Trajectory Planning with Safety Guaranty for a Multirotor based on the Forward and Backward Reachability Analysis

Planning a trajectory with guaranteed safety is a core part for a risk-free flight of a multirotor. If a trajectory planner only aims to ensure safety, it may generate trajectories which overly bypass risky regions and prevent the system from achieving specific missions. This work presents a robust trajectory planning algorithm which simultaneously guarantees the safety and reachability to the target state in the presence of unknown disturbances. We first characterize how the forward and backward reachable sets (FRSs and BRSs) are constructed by using Hamilton-Jacobi reachability analysis. Based on the analysis, we present analytic expressions for the reachable sets and then propose minimal ellipsoids which closely approximate the reachable sets. In the planning process, we optimize the reference trajectory to connect the FRSs and BRSs, while avoiding obstacles. By combining the FRSs and BRSs, we can guarantee that any state inside of the initial set reaches the target set. We validate the proposed algorithm through a simulation of traversing a narrow gap.