## 3 Problem Statement and Results
### Scenario Description and Formulation  
The general problem we are interested in deals with computing a trajectory tracing procedure to handle critical situations under which a vehicle circulating at high speeds has to avoid the collision against a suddenly appearing obstacle (another vehicle, a pedestrian or an animal) in the middle of the road. However, we think it is instructive to examine in detail alternative formulations, thus in this paper we will focus on the simplest version of the problem in order to get an understanding on how trajectories for a single vehicle evolve according to the specific optimization requirements we set, and how the behavioral restrictions to which the car's mobility is constrained affect the maneuver. As a straightforward version of the problem, we assume there is only one vehicle circulating on a _W_-meter-wide road (regardless of the number of lanes). This vehicle has to trace a trajectory in an interval of \(t_{f}\) seconds after the maneuver has begun (see Figure 1). During this interval, the vehicle will try to adjust its mobility evolution according to a set of constraints and optimizing a given functional. In this particular case, our goal would be to achieve a given degree of safety while keeping the maneuver as comfortable as possible. To combine these two features, we look for the maximization of the lateral distance with respect to the lateral boundaries of the road, while minimizing the final lateral speed at the end of the path, all within a given time interval of \(t_{f}\) seconds. Maximizing the lateral distance implies in this case leaving the maximum possible road width on both sides of the vehicle, thus minimizing the probability of colliding with the lateral protections. On the other hand, by simultaneously minimizing the lateral speed at \(t_{f}\), we can change the general appearance of the trajectory in such a way that the inertial impact on the mobility of the passengers is reduced when reaching the optimum lateral position. For the analysis of the problem, we first need to assume a model for the vehicle's motion, which is represented by the system of differential equations that describes the lateral movement of the car (only \(x\) axis, see Figure 1):  
\[\left\{ \begin{array}{l}\dot{x}_{1}\left(t\right)=v_{1}(t)\\ \dot{v}_{1}\left(t\right)=a_{1}(t)\end{array} \right. \tag{1}\]  
where \(x_{1}(t)\) refers to the position of vehicle 1 in axis \(x\) at time \(t\); \(v_{1}(t)\) denotes the current speed of vehicle 1 in axis \(x\) as a function of \(t\); and \(a_{1}(t)\) defines the lateral acceleration of vehicle 1 along time. We take \(a_{1}(t)\) as the control variable that we have to manipulate in real time in order to modify the trajectory traced by the vehicle under consideration. As was specified before, we assume that these mobility equations only govern the lateral displacement of the vehicle (we consider the longitudinal speed to be constant and fixed to 120 km/h during the whole trajectory). Furthermore, the car's mobility has to respect the physical limits imposed by the inertial laws of kinematics. This means that the vehicle can only turn at a maximum established acceleration, which depends on the longitudinal speed (the higher the speed, the harder the maneuver). Considering this, we can establish the mechanical restrictions of the problem at hand:  
Figure 1: Maximization of lateral distance after \(t_{f}\). 1. **Lateral acceleration restrictions**. The absolute value of the lateral acceleration cannot take a value higher than the limit \(c(v_{i})\) m/s\({}^{2}\), where \(v_{i}\) is the constant longitudinal speed of the vehicle and \(c(\cdot)\) is a function of the longitudinal speed. \[\left|a(t)\right|\leq c\left(v_{i}\right)\] (2)
2. **Lateral position restrictions**. The vehicle can only have a lateral displacement inside the width limits of the road. \[0<x_{1}\left(t\right)<W\] (3)  
The third and most important aspect of the statement of the problem is the functional we want to optimize.