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], where more detailed proofs are provided 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 H2 control problem is solved using LMIs and implemented within the scope of gain-scheduling control theory. In this document, Section A presents the stability analysis of the attitude inner-loop presented in [1, Section II.B], whereas Section B presents a more detailed version of the stability and performance guarantees for the LPV system, extending the results presented in [1, Section IV.A]. APPENDIX A INNER-LOOP DYNAMICS This appendix presents a possible inner-loop stabilization method, within the framework of feedback linearization and Lyapunov stability methods, that results in a second-order linear model for the pitch and roll angular motion, as well as a first-order linear model for the yaw angular velocity. Theorem 1 (Inner-loop Stability). Consider the control law given by next =S(ω )JB ω + JB Q −1(λ)[Q̇(λ)ω−K2 (λ̇ − e3 e T 3 uIL) −Π e3 K1Πe3 (λ − uIL)] (1) where K1 ∈ R 2 and K2 ∈ R 3 are positive definite diagonal matrices and the inner-loop input vector is denoted as uIL =