## 3 Background
### Wireless Network Characteristics  
According to the Shannon-Hartley capacity theorem [32], in a wireless channel \(c\), the maximum communication channel capacity \(C_{c}\) and the data throughput \(T_{c}\) is related to the SNR as follows:  
\[C_{c}=B\log_{2}(1+SNR)=B\log_{2}(1+10^{\frac{RSS_{dBm}-PN_{dBm}}{10}})\ \rm{[ Mb/s]} \tag{4}\] \[T_{c}=\alpha_{T}C_{c}\ \rm{[ Mb/s]} \tag{5}\]  
Here, \(B\) is the frequency bandwidth of the channel in MHz. The data throughput \(T_{c}\) is a factor (by \(\alpha_{T}\)) of the maximum capacity \(C_{c}\), because the capacity is determined at the \"physical\" layer in the open systems interconnection (OSI) model [33], whereas the throughput is determined at the \"data transport\" layer, which includes the data link, network and physical layers in the OSI model. As our objective is to implement our solution at the application level, we use a metric called \"goodput\" \(G_{c}\), which is defined as the application-level data throughput. The goodput of a network can be modeled with Equation (6) and is measured right at the \"application\" layer (the top-most layer in the OSI model). The goodput is characterized as follows:  
\[G_{c}(RSS)= \begin{cases}\alpha_{G}T_{c}(RSS)&\text{when }RSS<RSS_{\rm threshold}\\ G_{max}&\text{when }RSS\geq RSS_{\rm threshold}\end{cases}  \tag{6}\]  
\(\alpha_{G}\) is a parameter that determines the proportionality of the goodput with respect to the throughput. The model indicates that the goodput has a linear relationship with the maximum channel capacity \(C_{c}\). The goodput depends on the RSS and becomes saturated to a maximum level (\(G_{max}\)) for a particular link when the RSS reaches an upper threshold \(RSS_{\rm threshold}\) (which is different for different wireless device). This model is verified with the experimental data presented in Figure 2(a), where the \(RSS_{\rm threshold}\) is observed to be \(-55\) dBm. The lower threshold for the RSS is normally the sensitivity (\(R_{S}\)) of the wireless sensor, which in this case is \(-82\) dBm (for a Zyxel NWD2105 device operating in the 2.4 GHz 801.11b band). Figure 2(a) also shows a linear fit for the RSS and goodput relation using a minimum least squares error (MLSE) technique. The goodput model considered in this work is a more practical model than the Shannon capacity model in Equation (5). A network can make use of a relay node using either a relay-and-retransmit (RR) scheme or a flow-pipe (FP) scheme for data transport through the relay [16]. While the former is based on packet switching, the latter has the functionality of a radio-repeater. The total network goodput \(G_{t}\) of a robotic network having \(n\) nodes (the first being the server node, the \(n^{th}\) node being the client node and \(n-2\) relay nodes in between the server and the client nodes) is given by [34]:  
\[G_{t}= \begin{cases}\left[\sum\limits_{\begin{subarray}{c}i\in[1,n-1]\\ j+i+1\end{subarray}}\frac{1}{G_{ij}}\right]^{-1}=\begin{bmatrix}\sum\limits_{ \begin{subarray}{c}i\in[1,n-1]\\ j=i+1\end{subarray}}\left(\frac{1}{\alpha_{T}\alpha_{G}Blog_{2}(1+10^{\frac{BS \delta_{dBm}(i,j)-PN_{dBm}(i,j)}{10}})}\right)\end{bmatrix}^{-1}&\text{for the RR scheme}\\ \sum\limits_{\begin{subarray}{c}i\in[1,n-1]\\ j=i+1\end{subarray}}\{G_{ij}\}=\min\limits_{\begin{subarray}{c}i\in[1,n-1]\\ j=i+1\end{subarray}}(\alpha_{T}\alpha_{G}Blog_{2}(1+10^{\frac{BS\delta_{dBm}(i, j)-PN_{dBm}(i,j)}{10}}))\end{bmatrix}&\text{for the FP scheme}\end{cases}  \tag{7}\]  
where \(G_{ij}\) and \(RSS_{dBm}(i,j)\) represent the goodput of the link and the received signal strength (from node \(j\) to node \(i\)), respectively. In this work, we assume that we use one relay node and that the relay node uses the FP scheme for relaying the data from the server to the client. Therefore, Equation (7), for a single-relay SRC using the FP relaying scheme, becomes:  
\[G_{t}=\min(G(r,s),G(r,c)) \tag{8}\]  
Here, \(G(r,s)\) and \(G(r,c)\) are the goodput of the server-relay and relay-client links, respectively.