Geometrical Properties of Aircraft Equilibrium and Nonequilibrium Trajectory Arcs

This article is concerned with methods of computing a path in 3-DOF space that describes the desired translational motion of an aerial vehicle. Trajectory planning is an optimization problem which generates an optimal trajectory between two configurations in the state space, considering a given performance index (time, energy or distance). Its feasibility depends on the choice of the optimization method, the performance index and a number of constraints from various nature, the latter depending essentially on the vehicle itself (architecture, dynamics and actuation modes) and the environment in which the vehicle moves (endurance, airspeed, altitude, landing and takeoff modes, etc.) [3, 5, 12, 13]. The classical differential geometry curve theory is a study of 3D space curves with orthogonal coordinate systems attached to moving points on the space curve [1, 2, 4, 11]. The atmosphere is considered to be an isotropic and homogeneous medium, i.e. when there is no wind and the air density is constant with altitude. Classically, in motion planning and generation, methods such as continuous optimization and discrete search are sought. In [9] randomized motion planning algorithm by employing obstacle free guidance system as local planners in a probabilistic road-map framework are presented. In [13], path planning of autonomous fixed wing aircraft is based on a learning real-time A* search algorithm, considering only the motion on a horizontal plane. A family of trim trajectories in level flight is used in all these references to construct paths. In [7], plans are described as the concatenation of a number of well defined motion primitives selected from a finite library.