Trajectory Waypoint Spacing for Spline-Based Flight Plans

A new guidance optimization scheme for spacing waypoints on spline trajectories is proposed. This scheme, the bounded area minimization algorithm, examines sequences of 3 waypoints that sample a given spline trajectory at its constituent knot locations and moves the interior waypoint to a location on the spline trajectory that minimizes the bounded area, computed using Green’s theorem, between the trajectory and the straight-line paths (legs) that connect adjacent waypoints. For spline trajectories defined by more than 3 knots, the algorithm can be applied sequentially to cover the entire chain of knots. Five motion primitives were chosen to test the performance of the optimization scheme on piecewise cubic polynomial spline trajectories. Two of these motion primitives (sinusoid and exponential) as well as two real-world trajectories that have been flight-tested on an MD530F platform were then simulated in a full nonlinear rotorcraft flight dynamics simulator to quantify and compare the effects of optimized versus baseline waypoint spacing. The bounded area minimization algorithm was extremely effective at reducing cumulative cross track error when the waypoint spacing was large enough that the aircraft trajectories closely matched the straight-line waypoint legs used in the algorithm. A method of determining the smallest waypoint spacing at which the optimization algorithm is still beneficial was proposed based on varying the wavelength of the ADS-33 Slalom MTE. For densely sampled spline trajectories that do not have knot spacings large enough to realize the benefits of the optimization algorithm, knot removal can sometimes reduce the number of waypoints required to represent the trajectory while maintaining the cumulative cross track error. However, there is no guarantee that subsequent bounded area minimization results in better performance than the baseline waypoint spacing. Determination of waypoint spacing for densely sampled spline trajectories is a recommended area of future work. A practical benefit of the bounded area minimization algorithm is that it requires very little modification to in-service coupled waypoint guidance flight director and autopilot schemes on existing aircraft, which improves the likelihood of its adoption by reducing the effort required for formal qualification and certification.