# A High-Fidelity Energy Efficient Path Planner for Unmanned Airships
## 5 Trajectory Objectives
### Arrival Time  
The time to traverse a cell bounded by the nodes \(n\) and \(n^{\prime}\) can be approximated by,  
\[t\left(n,n^{\prime}\right)\approx\begin{cases}\frac{\partial\cdot d(n,n^{ \prime})}{\left|v_{pg}\right|}=\frac{\partial\cdot d(n,n^{\prime})}{\left|v_{ w,\parallel}+v_{w,\parallel}\right|},&v_{w,\parallel}\cdot v_{pg}>0\\ \infty,&v_{w,\parallel}\cdot v_{pg}\leq 0\end{cases} \tag{3}\]  
where \(o\) is the orientation ratio (described in Section 3) and \(d\) is the distance between nodes. This time estimate extends the work of [PERSON] et al. [5] by considering that the distances and vectors are three-dimensional, and the time estimate approaches infinity when the wind magnitude along the vehicle trajectory is equal to or above the vehicle's cruise velocity. The equation adjusts for the non-linear flight trajectory between nodes using the orientation ratio. It is assumed that, when \(v_{vw,\perp}=-v_{w,\perp}\), the vehicle will achieve an angle of sideslip (crab angle) to offset the lateral wind. This  
Figure 5: Vehicle velocity vector components relative to the ground and to the wind.  
is an important distinction as the lateral wind reduces the vehicle's available goal-oriented velocity, whereas the parallel component reduces the traversal time. It is also important to note that the true traversal time depends on other factors such as vehicle stabilization effort and trajectory tracking error, which require a complete dynamic model flight simulation for a more accurate approximation.  
Although time between nodes can be minimized directly, it does not provide the operator any feedback on the final travel time before selecting a weight, which makes weight selection difficult. In real-world applications, the travel time between grid nodes is time-dependent [46; 47]. In other words, the arrival time cost is a function of the progress the aircraft has made towards arriving to the goal node. Therefore, it is appropriate to non-dimensionalize time by a user-selectable time goal and the distance remaining to the final goal node. This also has the advantage of extending the arrival time factor to the time-dependent case where the remaining cost is dependent on the total cost so far (i.e., how on schedule the UAV is to meet a given estimated time of arrival). Cumulative time can be non-dimensionalized by considering the user-selectable arrival time to the goal node, \(t_{g}\). The time objective cost function, \(\mathbb{C}_{t}\), can thus be written as,  
\[\mathbb{C}_{t}\left(n,n^{\prime}\right)=\begin{cases}\left(\frac{t_{sum}(n)}{ 2t_{g}}\right)-\left(\frac{d_{x}-d_{r}(n)}{2d_{g}}\right)+\frac{1}{2},&t_{sum} <t_{g}\\ 1,&t_{sum}\geq t_{g}\end{cases} \tag{4}\]  
where \(d_{g}\) is the initial Euclidean distance from the start to the goal node while, \(d_{r}\) is the remaining Euclidean distance from the current node \(n\) to the goal node and \(t_{sum}\) is the total time spent from the start node to current node. When \(\mathbb{C}_{t}\) is less than one half, the UAV is ahead of schedule, and when it is greater than one half, the UAV is behind schedule. Equation (4) enables the trajectory planner to adapt to unforeseen environments. Since lighter-than-air vehicles do not need a forward velocity to maintain lift, the velocity of the UAV can be modified along the length of the path to successfully arrive at the goal node by the user-set time. Under this assumption, the optimal vehicle speed, \(v_{ot}\), for time at each node can be defined as,  
\[v_{ot}\left(n^{\prime}\right)=\begin{cases}\left(\frac{d_{r}(n)}{t_{g}-t_{sum} (n)}-v_{w,\parallel}\cdot v_{pg}\right),&t_{sum}<t_{g}\\ v_{v,\max},&t_{sum}\geq t_{g}\end{cases} \tag{5}\]  
If the current travel time has already exceeded the time goal, the vehicle speed for time is set to maximum vehicle velocity, \(v_{v,\max}\), for the remaining duration of the flight. Changing the weight for arrival time, \(W_{t}\), affects the choice of the path, but also changes the vehicle's velocity, which is further explained in Section 5.2.