## 2 Collision Avoidance System
### Decision Module
It should be noted that all the variables are known, which means that at every instant the algorithm assesses whether or not it is possible to overtake. However, the fact that no obstacle is detected in the left-hand lane does not necessarily mean the overtaking manoeuvre can be carried out. For instance, if no obstacle is detected in the left-hand lane in the laser's field of vision, in the most unfavourable extreme case, V3 could be found to be in the initial instant at a distance \(d_{laser}\) from V1. That most unfavourable distance is the minimum between the scanner's maximum range and the field of vision allowed by the obstacles. Geometrically it is possible to establish what the field of vision is by taking the fact that vehicle V2 is an obstacle that impedes part of the vision. For safety reasons, a vehicle is deemed to be recognized when data are received from at least half its front part. The laser's visibility distance is then defined by the following expression:  
\[d_{laser}=\frac{2\cdot w_{0}}{b_{obs}}\cdot d \tag{17}\]  
where \(b_{obs}\) is the obstacle width and d is the distance between the scanner and the obstacle. In this situation, in equation (A), \(d_{3}\) should be substituted for \(d_{laser}\) and a hypothesis must be made regarding the speed of V3, which, reasonably, should be related to the road's speed limit. Finally, the algorithm must estimate the lane-change manoeuvre. Different models exist for calculating the lane-change manoeuvre. For example, in [51] for carrying out a lane-change manoeuvre is calculated by taking into account the maximum lateral acceleration and over-accelerations defined in [52]. The distance travelled by the vehicle during that lane change is the product of the manoeuvre time multiplied by the speed, taken as constant, which gives the following expression:  
\[\left(v_{1}+v_{3}\right)\cdot t_{T}\leq d_{3} \tag{10}\]0  
where \(D_{LC\text{-}min1}\) is the minimum lane-change distance; v the longitudinal speed; \(T_{LC}\) is the minimum lane-change time; \(a_{max}\) is the maximum lateral acceleration; \(J_{max}\) is the maximum lateral over-acceleration and \(w_{0}\) is the maximum lateral displacement during the manoeuvre. Other works along the same lines are included in [53], where the lane-change manoeuvre approximates to a sinusoidal path:  
\[\left(v_{1}+v_{3}\right)\cdot t_{T}\leq d_{3} \tag{10}\]1  
In this case, the lane-change distance is given by the expression:  
\[\left(v_{1}+v_{3}\right)\cdot t_{T}\leq d_{3} \tag{10}\]2  
In the same way, the same authors propose the lane-change distance if this manoeuvre is modeled according to an exponential approximation, the result being:  
\[\left(v_{1}+v_{3}\right)\cdot t_{T}\leq d_{3} \tag{10}\]3\[\left(v_{1}+v_{3}\right)\cdot t_{T}\leq d_{3} \tag{10}\]4  
Other works along the lines of defining controllers for developing lane change or overtaking manoeuvres can be found in [54; 55; 56; 57; 58], among others. The above figures have been compared and validated by on-track tests because of the importance of that distance when computing the total distance required for the overtaking manoeuvre. These tests involved 4 drivers who carried out lane change manoeuvres at different speeds, measuring the distance travelled from the start of the steering wheel manoeuvre until reaching the parallel lane. To this end, a vehicle was instrumented with a Trimble R4 RTK GPS receiver, a L-CE Correvit non-contact speed sensor, an RMS FES 33 Gyroscopic platform (that provides accelerations and yaw angle) and a Bus Can Vector CANcaseXL interface (that provides steering wheel angle). The results for the 50 km/h case study (with a 2% maximum variation) are set out in \[\left(v_{1}+v_{3}\right)\cdot t_{T}\leq d_{3} \tag{10}\]5 A good correlation can be seen between the models and experimental data although with a slight underestimation of the distance by the models. This fact is taken into account by the decision module in order to adopt larger safety margins and theoretical and experimental results are combined.