# Game Theory-Based Interactive Control for Human-Machine Cooperative Driving
## 4 Vehicle Interactive Control Strategy
### Motion Prediction for HSCVs  
The future states of HSCVs and SVs are related to their current states and control inputs. In an HSCV, the automation system can obtain driver control commands \(u^{h}\) through sensors, and it will use the MPC controller to compensate for the driver's behavior based  
Figure 4: Graph of driving authority allocation strategy.  
on the allocated driving authority \(\lambda\) and surrounding environment information. Assuming that the prediction time domain and the control time domain are \(N_{p}\) and \(N_{c}\), respectively, the prediction Equation (14) is rewritten as follows:  
\[\mathbf{Z}(k)=\boldsymbol{\
abla}\boldsymbol{x}(k)+\lambda\boldsymbol{\Theta}_{ 1}\mathbf{U}^{h}(k)+(1-\lambda)\boldsymbol{\Theta}_{2}\mathbf{U}^{e}(k) \tag{15}\]  
It is worth noting that the prediction output of an HSCV without driver participation in driving control is similar to that of SVs, and its degradation form is as follows:  
\[\mathbf{Z}(k)=\boldsymbol{\
abla}\boldsymbol{x}(k)+\boldsymbol{\Theta}_{2} \mathbf{U}^{e}(k) \tag{16}\]  
The expansion of the above formula is as follows:  
\[\mathbf{U}^{h}(k)=\left\{\begin{array}{c}\mathbf{u}^{h}(k)\\ \mathbf{u}^{h}(k+1)\\ \mathbf{u}^{h}(k+2)\\ \vdots\\ \mathbf{u}^{h}(k+N_{u}-2)\\ \mathbf{u}^{h}(k+N_{u}-1)\end{array}\right\},\mathbf{U}^{e}(k)=\left\{ \begin{array}{c}\mathbf{u}^{e}(k)\\ \mathbf{u}^{e}(k+1)\\ \mathbf{u}^{e}(k+2)\\ \vdots\\ \mathbf{u}^{e}(k+N_{u}-2)\\ \mathbf{u}^{e}(k+N_{u}-1)\end{array}\right\},\mathbf{\Theta}_{1}(k)=\left[ \begin{array}{c}\mathbf{CB}\\ \mathbf{CB}+\mathbf{CAB}\\ \vdots\\ \sum_{i=0}^{N_{r}-1}\mathbf{C}\mathbf{A}^{i}\mathbf{B}\end{array}\right] \tag{17}\]  
The MPC controller in an HSCV will consider the control sequence amplitude and track the reference target to match the driver's operation command. At the same time, the driver's steering control and speed control commands are assumed to remain unchanged in the prediction time domain. Its cost function is defined as follows:  
\[\min_{\mathbf{u}}J(k)= \Big{[}\mathbf{Z}(k)-\mathbf{Z}_{ref}(k)\Big{]}^{\mathrm{T}} \mathbf{Q}\Big{[}\mathbf{Z}(k)-\mathbf{Z}_{ref}(k)\Big{]}+\mathbf{U}^{e}(k)^{ \mathrm{T}}\mathbf{R}\mathbf{U}^{e}(k)\] (18a) s.t. \[\mathbf{x}(k+1)=\mathbf{A}\mathbf{x}(k)+\lambda\mathbf{B}\mathbf{ u}^{h}(k)+(1-\lambda)\mathbf{B}\mathbf{u}^{e}(k) \tag{18b}\] \[\mathbf{z}(k)=\mathbf{C}\mathbf{x}(k)\] (18c) \[\left[\begin{array}{c}-\delta^{e}_{lim}\\ -\delta^{e}_{lim}\\ -\delta^{e}_{lim}\end{array}\right]\leqslant\mathbf{u}^{e}(k)\leqslant\left[ \begin{array}{c}\delta^{e}_{lim}\\ \delta^{e}_{lim}\end{array}\right]\] (18d) \[\left[\begin{array}{c}-\Delta\delta^{e}_{lim}\\ -\Delta\delta^{e}_{lim}\end{array}\right]\leqslant\Delta\mathbf{u}^{e}(k) \leqslant\left[\begin{array}{c}\Delta\delta^{e}_{lim}\\ \Delta\delta^{e}_{lim}\end{array}\right] \tag{18e}\]  
where \(\mathbf{Q}\) and \(\mathbf{R}\) are the output weight matrix and the control input weight matrix, respectively. \(\mathbf{Z}_{ref}\) is the target state matrix related to the reference trajectory, which consists of the reference heading angle \(\phi_{ref}\) and the reference lateral position \(Y_{ref}\). The maximum amplitude and maximum change rate of the vehicle's front wheel angle are represented by \(\delta^{e}_{lim}\) and \(\Delta\delta^{e}_{lim}\) respectively. \(\alpha^{e}_{lim}\) and \(\Delta\alpha^{e}_{lim}\) are used to set the upper and lower bounds of vehicle acceleration and limit the change rate of acceleration, respectively.  
After solving the above constrained optimization problem, the optimal control sequence \(\mathbf{U}^{e*}(k)\) can be obtained as follows:  
\[\mathbf{U}^{e*}(k)=\left[\mathbf{u}^{e*}(k),\mathbf{u}^{e*}(k+1),\cdots, \mathbf{u}^{e*}(k+N_{c}-1)\right]^{T} \tag{19}\]  
The first element in \(\mathbf{U}^{e*}(k)\) will be selected to control the movement of the vehicle, thus assisting the driver in driving the vehicle. In terms of driverless vehicles, many related motion prediction methods have been developed, which can be referred to in [31].