Trajectory Generation and Regeneration for Constrained Differentially Flat Control Systems

Abstract

In this study, differential flatness principle is applied in real time optimal flight trajectory generation. This principle allows formulating the desired output trajectory through Bspline parameterization. Integrating these methodologies with sequential quadratic programming, an optimal feasible trajectory that meets all the given and dynamical limits, is generated. Through this fashion, it is guaranteed to generate dynamically feasible trajectories passing, as closely as possible, by given waypoints which guide the vehicle to track its intent. For the simulation purpose, this methodology is applied to two under-actuated vehicle models (quadrotors and Dubins' airplanes)and their maneuverability for a given mission is compared to show the validity of the integrated methodologies. While the majority of similar methodologies focus on an uninformed search for a dynamically feasible trajectory through an outer checking loop, the methodology provided in this paper benefits from a guided search for a feasible solution that an optimization algorithm provides. The advantage of this study over the ones that do benefit from an optimization algorithm is that the constraints on key dynamical states of the vehicles are strictly considered in a continuous manner instead of sampling hypersurfaces. To do so, the geometrical feature of the B-splines is utilized and the constraints are checked at times in which the dynamical state of interest reaches to an extremum. This way, the constraints do not necessarily have to be linear or representable with polynomials. In this work, it is shown that these critical times are roots of polynomials with unique sets of coefficients. Moreover, the local property of B-splines is utilized for instantaneous regeneration of the trajectory without distorting the entire path and the continuity when the vehicle needs a rapid update in trajectory plan or collision avoidance. Through the knot insertion algorithm, it is shown that it is always possible to design a new trajectory that deviates from the old one before the vehicle reaches the deviation point.