The geometric convexity on SE(3) and its application to the formation tracking in multi-vehicle systems

This paper considers the problem of geometric convexity on special Euclidean group SE(3) and its application to the formation tracking in multi-vehicle systems. Motivated by the convex hull on Euclidean space, the specific expression of geodesic connecting two points on SE(3) is obtained based on the Pontryagin's Minimum Principle. Then it is extended to the geometric convex combination for multiple points, based on which the virtual systems on SE(3) is given to be used in the application of formation tracking problem. In light of the geometric convex combination and virtual systems on SE(3), a consensus-based tracking protocol is proposed to guarantee that the formation is achieved under the directed acyclic graphs. Finally, two numerical examples are provided to demonstrate the validity of the theoretical results.