## 2 Model Formulation for Handling Intersection Scenarios
The objective (4a) is transformed using (3) as  
\[\big{(}a_{1}(k)-a_{L}(k)\big{)}^{2} =\big{(}a_{K}(k)+\Delta(k)-a_{L}(k)\big{)}^{2}=\Delta(k)^{2}+2f(k )\Delta(k)+(a_{K}(k)^{2}+a_{L}(k)^{2}-2a_{L}(k)a_{K}(k)), \tag{5}\]  
where \(f\) (a vector) is formed as \(f(k)=a_{K}(k)-a_{L}(k)\). Since \((a_{K}(k)^{2}+a_{L}(k)^{2}-2a_{L}(k)a_{K}(k))\) is independent of \(\Delta(k)\), it is omitted in the rest of the optimization problem. The constraint for collision avoidance (4b) is formed to keep \(s_{safe}\) between the vehicles. The distance is measured in the sense of the longitudinal displacements of the vehicles on their routes. Geometrically, the quadratic constraints (4b) represent that the trajectories of the autonomous vehicle \(s_{1}\) and the related surrounding vehicles must be out of a circle. The radius of the circle is defined by \(s_{safe}\); see Figure 1a. For example, it there are two surrounding vehicles and the autonomous vehicle the trajectories must be out of a sphere, whose radius is represented by \(s_{safe}\); see Figure 1b. Although the circle perfectly describes the region of = state-space that must be avoided, it leads to a quadratic constraint in the optimization problem (4). Since the optimization problem must be solved at each \(k\) time step during the motion of the vehicles, the quadratic constraint can make the solution of the optimization more complex, and it can make  
Figure 1: Geometrical illustration of the quadratic constraints. the real-time solution more difficult. This motivates the transformation of the quadratic constraints into linear constraints. In this paper it is performed through the approximation of the avoidable regions of circles to the avoidable regions of half-planes. The method of the approximation is illustrated in Figure 2. First, the tangent lines to the circle from the actual state \( \begin{bmatrix}s_{1}(k)&s_{j}(k)\end{bmatrix} ^{\mathsf{T}}\) are assigned. The avoidable half-plane is determined by the region between the tangent lines; i.e., the trajectories must be out of it. Second, two linear inequality constraints are specified, which represent that the trajectory of the state must be out of the avoidable half-plane:  
\[ \begin{bmatrix}s_{T1,1}(k)\\ s_{T1,j}(k)\end{bmatrix} ^{\mathsf{T}} \begin{bmatrix}s_{1}(k)\\ s_{j}(k)\end{bmatrix} \leq \begin{bmatrix}s_{T1,1}(k)\\ s_{T1,j}(k)\end{bmatrix} ^{\mathsf{T}} \begin{bmatrix}s_{1}(k+1)\\ s_{j}(k+1)\end{bmatrix} ,\] (6a) or \[ \begin{bmatrix}s_{T2,1}(k)\\ s_{T2,j}(k)\end{bmatrix} ^{\mathsf{T}} \begin{bmatrix}s_{1}(k)\\ s_{j}(k)\end{bmatrix} \geq \begin{bmatrix}s_{T2,1}(k)\\ s_{T2,j}(k)\end{bmatrix} ^{\mathsf{T}} \begin{bmatrix}s_{1}(k+1)\\ s_{j}(k+1)\end{bmatrix} , \tag{6b}\]  
where \( \begin{bmatrix}s_{T1,1}(k)&s_{T1,j}(k)\end{bmatrix} \), \( \begin{bmatrix}s_{T2,1}(k)&s_{T2,j}(k)\end{bmatrix} \) are the tangent points on the circle at time step \(k\). Note that the linear constraints result in an outer approximation of the avoidable set. There are regions of the half-space which cannot be reached due to the linear constraints, but they are out of the circle. The longitudinal displacement of the autonomous vehicle at \(k+1\) in (6) is transformed to express the linear constraints in terms of \(\Delta\). The transformation is based on the motion equation (1) and the relation of actuation separation (3). Moreover, the motions of surrounding vehicles for \(k+1\) with their actual velocity \(v_{j}(k)\) are predicted this way:  
\[s_{1}(k+1) =s_{1}(k)+Tv_{1}(k)+\frac{T^{2}}{2}a_{K}(k)+\frac{T^{2}}{2}\Delta (k), \tag{7a}\] \[s_{j}(k+1) =s_{j}(k)+Tv_{j}(k), \tag{7b}\]  
which can be substituted into (6). It leads to the linear constraints  
\[ \begin{bmatrix}s_{T1,1}(k)\\ s_{T1,j}(k)\end{bmatrix} ^{\mathsf{T}} \begin{bmatrix}-Tv_{1}(k)-\frac{T^{2}}{2} a_{K}(k)\\ -Tv_{j}(k)\end{bmatrix} \leq\frac{T^{2}}{2} \begin{bmatrix}s_{T1,1}(k)\\ 0\end{bmatrix} ^{\mathsf{T}}\Delta(k),\] (8a) or \[ \begin{bmatrix}s_{T2,1}(k)\\ s_{T2,j}(k)\end{bmatrix} ^{\mathsf{T}}