## Main Design of DERM
### Sink Location Calibration  
As stated above, the sink location estimated by the mobility pattern may deviate from the real one, which will lead to inefficient routing. A straightforward solution is that the sink regularly broadcasts its location to the entire network. However, it will bring tremendous control overhead. Therefore, we introduce a location calibration method with which the updated location information is forwarded by a node (e.g., node \(i\)) only when the route length is enlarged beyond a threshold due to the stale information [11]. Let \(d_{updt}\) denote the estimated routing path length of a packet when the updated sink location is used for forwarding, and let \(d_{prev}\) denote that when the previous location information before updating is used. The method ensures \(d_{prev}/d_{updt}\leq\alpha\), where \(\alpha\) is the route length dilation threshold and \(\alpha>1\). \(d_{updt}\) and \(d_{prev}\) are estimated as  
\[d_{updt}=dist(i,L_{updt}), \tag{12}\]  
\[d_{prev}=dist(i,L_{prev})+dist(D_{prev},L_{updt}). \tag{13}\]  
\(d_{updt}\) is calculated as the shortest distance from node \(i\) to the travel path within the delay constraint, termed \(L_{updt}\), which is predicted based on the updated sink location. To compute \(d_{prev}\), we first calculate the distance from \(i\) to the travel path \(L_{prev}\) predicted based on the previous location, and obtain the corresponding virtual destination \(D_{prev}\). If \(D_{prev}\) is not on the path \(L_{updt}\), to ensure on-time delivery, the packet should be further transferred to \(L_{updt}\) via the track routing introduced in Section 4.4. Thus, \(d_{prev}\) includes an additional item, \(dist(D_{prev},L_{updt})\). Figure 5 shows an example of the sink location calibration in DERM. The sink is currently located at \(A\) deviated from the estimated location \(B\) for \(\Delta\) meters. The travel path length within the delay constraint is \(l\). The sink broadcasts its current location information. Each node receiving the updated information will calculate its \(d_{updt}\) and \(d_{prev}\) based on Equations (12) and (13). Figure 5 depicts \(d_{updt}\) and \(d_{prev}\) of three nodes, i.e., \(N_{1}\) in Area I, \(N_{2}\) in Area III, and \(N_{3}\) in Area V. Take node \(N_{3}\) as an example. Its \(d_{updt}\) is \(|N_{3}C|\) and \(d_{prev}\) is \(|N_{3}E|+|CE|\). If \(d_{prev}/d_{updt}>\alpha\), the node will update the sink location and forward it. Without loss of generality, the coordinate of \(A\) is set as \((0,0)\). A node whose coordinate is \((x,y)\) will be required to forward the updated location if satisfying the following conditions:  
\[\left\{ \begin{array}{ll}\sqrt{\left(x-\Delta\right)^{2}+y^{2}}\bigg{/} \sqrt{\left(x+y\right)^{2}}>\alpha,&\qquad\left(\text{Area I}:x\leq 0\right),\\ \sqrt{\left(x-\Delta\right)^{2}+y^{2}}\bigg{/}|y|>\alpha,&\qquad\left(\text{ Area II}:0<x\leq\Delta\right),\\ \left(x-l+|y|\right)\bigg{/}\sqrt{\left(x-l\right)^{2}+y^{2}}>\alpha,&\qquad \left(\text{Area IV}:l<x\leq l+\Delta\right),\\ \left(\sqrt{\left(x-l-\Delta\right)^{2}+y^{2}}+\Delta\right)\bigg{/}\sqrt{ \left(x-l\right)^{2}+y^{2}}>\alpha,&\left(\text{Area V}:x>l+\Delta\right). \end{array} \right. \tag{14}\]  
From Equation (14), we can sketch out the location updating area (e.g., the shaded area in Figure 5). It can be observed that: (1) exploiting the mobility for delay-aware routing can effectively reduce the overhead of location calibration, compared to the immediate multi-hop routing. For instance, nodes in Area III need not update the sink location as their routing paths are not affected by the estimation error; (2) the updated area is bounded, and its size is related to \(\Delta\) and \(\alpha\). Only if the deviation \(\Delta\) does exist and incurs route dilation at a node beyond the threshold \(\alpha\), the updating information will be relayed. These two facts indicate that the introduced method can deal with the sink movement deviations and guarantee energy-efficient routing in DERM at a very low cost.