Flocking behaviors of a modified Cucker–Smale model on Riemannian manifolds with attractive–repulsive force

This article proposes a modified Cucker–Smale model in view of a double-integrator multiagent dynamic system on complete Riemannian manifolds with an attractive–repulsive force. Then, its flocking behaviors, such as collision avoidance, velocity alignment, and aggregation, are researched. On the basis of covariant derivative and parallel transport, the original Cucker–Smale model on complete Riemannian manifolds can reach velocity alignment. By using the logarithm map on Riemannian manifolds, an attractive–repulsive force is added, which enables all agents to achieve collision avoidance and aggregation on Riemannian manifolds. The proposed model is consistent with the related model on the Euclidean space. Moreover, five specific complete Riemannian manifolds are taken into consideration: the unit sphere, the hyperboloid, the infinite cylinder, the unit circle, and the special orthogonal group. After giving the corresponding covariant derivatives, parallel transports, and logarithm maps, the explicit forms of the proposed model on those manifolds are shown. In the meantime, simulations are provided to verify the theoretical conclusions for all the aforementioned manifolds.