LiDAR-based Control of Autonomous Rotorcraft for the Inspection of Pier-like Structures: Proofs

This is a complementary document to the paper presented in [1], to provide more detailed proofs for some results. The main paper addresses the problem of trajectory tracking control of autonomous rotorcraft in operation scenarios where only relative position measurements obtained from LiDAR sensors are possible. The proposed approach defines an alternative kinematic model, directly based on LiDAR measurements, and uses a trajectory-dependent error space to express the dynamic model of the vehicle. An LPV representation with piecewise affine dependence on the parameters is adopted to describe the error dynamics over a set of predefined operating regions, and a continuous-time $H_2$ control problem is solved using LMIs and implemented within the scope of gain-scheduling control theory.