Modelling of aircraft trajectories for emergency landing using kinematoid chains

Most methods that compute trajectories for un- or low-powered flight operate under simplifying assumptions such as constant curve radii and wind conditions. Likewise, changes of air density with altitude that lead to significant differences between equivalent airspeed (EAS) und true airspeed (TAS) are often not considered. Some approaches are based on Dubins paths, which are introduced in Dubins LE (Am J Math 79: 497, 1957). They combine three sections to form a trajectory, which is the shortest from a given start to an end position (In the original work, the position extended by the heading, is referred to as configuration. Since configuration has a different connotation in our context, we use the term state, which can contain other parameters in addition to the position and the heading, e.g. orientation, configuration of landing flaps, landing gear, etc.). A maximum distant landing spot can be reached this way. Often, the targeted landing spot is closer to the aircraft. If it is approached using a Dubins Path, the excess height must be dealt with. Here, we present a method addressing the problem directly, namely finding a trajectory which reduces the excess height over its entire length. Furthermore, it takes spatial and temporal changes of wind and air density into account. Several conditions influence the final shape of the trajectory. For example, avoidance of obstacles and predefined areas is easily achieved. Our method is motivated by kinematic chains, which are used in robotics and computer animation. We extend and modify this principle by incrementally transferring start and end states of the trajectory, modelled as state vectors, into each other. The resulting intermediate states form ends of chain links. To connect initial and final states through the resulting chain, we solve the inverse kinematic problem known from robotics. We extend it by several conditions, which are derived from the flight mechanical characteristics of the modeled aircraft on the one hand and from the desired properties of the trajectory on the other. Using practical examples, we will show the performance of this method, which we have efficiently implemented on off-the-shelf hardware. The method is suitable for systems that assist in the event of engine failures as well as for modeling planned un- or low-powered flights like continuous descent approaches or return flights of space gliders. © 2023, The Author(s).