# Reactive Planning-Driven Approach to Online UAVs Mission Rerouting and Rescheduling
## 3 Approach to UAV Fleet Online Routing
The forecasted weather conditions are described by the set \(\mathbb{F}\) of pairs composed of direction \(\theta\) and wind speed \(vw\ (\theta,vw)\in\mathcal{F}\), i.e., \(\mathbb{F}\) is defined as follows: \(\mathbb{F}=\{(\theta,vw)|\theta\in[0^{\circ},360^{\circ}),vw\in[0,\ \mathcal{F}(\theta)]\}\). Where \(Z(\theta)\) is a function whose values determine the maximum forecasted wind speed for a given direction \(\theta\). The weather conditions determine the admissibility of the adopted sub-mission's plan \({}^{l}S\), i.e., they determine whether the batteries of UAVs will be prematurely discharged during implementation. A function \(Y_{k,l}(\theta)\) determines the borderline wind speed (for a given direction \(\theta\)), which guarantees the successful completion of the delivery plan by the \(U_{k}\) during the sub-mission \({}^{l}S\) in the distribution network \(G\): \(Y_{k,l}(\theta)=\max\Gamma_{k,l}(\theta)\), where \(\Gamma_{k,l}(\theta)\)--the set of the wind speed's values \(vw\) for a given direction \(\theta\), for which the batteries of \(U_{k}\) are not discharged. The sub-mission's plan \({}^{l}S\) is assumed [12] to be resistant to the forecast weather conditions, \(\mathbb{F}\), if the boundary wind \(Y_{k,l}(\theta)\) of all \(U_{k}\in{}^{l}\mathcal{U}\) in any direction \(\theta\) does not exceed the forecasted value \(Z(\theta)\): \(\forall_{U_{k}\in{}^{l}\mathcal{U}}\psi_{\theta\in[0^{\circ},360^{\circ})}Y_{k,l }(\theta)\geq\mathcal{F}(\theta)\). The implementation of the designated mission \(S\) may be subject to various disturbances \(IS\). Among them, there are sudden changes in the weather (beyond the expected \(\mathcal{F}^{*}(\theta)\) range and changes in orders \(Z\)), order changes (increased/decreased amount of ordered deliveries \(Z^{*}\)), changes to the number of delivery points served (changing the structure of the network \(G^{*}\)). The UAVs' fleet, when performing the mission plan \(S\), meets a disturbance \(IS(t^{*})\)--covering one of the cases: the weather \(\mathcal{F}^{*}(\theta)\), the network \(G^{*}\), orders \(Z^{*}\), the time \(t^{*}\). In such situations, it becomes necessary to answer to the question: _Does a re-route planexit for mission \(S^{*}\) that guarantees timely deliveries in a given time horizon \(H\) and at acceptable battery levels?_  
In order to provide an example to illustrate the proposed approach, let us consider a distribution network (from Figure 1a) covering an area of 100 km\({}^{2}\) and containing two bases, \(NB=\{N_{1},N_{2}\}\), and 38 delivery points \(ND=\{N_{3},\ldots,N_{40}\}\). The goods are delivered by a fleet of 4 UAVs (\(\mathcal{U}=\{U_{1},\ldots,U_{4}\}\)) divided into two sub-fleets (\(\mathcal{U}^{1}=\{U_{1},U_{2}\}\) and \(\mathcal{U}^{2}=\{U_{3},U_{4}\}\)) serving the bases \(N_{1}\) and \(N_{2}\), respectively. The technical parameters of the UAVs are shown in the table in Figure 1c. The weight of individual orders is: \(z_{3}\div z_{6}=5,\ z_{7}=10,z_{8}=z_{9}=15\), \(z_{10}\div z_{16}=5\), \(z_{17}=10\), \(z_{18}=z_{19}=15\), \(z_{20}\div z_{26}=5\), \(z_{27}=10\), \(z_{28}=15\), \(z_{30}\div z_{36}=5\), \(z_{37}=10\), \(z_{38}=z_{39}=15\), \(z_{40}=5\). These assumptions are used to seek the UAVs mission 5 plan guaranteeing timely deliveries (within the time horizon of 2.5 h) for the required amount of goods. The example of an admissible solution is presented in Figure 2. It consists of six sub-missions: \(S=\left({}^{1}S,{}^{2}S,\ldots,{}^{6}S\right)\) following the wind speed constraints of \(<\)9 m/s.