Use of Conformal Mapping in the Field of Flying Vehicle Route Planning

In this paper, a novel method for designing the flight paths of an aircraft is presented based on the concept of conformal mapping. Here, a low-altitude route-planning problem has been considered. In this problem, maintaining the control effort to reduce aircraft's altitude and increasing the speed with the limitations of Terrain Following (TF) and Terrain Avoidance (TA) issues, is the main strategy of this performance maneuver. In the proposed approach, attempts are made to convert the real space including terrains and obstacles, in which their data are provided using a digital elevation map, into a pseudo obstacle-free virtual space with no barriers and altitude constraints. In this regard, the concept of conformal mapping has been used as a facilitating mathematical tool for this problem-solving space transformation. The transformation of the problem-solving spaces under the mapping leads to solving the problem of dynamic reflection, the performance criterion, and the real altitude constraints in the virtual space. It is noteworthy that in designing a path in a newly converted space, the effect of barriers on the formation of flight routes is somehow included in the equations expressed in the virtual space. The results of multiple case studies and numerical optimizations performed for 2D geometrical terrains and obstacles show that the proposed approach is more consistent with the basic flight concepts as well as real-world applications.