Exponential Stability of Formations for Reduced Attitudes: A Coordinates Free Approach

In this work, a methodology based on differential geometry techniques is proposed to investigate exponential stability of a formation for reduced attitudes. By the proposed method, there is no need in finding any relative coordinates, which is typically needed but shown to be difficult when the formation problem is evolving in a non-Euclidean space. In this paper, the desired formation is treated as an embedding submanifold in (S
²
)
^N
and by using the rotation symmetries owned by the attitude dynamics its stability is directly examined. Moreover, such a method turns out to be coordinates free, namely, exponential stability of a formation can be completely determined by just investigating anyone equilibrium which can result in the formation under any local chart of (S
²
)
^N
. This greatly simplifies the stability analysis for the formation problems.