Formation Control via Global Orientation Alignment

Abstract

In Chap. 6, formation controls via orientation alignment, including orientation estimation and orientation control, were presented. However, the algorithms are valid only for a quasi-global convergence when the initial orientation angles are restricted to the condition $\max _{i \in \mathcal {V}} \theta _i(t_0) - \min _{i \in \mathcal {V}} \theta _i(t_0) < \pi$. In this chapter, we would like to remove this restriction such that the orientation alignment could be achieved for almost all initial conditions. It will be shown that the orientation alignment could be done for almost any initial orientation angles with more information exchanges between neighboring nodes, and with more computational load in each node. Thus, there is a cost in implementing a global orientation alignment algorithm. But, since all the measurements are relative and information exchanges take place between neighboring agents, it is still a distributed control law. For a global convergence, we utilize virtual variables that transform a non-convex circle or a sphere to the linear Euclidean space. The global orientation alignment problem is defined in a non-convex circle or a sphere; for the convergence analysis, we conduct analysis in the Euclidean space by way of using virtual variables. Then, after analyzing and designing the control law in the Euclidean space, we again transform the control law into the circle or the spherical space.