Emergent behaviors of rotation matrix flocks

Abstract: We derive an explicit form for the Cucker-Smale (CS) model on the special orthogonal group $\mathrm{SO}(3)$ by identifying closed form expressions for geometric quantities such as covariant derivative and parallel transport in exponential coordinates. We study the emergent dynamics of the model by using a Lyapunov functional approach and La Salle's invariance principle. Specifically, we show that velocity alignment emerges from some admissible class of initial data, under suitable assumptions on the communication weight function. We characterize the $\omega$-limit set of the dynamical system and identify a dichotomy in the asymptotic behavior of solutions. Several numerical examples are provided to support the analytical results.