## 2 Problem Formulation  
In this paper, we address the problem of multidomain path planning and trajectory generation. We consider the cases of an HAUV moving underwater, aerially, and between the two media (water and air). The vehicle is required to follow a low-cost, safe, and smooth trajectory within its motion capabilities when random obstacles and disruptions exist. \begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline
**Variable** & **Description** & **Variable** & **Description** \\ \hline \(P\) & The whole path. & \(\Phi\{p_{1},p_{2},\ldots p_{i},\ldots,p_{n}\}\) & \(\Phi\) is denoted as the set of target spots the while \(p_{i}\) is coordinate of the \(i_{th}\) target spot in path. \\ \hline \(\text{sub}\ P_{i}\) & The \(i_{th}\) sub path in \(P\), each sub path contains 5–10 path segment, which denote the length of path the HAUV can detect when path re-planning. & \(\Gamma_{i}\) & The path segment between \(p_{i}\) and \(p_{i+1}\). \\ \hline \(D_{i}\) & The length of \(\Gamma_{i}\). & \(\xi_{i}\) & The overall weighted edge of \(i_{th}\) path segment. \\ \hline \(\xi_{surf/air/sea/c}\) & The overall weighted edge of surface/air/water/current fields. & \(v_{i}\) & The velocity with which HAUV move from \(p_{i}\) to \(p_{i+1}\). \\ \hline \(a_{i}\) & The acceleration with which HAUV move from \(p_{i}\) to \(p_{i+1}\). & \(v_{\text{max}}^{j}\) & The maximum velocity with which HAUV move from \(p_{i}\) to \(p_{i+1}\). \\ \hline \(a_{\text{max}}^{i}\) & The maximum acceleration with which HAUV move from \(p_{i}\) to \(p_{i+1}\). & \(OBS\{obs_{1},obs_{2},\ldots,obs_{k}\}\) & \begin{tabular}{} \end{tabular} \\ \(a_{\text{max}}^{i}\) & The maximum acceleration with which HAUV move from \(p_{i}\) to \(p_{i+1}\). & \(OBS\{obs_{1},obs_{2},\ldots,obs_{k}\}\) & \
\begin{tabular}{} \end{tabular} \\ \(a_{\text{max}}^{i}\) & The maximum acceleration with which HAUV move from \(p_{i}\) to \(p_{i+1}\). & \(OBS\{obs_{1},obs_{2},\ldots,obs_{k}\}\) &
\begin{tabular}{} \end{tabular} \\ \(obs_{j}\) is the set of all spots’ coordinate of the \(j_{th}\) obstacle. & \(CIR_{s}^{1}/SPH_{s}^{1}\) & The minimum circumscribed circle/sphere of \(obs_{s}\). \\ \hline \(CIR_{s}^{2}/SPH_{s}^{2}\) & The collision free circle/sphere out of \(obs_{s}\), which is an extension of \(CIR_{s}^{1}/SPH_{s}^{1}\) & \(D_{obs}^{j}\) & The shortest distance between the obstacle \(obs_{s}\) and \(SPH_{s}^{2}\). \\ \hline \(D_{mi}\) & The minimum distance between \(P_{m+i}\) and \(obs_{s}\). & \(tertree\) & KD-tree constructed by terrain map. \\ \hline \(obtree\) & KD-tree constructed by obstacles. & \(F_{t}^{x,y,z}\) & The function of trajectory. \\ \hline \(\alpha\) & The control parameter of scapegoat tree. & \(TE\) & The spots set of terrain map. (The barriers.) \\ \hline \(\Omega\{s_{1},s_{2},\ldots,s_{m}\}\) & \(\Omega\) is denoted as map of informative spots while \(\{s_{1},s_{2},\ldots,s_{m}\}\) are the coordinates of those spots. & \(Y_{air/sea}\) & The force HAUV need to overcome without current fields. \\ \hline \(Y_{air/sea}\) & The force HAUV need to overcome with current fields. & \(v_{c}\) & The velocity of currents. \\ \hline \(S_{air}\) & wing area of the HAUV. & \(\rho_{air/sea}\) & Density of air/sea water. \\ \hline \(C_{air/sea}\) & Coefficient of lift/fraction in air/sea water. & \(veh_{L}\) & The length of HAUV. \\ \hline \(COV_{i}\) & The covariance of the \(i_{th}\) informative spots group. & \(e_{x,y,z}\) & The expectation of the \(i_{th}\) informative spots group. \\ \hline \(x_{\text{max}}/y_{\text{max}}/h_{\text{max}}\) & The size of terrain map in simulation. & \(\eta\) & The control parameter of current fields checking. \\ \hline \end{tabular}
\end{table} 
Table 1: Nomenclature. For \(n\) clusters of information spots, our goal is to pick one target spot from each group, and minimize the energy cost for all spots traversed. The whole mission is required to conform to all constraints in static and dynamic environments, and the HAUV's motion capabilities.