## 2 Model Formulation for Handling Intersection Scenarios
\begin{bmatrix}-Tv_{1}(k)-\frac{T^{2}}{2} a_{K}(k)\\ -Tv_{j}(k)\end{bmatrix} \geq\frac{T^{2}}{2} \begin{bmatrix}s_{T2,1}(k)\\ 0\end{bmatrix} ^{\mathsf{T}}\Delta(k). \tag{8b}\]  
Figure 2: Illustration of constraint approximation. Figure 2 illustrates that the reachable set for the state \(\big{[}s_{1}(k+1)\quad s_{j}(k+1)\big{]}^{T}\) is non-convex, which means that (8) formulates disjunctive inequalities. However, each constraint in (8) leads to convex reachable sets, which means that the optimization problem can be divided, as it is proposed below. Another constraint in the optimization (4) is on the velocity of the autonomous vehicle; see (4c). In the case of this constraint, \(v_{i}(k+1)\) is expressed in terms of \(\Delta\) using the motion equation (1) and the relation of actuation division (3). The linear inequality constraint is formed as  
\[0 \leq v_{1}(k)+Ta_{K}(k)+T\Delta(k),\quad, \tag{9a}\] \[v_{max} \geq v_{1}(k)+Ta_{K}(k)+T\Delta(k), \tag{9b}\]  
which leads to  
\[ \begin{bmatrix}-1\\ 1\end{bmatrix} \Delta(k)\leq \begin{bmatrix}\frac{v_{1}(k)}{T}+a_{K}(k)\\ \frac{v_{max}-v_{1}(k)}{T}-a_{K}(k)\end{bmatrix} , \tag{10}\]  
where \(v_{max}\) is the maximum velocity of the autonomous vehicle. In an urban environment it is determined by the velocity regulations, and in the case of cornering maneuvers at intersections, the curvature avoids skidding. The last constraint in the optimization problem (4) is the limitation of the resulting optimization variable; see (4d). The value of \(\Delta\) is limited by the bounds of \(a_{1}(k)\), such as \(a_{min}\) and \(a_{max}\), which represent full braking and throttle. Since \(a_{1}(k)\) is also influenced by \(a_{K}(k)\), the constraints on \(\Delta(k)\) are formed as  
\[a_{min}-a_{K}(k) \leq \Delta(k), \tag{11a}\] \[a_{max}-a_{K}(k) \geq \Delta(k), \tag{11b}\]  
which leads to the constraint  
\[ \begin{bmatrix}-1\\ 1\end{bmatrix} \Delta(k)\leq \begin{bmatrix}a_{K}(k)-a_{min}\\ a_{max}-a_{K}(k)\end{bmatrix} . \tag{12}\]  
The optimization task (4) using (5), (8), (10) and (12) is reformulated as  
\[\min_{\Delta(k)} \ \ \Delta(k)^{2}+2f^{T}\Delta(k)\] (13a) subject to \[ \begin{bmatrix}-1\\ 1\end{bmatrix} \Delta(k)\leq \begin{bmatrix}\frac{v_{1}(k)}{T}+a_{K}(k)\\ \frac{v_{max}-v_{1}(k)}{T}-a_{K}(k)\end{bmatrix} \] (13b) and \[ \begin{bmatrix}-1\\ 1\end{bmatrix} \Delta(k)\leq \begin{bmatrix}a_{K}(k)-a_{min}\\ a_{max}-a_{K}(k)\end{bmatrix} \] (13c) and \[ \begin{bmatrix}s_{T1,1}(k)\\ s_{T1,j}(k)\end{bmatrix} ^{T} \begin{bmatrix}-Tv_{1}(k)-\frac{T^{2}}{2}a_{K}(k) \\ -Tv_{j}(k)\end{bmatrix} \leq\frac{T^{2}}{2} \begin{bmatrix}s_{T1,1}(k)\\ 0\end{bmatrix} ^{T}\Delta(k),\ \ \forall j\in n_{s}\] (13d) or \[ \begin{bmatrix}s_{T2,1}(k)\\ s_{T2,j}(k)\end{bmatrix} ^{T} \begin{bmatrix}-Tv_{1}(k)-\frac{T^{2}}{2}a_{K}(k) \\ -Tv_{j}(k)\end{bmatrix} \geq\frac{T^{2}}{2} \begin{bmatrix}s_{T2,1}(k)\\ 0\end{bmatrix} ^{T}\Delta(k),\ \ \forall j\in n_{s}. \tag{13e}\]  
The quadratic optimization in (13) contains disjunctive inequalities. This means that the optimization task for the solution can be reformulated to a mixed-integer optimization problem [48]. In practice, the optimization problem (13) can be solved as a set of quadratic optimization problems, which has only \"and\" conditions between the constraints. Thus, in the case of \(n_{s}\) surrounding vehicles, \(2^{n_{s}}\) distinct constrained optimization problems can be formed. Since the objective function for each problem is the same, the global minimum solution can be found through a comparison of their solutions. Finally, the solution is yielded by \(\Delta(k)\), which leads to the minimum value of the objective function, considering all of the optimization tasks. The optimization problem during the motion of the autonomous vehicle is continuously solved.