Spatio-Temporal Trajectory Optimization via Feature Space Collocation for Large-Scale Quadrotor Flights

Optimizing large-scale single-quadrotor trajectories over extended distances and durations is challenging due to intricate spatio-temporal coupling. Existing methods decouple the problem into interleaved spatial and temporal steps to reduce complexity, causing optimality loss and low success rates under constraints and at scale. This article proposes a method that leverages spatio-temporal coupling to generate large-scale trajectories connecting hundreds of waypoints with guaranteed optimality and high success rates. The method initiates by constructing a feature space where a feature-explicit formulation (FEF) of the spatio-temporal trajectory optimization problem is established. In a bilevel framework, the lower-level solves spatial optimization, while the upper-level addresses time allocation. By virtue of feature explicitness, the lower-level solution of FEF reveals dependence between the spatial and temporal features of the equality constrained optimal trajectory, and is referred to as the feature flow. Enforcing feature flow in the inequality constrained time allocation problem effectively turns the bilevel problem into a single-level temporal feature optimization problem. A key contribution is the analytic derivation of bilevel gradients through feature flow, enabling simultaneous spatio-temporal search during temporal optimization. The resulting method is guaranteed to converge to a local optimum under a mild tightening of constraint boundaries. Additionally, FEF is stable and efficient. Numerical experiments show that the proposed method outperforms existing bilevel approaches in terms of optimality, stability, and efficiency. It reduces trajectory cost by 29% in highly constrained settings and optimizes large-scale trajectories of up to 500 waypoints with success rates above 90%.