Reachable Set of a Planar Vertical Take-Off and Landing Model With Applications to Quadcopters

Quadcopters are increasingly popular for robotics applications. Being able to efficiently calculate the set of positions reachable by a quadcopter within a time budget enables collision avoidance and pursuit-evasion strategies. This article examines the set of positions that a quadcopter can reach within a specified time limit using a planar vertical take-off and landing (PVTOL) model for 2-D quadcopter dynamics. This popular model is used to determine the set of candidate optimal control sequences to build the full 3-D reachable set at final time$T$in$(x,z,\theta)$phase space or rotated to form the reachable set in$(x,y,z)$Cartesian space. We calculate the analytic equations that exactly bound the set of positions reachable in a given time horizon for all initial conditions. We use these bounds to: escape a bounded region, avoid a collision, find the collision set, determine the closest point to the reachable set, reach a goal$(x,z)$in the reachable set, and for drone countermeasures.