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The control of complex non-linear systems can be aided by modeling each system as a collection of simplified hybrid modes, with each mode representing a particular operating regime defined by the system dynamics or by a region of the state space in which the system operates. Guarantees on the safety and performance of such hybrid systems can still be challenging to generate, however. Reachability analysis using a dynamic game formulation with Hamilton-Jacobi methods provides a useful way to generate these types of guarantees, and the technique is flexible enough to analyze a wide variety of systems. This paper presents two applications of reachable sets, both focused on guaranteeing the safety and performance of robotic aerial vehicles. In the first example, reachable sets are used to design and implement a backflip maneuver for a quadrotor helicopter. In the second, reachability analysis is used to design a decentralized collision avoidance algorithm for multiple quadrotors. The theory for both examples is explained, and successful experimental results are presented from flight tests on the STARMAC quadrotor helicopter platform.

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Abstract-For many applications, the control of a complex nonlinear system can be greatly aided by modeling the system as a collection of simplified hybrid modes, each representing a particular operating regime or portion of the state space. An example of this is the decomposition of complex aerobatic flights into sequences of discrete maneuvers, an approach that has proven very successful for both human piloted and autonomously controlled aircraft. However, a critical component of designing such control systems for autonomous flight is to ensure the safety and feasibility of transitions between these maneuvers. This work presents a hybrid dynamics framework for the design of guaranteed safe switching regions in performing an autonomous backflip by a quadrotor helicopter. The regions are constructed using reachable sets calculated via a HamiltonJacobi differential game formulation, and experimental results are presented from flight tests on the STARMAC quadrotor platform.