This paper addresses the issue of bearing-constrained formation control (BCFC) of unmanned aerial vehicle (UAV) swarms. Unlike existing methods, the proposed method operates independently of absolute positions and the global coordinate system, relying solely on the relative orientations in local coordinate systems for each UAV. A distributed orientation estimator is developed on \(\textsf{SO}(3)\) . Employing the estimator, a BCFC controller is designed to achieve both target formation and maneuverability. The stability of the closed-loop system is established using cascade system theory. Numerical simulations are presented to illustrate the effectiveness of the proposed method.