# A High-Fidelity Energy Efficient Path Planner for Unmanned Airships
## 6 Numerical Simulations  
### Trajectory Planning and Test Setup  
Once a multi-objective cost function field is formulated, the desired trajectory through the field is determined using 3D grid search techniques. A terrain map nodal list is generated based on an rectangular grid representation of 2 km \(\times\) 2 km \(\times\) 1 km with a nodal resolution of 50 m \(\times\) 50 m \(\times\) 10 m for a total of 160 \(\times\) 10\({}^{3}\) nodes. This resolution was chosen because it is roughly the same as 0.75 arc-sec digital elevation maps that are publicly available from GeoBase. To reduce the computation time, the connections between nodes are generated using the vector neighbor operator [5]. It provides an average horizontal angle increment and allows for flight path angles of some predetermined values to connect successive path nodes as they may not lie within adjacent grid cells. This technique avoids the exhaustive search within all the nodes and provides a better scalability to the search space. As it is illustrated in Figure 7, the black arrow represents the UAV direction while the gray lines represent possible UAV directions (assuming that the nodes exist and are not outside the search environment). Redundant connections are eliminated (for example, [2; 2; 2] is excluded because it is a multiple of [1; 1; 1]). Furthermore, connections that create an angle greater than the critical turning or pitching angles of the UAV described in Section 3 were also omitted. The field strengths for obstacles are computed using (15) to (17). The field strengths for time and energy consumption are computed on demand as they depend on the nodal approach direction, wind conditions and other flight characteristics. The multi-objective cost function is minimized using the Dijkstra algorithm [51] and the computation time, total trip time, energy consumption and the distance to the closest obstacle along the path are recorded. In the implementation of the planner, the WVF data would be recomputed frequently to capture changes over time and could be supplemented by publicly available weather forecasts for added confidence. Figure 6: Definition of term \(\mathbb{C}_{a}^{o}\) in relation to buildings. Figure 7: Vector neighbor operator. As mentioned in Section 1, many planners described in the literature consider only uniform WVFs or assume constant vehicle velocity paths. Additionally, planners that do include variable WVFs were found to have a coarse terrain resolution (>1 km). The low resolution of a planner can oversimplify weather information between nodes reducing the planning problem to the shortest straight line path in a uniform wind field case. The following four test cases were identified to illustrate the importance of a fine terrain nodal resolution, spatial WVFs, and temporal vehicle velocities. **Case 1, Low terrain resolution:** This case emulates low terrain resolution planning by constructing a trajectory that consists of the shortest straight line segment between the current node and the goal node that respects the vehicle constraints and avoids known obstacles. The cost function (1) is modified to \(\mathbb{C}\left(n,n^{\prime}\right)=d_{r}\left(n\right)+W_{a}\cdot\mathbb{C}_{ a}\left(n,n^{\prime}\right)\). This case uses the spatial WVF, and the planner optimizes the velocity along the path. **Case 2, Constant velocity:** The optimal velocity Equation (13) is ignored, and the vehicle velocity is held at a constant value equal to the distance between the start and goal nodes divided by the goal time. This case uses the spatial WVF. **Case 3, Uniform WVF:** The planner assumes a constant and uniform WVF equal to the average bulk flow direction and magnitude at all nodes (computed from the spatial WVF). The planner optimizes the velocity and the path. **Case 4, Complete planner:** This is the full trajectory planner proposed in this work. It is applied as outlined in Section 5 using fine nodal resolution. This case uses the spatial WVF, and the planner optimizes the velocity and the path.