  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.
