Error-State LQR Geofencing Tracking Control for Underactuated Quadrotor Systems

Safety is a critical concern for autonomous quadrotor systems, which can be compromised by geofences such as narrow terrains and physiographic barriers. Moreover, the maneuverability of quadrotors within geofences is significantly impacted by the underactuated system properties. This article focuses on designing a constructive error-state linear quadratic regulation (LQR) geofencing tracking controller for underactuated quadrotor systems. In particular, the translational and rotational information of quadrotors are unified in one feedback loop to achieve full-state feedback control, while still maintaining external motion constraints and inherent input saturation. First, the rotational error is expressed in the Lie algebra using the logarithmic map of
$\text{SO}(3)$
, enabling attitude errors to be operated in a vector space. Then, the error-state dynamics are linearized at each time step based on the current information, far from the singularity. Furthermore, an optimal LQR control strategy is proposed to directly control the quadrotor's force and torque without requiring parameter tuning. The Lyapunov stability theory is used to rigorously prove the asymptotic stability of the proposed controller. Exponential control barrier functions based on superellipsoid envelopes are employed to solve the position constraint of a higher relative degree and input saturation. Ultimately, the effectiveness of the proposed safe tracking control scheme is demonstrated through numerical simulations and real-world experiments.