_{i}}\), \(J_{s-y}^{V_{i}}\), \(J_{s-lc}^{V_{i}}\), respectively, denote the longitudinal, lateral, and lane change threats in the surrounding environment of vehicle \(V_{i}\). In terms of longitudinal threat \(J_{s-x}^{V_{i}}\), the relative longitudinal speed and distance between vehicle \(V_{i}\) and LV are used to design its cost function. \[J_{s-x}^{V_{i}} =\alpha_{v-x}^{V_{i}}J^{V_{i}}\Big{(}v_{x}^{LV}-v_{x}^{V_{i}} \Big{)}^{2}+\frac{\sigma_{s-x}^{V_{i}}}{\Xi_{x}^{n}} \tag{22a}\] \[\eta^{V_{i}} =0.5+0.5\,\mathrm{sgn}\Big{(}v_{x}^{V_{i}}-v_{x}^{LV}\Big{)}\] (22b) \[\Xi_{x} =2+\tanh\Bigg{[}\sigma_{\Xi}\Bigg{(}\frac{\big{(}X^{LV}-X^{V_{i}} \big{)}^{2}}{\big{(}L_{x}\big{)}^{2}}-1\Bigg{)}\Bigg{]} \tag{22c}\]  
The longitudinal velocities of LV and \(V_{i}\) are expressed as \(v_{x}^{LV}\) and \(v_{x}^{V_{i}}\), respectively, and their positions in global coordinates are \(\big{(}X^{LV},Y^{LV}\big{)}\) and \(\big{(}X^{V_{i}},Y^{V_{i}}\big{)}\), respectively. \(\alpha_{v-x}^{V_{i}}\) and \(\alpha_{s-x}^{V_{i}}\) are weight coefficients. \(\eta^{V_{i}}\) is the relative speed switching function. Only when \(v_{x}^{V_{i}}\) is greater than \(v_{x,\sigma}^{LV}\), the value of \(\eta^{V_{i}}\) is 1; otherwise, it is 0. The parameters \(n\) and \(\sigma_{\Xi}\) are 4 and 1.5, respectively, and \(L_{x}\) is the longitudinal safety threshold. The cost function of lateral threat \(J_{s-y}^{V_{i}}\) is related to the distance between the vehicle \(V_{i}\) and NV, and it is defined by the following: \begin{table}
\begin{tabular}{c c c c} \hline
**Level of** & **High Level** & **Moderate Level** & **Low Level** \\
**Aggressiveness** & & & \\ \hline Coefficient: \(k^{V_{i}}\) & 0.9 & 0.5 & 0.1 \\ \hline \end{tabular}
\end{table} 
Table 1: Weight coefficient of driving style cost function. \[J_{s-y}^{V_{i}} =\beta_{\sigma}\omega_{s-y}^{V_{i}}\eta_{\sigma}^{V_{i}}\Big{(}X^{V _{i}}-X_{\sigma}^{NV}-L_{y1}\Big{)}^{2} \tag{23a}\] \[\eta_{\sigma}^{V_{i}} =0.5+0.5\,\mathrm{sgn}\Big{(}X^{V_{i}}-X_{\sigma}^{NV}-L_{y1}\Big{)} \tag{23b}\]  
where NV is in front of \(V_{i}\) and their longitudinal positions are expressed as \(X_{\sigma}^{NV}\) and \(X^{V_{i}}\), respectively, and \(X^{V_{i}}\leq X_{\sigma}^{NV}\). \(\sigma\in\{1,2,3\}\) is the index used to identify adjacent lanes. \(\beta_{\sigma}\) is a lane change intention judgment, and the value of \(\beta_{\sigma}\) is 1 only if one of the vehicles wants to change lanes into the lane of the other vehicle; otherwise, it is 0. \(\sigma_{s-y}^{V_{i}}\) are the weighting coefficients. \(\eta_{\sigma}^{V_{i}}\) is a switch function. \(L_{y1}\) represents the lateral safety threshold.