Towards Optimizing a Convex Cover of Collision-Free Space for Trajectory Generation

TL;DR

An online iterative algorithm to optimize a convex cover to under-approximate the free space for autonomous navigation to delineate Safe Flight Corridors to validate the effectiveness of the proposed algorithm using a range of parameterized environments and show its applications for two-stage motion planning.

Abstract

We propose an online iterative algorithm to optimize a convex cover to under-approximate the free space for autonomous navigation to delineate Safe Flight Corridors (SFC). The convex cover consists of a set of polytopes such that the union of the polytopes represents obstacle-free space, allowing us to find trajectories for robots that lie within the convex cover. In order to find the SFC that facilitates trajectory optimization, we iteratively find overlapping polytopes of maximum volumes that include specified waypoints initialized by a geometric or kinematic planner. Constraints at waypoints appear in two alternating stages of a joint optimization problem, which is solved by a novel heuristic-based iterative algorithm with partially distributed variables. We validate the effectiveness of our proposed algorithm using a range of parameterized environments and show its applications for two-stage motion planning.

Figures (10)

• Figure 1: The influence of convex cover on trajectory generation. (a) Seed points (blue) determine the volume of the largest polytope (red) within a map (obstacles in grey). The largest-volume polytope can be identified by selecting the seed point at the orange star. (b) Polytopes may not always cover a large part of the trajectory, even if they have a large volume. (c, d) Given an initial path (black), the quality of the final trajectory (green) depends strongly upon which convex cover was used to optimize it.
• Figure 2: SFC generation pipeline of previous works and our method. Given waypoints from the path planning, the SFC generator selects seed points or lines from waypoints and outputs overlapping polytopes. The proposed method novelly formulates this process as a joint optimization problem to generate a better back-end trajectory downstream.
• Figure 3: Polynomial trajectories can take on various shapes by changing time allocations.
• Figure 4: Demonstration of geometric constraints for an ellipsoid (blue). (a) The general formulation of inner and outer Löwner-John ellipsoid. (b) The Inner Löwner-John ellipsoid with outer line segment constraints.
• Figure 5: Visualization of ellipsoids (blue) and polytopes (red) during 4 iterations. The black line segments represent the path, with waypoints in magenta. The green curve is the trajectory generated within polytopes.