Intrinsic Sampling-Aware Discrete-Time Geometric Backstepping for Aggressive Quadrotor Tracking on SO(3)

This paper addresses a well-recognized inconsistency arising in sampled implementations of nonlinear geometric control: continuous-time (CT) Lya- punov stability does not, in general, guarantee a decreas- ing discrete-time (DT) Lyapunov difference under non- negligible sampling effects, potentially leading to dis- crepancies between predicted and implemented closed- loop behavior. To resolve this issue for quadrotor trajec- tory tracking, an intrinsic DT geometric backstepping framework is developed on SO(3) that explicitly accounts for sampling effects while preserving the Lie-group struc- ture of the attitude dynamics. The proposed framework addresses the noncausal dependence inherent in DT backstepping through a structure-preserving geometric predictor that enables causal one-step matching directly on SO(3). Geometric consistency is preserved through a Lie-group formulation, while a rigorous DT Lyapunov difference analysis explicitly characterizes sampling- dependent stability properties arising from finite-step residual effects. The analysis establishes global exponen- tial stability of the translational subsystem and Semi- Globally Uniformly Ultimately Bounded (SGUUB) of the rotational tracking errors with sampling-dependent margins. Simulations demonstrate accurate tracking under aggressive maneuvers, machine-precision preser- vation of SO(3) orthogonality, and closed-loop behavior consistent with theoretical sampling bounds, supporting the effectiveness of intrinsic DT design for achieving sampling-consistent closed-loop behavior under practi- cal sampling conditions.