## 3 Multimetric Map-Aware Routing Protocol (MMMR)
### Metrics Evaluation
This means that the best forwarding node will get a score notably higher than the others. This way, we highlight those good candidates, so that they have more chances to be selected as the next forwarding nodes. \[u_{1,i}(d_{i})=e^{-\left(\frac{d_{i}}{d_{ref}}\right)^{\alpha}} \tag{3}\]  
**Trajectory**: A communication link in a VANET remains operative for a short time, due to the high speed of the vehicles. If a geographic protocol does not consider the moving direction of the nodes, the current node could make wrong forwarding decisions based only on the distance and send packets to vehicles that were actually going away from the destination. Therefore, packet losses could increase. For that reason, taking into account the moving direction of vehicles is an important feature in VANETs. Hence, we decided to design a metric to obtain an accurate measure of the trajectory of the vehicles in VANETs. We define the trajectory metric of node \(i\) as a comparison of the metric of the future distance to the destination in time \(t\) of that node, _i.e._, \(u_{1,i}(d(t))\), with respect to the metric of the current distance to the destination in \(t=0\), _i.e._, \(u_{1,i}(d(0))\). We can see the result in Equation (4) and a drawing in Figure 1(a). That is, we prefer those forwarding nodes that get to the destination sooner. Therefore, consecutive values of the metric of the distance (_i.e._, \(u_{1,i}(d_{i})\)) for a moving node will grow throughout time, given that the node gets closer to the destination. According to Equation (5) and Figure 1(a), the distance to the destination of the node in \(t\) (_i.e._, \(d(t)\)) is computed from its current position in \(t=0\) (_i.e._, \(\vec{x_{l}}\)), the position of the destination (_i.e._, \(\vec{x_{l}}\)) and the average speed, \(\vec{v_{i}}\). The average speed, \(\vec{v_{i}}\), is computed from two consecutive positions (see Figure 1(b)). \(\|\cdot\|\) refers to the module function of a vector. The future position in \(t\) is \(\vec{x}+\vec{v}_{i}\cdot t\). Using the speed of the node (_i.e._, \(\vec{v_{i}}\)), we can give a higher score to nodes that will be closer to the  
Figure 2: Trajectory towards the access point (AP) from two consecutive geographic positions. destination sooner (AP). The idea is that with a higher speed, a node may arrive sooner to the destination, given that the distance to the destination decreases. \[u_{2,i}(d(t),d(0))=\frac{u_{1,i}(d(t))}{u_{1,i}(d(0))} \tag{4}\]  
\[d(t)=\parallel\vec{x}+\vec{v}_{i}\cdot t-\vec{x}_{D}\parallel \tag{5}\]  
Substituting the expression of the metric of the distance expressed in Equation (3) into the metric of the trajectory defined in Equation (4), we obtain:  
\[u_{2,i}(d(t),d(0))=\frac{e^{-\left(\frac{d(t)}{d_{ref}}\right)^{\alpha}}}{e^{- \left(\frac{d(0)}{d_{ref}}\right)^{\alpha}}}=e^{-\frac{1}{d_{ref}}\alpha\left( d(t)^{\alpha}-d(0)^{\alpha}\right)} \tag{6}\]  
Then, we compute the metric of the trajectory, \(u_{2,i}(d(t),d(0))\), using Equation (8), where we define \(\Delta_{d}(t)^{\alpha}\) as a measure of the variation of the distance in a time, \(t\), to the power of \(\alpha\) with respect to the current distance in \(t=0\) to the power of \(\alpha\). \[\Delta_{d}(t)^{\alpha}=d(t)^{\alpha}-d(0)^{\alpha} \tag{7}\]  
\[u_{2,i}(d(t),d(0))=e^{-\left(\frac{\Delta_{d}(t)}{d_{ref}}\right)^{\alpha}} \tag{8}\]  
Again, the use of an exponential function helps us to emphasize good nodes compared to the others. A higher qualification in \(u_{2,i}(d(t),d(0))\) is obtained if the node presents a faster decrease of its distance to the destination. Next, using a Taylor approximation in the estimation of the future position, \(d(t)\), we obtain Equation (9), where \(O(t^{2})\) represents the upper order terms:  
\[d(t)^{\alpha} = d(0)^{\alpha}+\frac{d}{dt}d(t)^{\alpha}\bigg{|}_{t=0}\cdot t+O(t ^{2}) \tag{9}\]  
To facilitate the calculation, we write Equation (9) in terms of the well-known expression, \(d(t)^{2}\) (_i.e._, with \(\alpha=2\)). This results in the computation of \(\frac{d}{dt}(d(t)^{2})^{\frac{\alpha}{2}}\).