## 4 RoCoMAR Algorithm
### Link Quality Gain by using Robotic Relays  
One of major advantages of applying RoCoMAR to MSN is that RoCoMAR can greatly improve the quality of links in a mobile environment by using multiple robotic nodes. In order to gain analytical insight into the link quality gain of RoCoMAR, we consider a simple numerical model. To facilitate discussion and simplify the problem, we assume that the network is unsaturated, and hence the packet loss due to inter-node interference is negligible or can be recovered by using ARQ (Automatic Repeated reQuest) or other MAC technologies without significantly impairing the network throughput and packet delivery ratio. As shown in [35], the received signal strength is dominated by the distance from the transmitter to the receiver. More specifically, as the distance between the transmitter and receiver becomes longer, the minimum energy required for a successful data packet transmission sharply increases. The average reception power, \(P_{r}\), at a receiver from the transmitter with the distance \(d\) in free space is expressed by the Friis equation as \(P_{r}=\frac{P_{t}G_{t}G_{r}\lambda^{2}}{(4\pi d)^{2}}\), where \(P_{t}\) is transmitted power, \(G_{t}\) and \(G_{r}\) are the transmitter and receiver antenna gain respectively, both of which are assumed to be 1 for simplicity. \(\lambda\) is the wavelength of signal carrier in meters. Then, the received signal to noise ratio, \(SNR\), in an AWGN (Additive White Gaussian Noise) channel, becomes  
\[SNR=\frac{P_{r}}{P_{n}}=\frac{P_{t}\lambda^{2}}{P_{n}(4\pi d)^{2}} \tag{3}\]  
where \(P_{n}=WKT\) is thermal noise power [36], in which \(K=1.38\times 10^{-23}J/K\) is Boltzmann constant, \(T\) is room temperature in Kelvins (typically standard temperature of \(290K\)) and \(W\) is signal bandwidth in Hertz. Assuming that DBPSK (Differential Binary Phase Shift Keying) is used as the modulation scheme, as discussed in [37], the bit error probability, \(P_{be}\), for the AWGN channel is given by \[P_{be}=\frac{1}{2}e^{-\frac{E_{b}}{N_{0}}} \tag{4}\]  
where \(E_{b}\) is the energy per bit and \(N_{0}\) is the noise power spectral density. Depending on modulation scheme, \(E_{b}/N_{0}=\frac{SNR}{M}\), where \(M\) is the number of information bits per symbol. For DBPSK modulation scheme, \(M=1\). Then, from Equations (3) and (4), bit error probability, \(P_{be}\), becomes  
\[P_{be}=\frac{1}{2}e^{-SNR}=\frac{1}{2}e^{-\frac{B\lambda^{2}}{WKT(4\pi d)^{2}}}= \frac{1}{2}e^{-\frac{\Gamma}{d^{2}}} \tag{5}\]  
where \(\Gamma=\frac{P_{\lambda}2}{WKT(4\pi)^{2}}\). Let \(m\) denote the length of a data packet in bits. Also, assume that bit errors in a packet follows an uniform distribution with the probability of \(P_{be}\) for simplicity. Then, from Equation (5), the probability of a successful packet transmission, \(P_{ps}\), can be obtained as  
\[P_{ps}=(1-P_{e})^{m}=(1-\frac{1}{2}e^{-\frac{\Gamma}{d^{2}}})^{m} \tag{6}\]  
Also, let \(R_{u,v}\) denote the packet delivery ratio (PDR) of the link between arbitrary node \(u\) and \(v\). Then, for long-term packet transmissions, \(R_{u,v}\) can be approximated by the probability of a successful packet transmission, \(P_{ps}\). That is  
\[R_{u,v}\approx\left(1-\frac{1}{2}e^{-\frac{\Gamma}{d^{2}_{(u,v)}}}\right)^{m} \tag{7}\]  
where \(d_{(u,v)}\) represents the distance between node \(u\) and node \(v\). Now, consider a route that has \(n\) links _i.e._, there are \(n+1\) nodes on the route. Assume that the \(i^{th}\) link is formed by node \(i\) and node \((i+1)\) on the route, where \(1\leq i\leq n\). Then, the end-to-end PDR of the route is  
\[R_{ete}=\prod_{i=1}^{n}R_{i,i+1} \tag{8}\]  
where \(1\leq i\leq n\). Suppose that \(j^{th}\) link on the route has the lowest quality and this link is reinforced by \(l\) robotic nodes (where \(1\leq j\leq n\)). Then, the original link quality of the \(j^{th}\) link is \(P_{j,j+1}^{ori}=\left(1-\frac{1}{2}e^{-\frac{\Gamma}{d^{2}_{(j,j+1)}}}\right) ^{m}\).