## 2 Model Formulation for Handling Intersection Scenarios  
The goal of this section is to formulate the collision-free motion of the vehicles in intersection scenarios. The formulation includes the longitudinal motion model of the vehicles and the constraints on their motion to avoid collisions. The longitudinal motion of the vehicles is formulated through the simplified kinematics of vehicles:  
\[v_{i}(k+1) =v_{i}(k)+Ta_{i}(k), \tag{1}\] \[s_{i}(k+1) =s_{i}(k)+Tv_{i}(k)+\frac{T^{2}}{2}a_{i}(k), \tag{2}\]  
where \({}_{i}\) index represents the id of the vehicle, \(n\) is the number of vehicles, \(v_{i}\) is longitudinal velocity and \(s_{i}\) is longitudinal displacement. \(a_{i}\) represents the longitudinal acceleration of the vehicle, which is handled as a control input command, and \(T\) is the time step of the discrete motion model. The longitudinal displacement is related to the center point of the intersection, and thus, it is defined as \(s_{i}=0\) for all \(i\) at the center point. The longitudinal displacement of the approaching vehicle has a negative value, and the displacement of the vehicle moving away has a positive value. In the environment model of the intersection, the autonomous ego vehicle is numbered 1. Thus, for example, in the case of \(n=5\) in the environment, the ego vehicle and four surrounding vehicles are incorporated. The control input of the autonomous vehicles is separated into two elements, as has been introduced in Section 1 as  
\[a_{1}(k)=a_{K}(k)+\Delta(k), \tag{3}\]  
where \(a_{K}\) is the control input command of the robust longitudinal controller and \(\Delta(k)\) is the additional input from the supervisor in the model. The purpose of the supervisor in the collision-free motion model is to select \(\Delta(k)\) for the autonomous ego vehicle with the following objectives and constraints. * The objective of the selection is to minimize the difference between the current control input \(a_{1}(k)\) and the output of the learning-based controller \(a_{L}(k)\). The aim of the minimization is to preserve the performance level of the learning-based controller, if possible. * Some of the vehicles can intersect their routes at the intersection. Thus, an intervention with \(a_{1}(k)\) must provide motion for the autonomous vehicle, with which a safe distance \(s_{safe}\) between the autonomous vehicle and the other surrounding vehicles with intersecting routes can be guaranteed. Since the number of vehicles \(n\) can be high, a limited amount of surrounding vehicle motion must be simultaneously considered; i.e., \(n_{s}\) represents the number of vehicles which are incorporated into the design process. It is a constraint in the selection of \(\Delta\). * The velocity of the vehicles must be inside of a predefined range. The upper bound of the range is defined by the speed limit \(v_{max}\) or the curvature of the intersection. The lower bound is represented by the stopping of the vehicle. Thus, it is necessary to select \(\Delta\) for all vehicles to keep velocities inside of the range. It is also a constraint in the selection process. The selection process of \(\Delta\) for the autonomous ego vehicle is formed as an optimization problem as follows:  
\[\min_{\Delta(k)} \big{(}a_{1}(k)-a_{L}(k)\big{)}^{2}\] (4a) subject to \[(s_{1}(k+1)-s_{j}(k+1))^{2} \geq s_{safe},\qquad\forall j\in n_{s}, \tag{4b}\] \[0\leq v_{1}(k+1) \leq v_{max},\] (4c) \[\Delta \in\mathbf{\Delta}, \tag{4d}\]  
where \(j\) represents the surrounding vehicles, whose motion can be in conflict with the autonomous vehicle; i.e., their routes are intersected. \(\mathbf{\Delta}\) represents the domain of the optimization variable. In the optimization problem the kinematics of the vehicle motion (1) is considered through the formulation of the constraints, the separation of the control input (3) is involved in the objective function.