Fast Spline Trajectory Planning: Minimum Snap and Beyond

In this paper, we study spline trajectory generation via the solution of two optimisation problems: (i) a quadratic program (QP) with linear equality constraints and (ii) a nonlinear and nonconvex optimisation program. We propose an efficient algorithm to solve (i), which we then leverage to use in an iterative algorithm to solve (ii). Both the first algorithm and each iteration of the second algorithm have linear computational complexity in the number of spline segments. The scaling of each algorithm is such that we are able to solve the two problems faster than state-of-the-art methods and in times amenable to real-time trajectory generation requirements. The trajectories we generate are applicable to differentially flat systems, a broad class of mechanical systems, which we demonstrate by planning trajectories for a quadrotor.