# Adaptive Path Planning for UAV-Based Pollution Sampling
## 3 Materials and Methods
### Problem Statement  
The performance of the GPP is heavily influenced by the overarching mission scenario \(\mathcal{S}\), which is defined by mission parameters like measurement profiles, the positioning of human-specified waypoints, airspeed between waypoints, and other factors. They directly impact key criteria, including total flight distance, flight endurance, overall energy consumption, and the quality of the collected measurement data. As a result, these parameters must be carefully calibrated to align with the mission objectives. The primary goal of the optimization process is to identify the optimal mission scenario \(\mathcal{S}\) and the corresponding optimal values of mission parameters \(\mathbf{p}\) to determine the most effective global path \(\mathcal{P}^{G}\). This challenge represents a multi-objective optimization problem, which can be formulated as follows:  
\[ \begin{array}{l}\text{minimize}\ \mathbf{C}(\mathcal{S},\mathbf{p})=\big{[}c_{1 }(\mathcal{S},\mathbf{p}),c_{2}(\mathcal{S},\mathbf{p}),\ldots,c_{n}(\mathcal{ S},\mathbf{p}),\ldots,c_{N}(\mathcal{S},\mathbf{p})\big{]},\\ \text{subject to}\ \Omega(\mathcal{S},\mathbf{p}).\end{array}  \tag{1}\]  
In this context, \(\mathcal{S}\) and \(\mathbf{p}\) serve as decision variables, as defined earlier. The function \(c_{n}\) represents the \(n\)-th objective criterion within the set of non-conflicting goals, where \(n=1,2,\ldots,N\). The boundaries and constraints, collectively denoted by \(\Omega\), arise from general mission requirements, legal regulations, expert knowledge, and other factors. To simplify the problem in certain scenarios, the mission scenario \(\mathcal{S}\) can be determined using a subset of decision variables associated with the mission parameter \(\mathbf{p}\) represented in a generalized vector form as follows:  
\[\mathbf{p}=\big{[}\mathbf{w}_{1}\quad\mathbf{w}_{2}\quad\cdots\quad\mathbf{w} _{n}\quad\cdots\quad\mathbf{w}_{N_{G}}\quad\mathbf{g}\big{]}, \tag{2}\]  
where \(\mathbf{w}_{n}\) represents the \(n\)-th vector of waypoints, with \(n=1,2,\ldots,N_{G}\), and \(\mathbf{g}\) denotes the vector of global mission-specific settings. \(N_{G}\) represents the total count of waypoints optimized by the GPP. The parameters in \(\mathbf{w}_{n}\) are specified individually for each waypoint, whereas \(\mathbf{g}\) contains parameters that apply universally to the entire mission. Each _global waypoint_\(w\) is described as  
\[\mathbf{w}=\big{[}w_{x}\quad w_{y}\quad w_{z}\quad w_{x}\big{]}. \tag{3}\]In this representation, \(w_{x}\), \(w_{y}\), and \(w_{z}\) define the desired positions of the UAV relative to the inertial reference frame \(\mathcal{F}^{i}\). The variable \(w_{x}\) specifies the UAV's heading within \(\mathcal{F}^{i}\). Consequently, each waypoint \(\mathbf{w}\) represents a position characterized by four degrees of freedom. The vector \(\mathbf{g}\), in turn, is highly scenario-dependent and varies based on the specific mission \(\mathcal{S}\). However, in a general context, it can be expressed as  
\[\mathbf{g}= \begin{bmatrix}g_{1}&g_{2}&\cdots&g_{n}&\cdots&g_{N_{g}}\end{bmatrix} , \tag{4}\]  
where \(g_{n}\) represents the \(n\)-th parameter within a total of \(N_{g}\) global mission parameters. An infinite number of local and global extrema are often exhibited by multi-objective problems. As a result, it is necessary to examine a set of solutions, each of which satisfies the defined objectives. To address this complexity, the widely used concept of Pareto optimality is employed [31]. A Pareto-optimal solution exists when no alternative solution can improve at least one objective without negatively impacting others. Here, the optimization process focuses on estimating basic measures as objectives, discussed in Section 4.3. Alternatively, the global criterion method can be used to convert the multi-objective cost function (1) into a single-objective problem [31].