# Energy Reduction Multipath Routing Protocol for MANET Using Recoil Technique
## 3 Proposed Protocol (AOMDV-ER)
### Computation of Recoil off time  
The Figure 5b shows the recoil time calculation to nodes which are inside the curve. First, it is necessary to find out whether point \(P\)\(\left(x_{p},y_{p}\right)\) is inside the curve or not. The calculation related to this carried out as follows  
\[c^{2}=\ a^{2}+b^{2},a=\ x_{2}-\ x_{1},\ b=\ y_{2}-\ y_{1}\]  
The slope of SD line \(m=b/a\) and the equation can be written as  
\[y-y_{1}=m(x-x_{1}) \tag{1}\]  
\[y-y_{1}=\frac{b}{a}(x-x_{1})\]  
\[bx-ay-(bx_{1}-ay_{1})=0 \tag{2}\]  
The perpendicular distance of \(P\)\(\left(x_{p},\ y_{p}\right)\) from SD line can be express as under  
\[p=\frac{\left|b\ x_{p}-\ ay_{p}-bx_{1}+ay_{1}\right|}{\sqrt{(a^{2}+b^{2})}}= \ \frac{\left|a\left(y_{1}-y_{p}\right)-b\left(x_{1}-x_{p}\right)\right|}{c} \tag{3}\]  
The equation of line perpendicular to SD line and passing through point \(P\) can be given as  
\[y-y_{p}=-\ \frac{a}{b}\left(x-x_{p}\right)\]  
\[ax+by-\left(ax_{p}+by_{p}\right)=0 \tag{4}\]  
The value of e is the perpendicular distance on line (4) from point S.  
Figure 5: (**a**) Recoiled off time distribution to nodes 1, 2 and 3; (**b**) Recoiled off time calculation; (**c**) Recoiled off time calculation for an arbitrary node N.  
\[e=\ \frac{a(x_{p}-x_{1})+b(y_{p}-y_{1})}{c} \tag{5}\]  
The shape of the curve is approximately an ellipse, hence the function \(f(e)\), which gives the extreme value for point P in order to be inside the ellipse. The center coordinate (C\({}_{1}\)) of an ellipse, equation of an ellipse and value of the \(f(e)\) can be calculated as below. \[\mathrm{C}_{1}\ \left(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2}\right)\]  
Equation of an ellipse  
\[y-y_{1}=m(x-x_{1}) \tag{1}\]0  
\[y-y_{1}=m(x-x_{1}) \tag{1}\]1  
For the successful transfer of a packet, it is therefore required that \(\frac{1}{(1-e_{i,i+1})}\) is the mean number of transmission. Thus, the effective transmission energy between nodes i and i+1, (including the effect of link error) is given by Equation (13)  
\[y-y_{1}=m(x-x_{1}) \tag{1}\]2  
The link error rate changes frequently, hence probability distributions with random variables have been taken into consideration, which is given by Equation (12) and the similar case is for the transmission energy between the two corresponding nodes. However, the prediction based link lifetime of a node can be estimated before the link error occurs between nodes, which are expressed by Equation (13). \[y-y_{1}=m(x-x_{1}) \tag{1}\]3 \[y-y_{1}=m(x-x_{1}) \tag{1}\]4  
The distance and the transmission power are the determinant factors for computing the strength of the received signal. Consider the points P, A, B, E and M as a few locations of node P. The relative speed and the distance determines the link lifetime of node in network. The node P moves from A to B in \(\Delta t_{2}\) time interval and from B to E in \(\Delta t_{1}\) time interval (\(\Delta t_{1}=\ \Delta t_{2}=\Delta t\)). Similarly, several such reference points are taken into consideration in order to determine the link lifetime and the relative speed. The relative speed can be express as follows with the help of Figure 6b, (using cosine rule)  
\[y-y_{1}=m(x-x_{1}) \tag{1}\]5  
\[y-y_{1}=m(x-x_{1}) \tag{1}\]6  
where \(d_{i}\), \(d_{j}\) and \(d_{k}\) values can be calculated from Equation (5). However, \(cos\alpha=-cos\beta\), hence Equations (15) and (16) can be simplified as  
\[y-y_{1}=m(x-x_{1}) \tag{1}\]7 \[y-y_{1}=m(x-x_{1}) \tag{1}\]8  
Therefore, the value of speed is  
\[y-y_{1}=m(x-x_{1}) \tag{1}\]9  
Now computing the link lifetime \(T_{L}\\[y-y_{1}=\frac{b}{a}(x-x_{1})\]0  
where \(\theta\) represents angle \(\angle\)BSD. The value of the node speed and link lifetime are expressed by the above Equations (17) and (18). Unlike AODV, the maintenance operation can be done at local level when link error is encountered which reduces the extra overhead in the network.