Construction of control barrier function and $C^2$ reference trajectory for constrained attitude maneuvers

Constrained attitude maneuvers have numerous applications in robotics and aerospace. In our previous work, a general framework to this problem was proposed with resolution completeness guarantee. However, a smooth reference trajectory and a low-level safety-critical controller were lacking. In this work, we propose a novel construction of a $C^2$ continuous reference trajectory based on B\'ezier curves on $SO(3)$ that evolves within predetermined cells and eliminates previous stop-and-go behavior. Moreover, we propose a novel zeroing control barrier function on $SO(3)$ that provides a safety certificate over a set of overlapping cells on $SO(3)$ while avoiding nonsmooth analysis. The safety certificate is given as a linear constraint on the control input and implemented in real-time. A remedy is proposed to handle the states where the coefficient of the control input in the linear constraint vanishes. Numerical simulations are given to verify the advantages of the proposed method.