System Analysis of Acyclic Triangle Formation Control Initially in Collinearity

This paper focuses on the analysis of acyclic triangle formation control system under the gradient-like control law, in collinear situations (including collocations). The concept of congruent triangle formation and a 3-dimensional Triangle Formation Phase Space (or TP-space) are proposed to facilitate the visualization of the trajectories, which is usually within six dimensional space, of forming a triangle formation to its prescribed shape in the plane. The equilibrium manifold of the acyclic triangle formation under the gradient-like control law is mapped to the TP-space, which makes it easier to view the dynamic analysis results given by Cao[6]. The TP-space variables also find its advantage in the stability analysis of the equilibria of acyclic triangle formation under gradient-like control law, by which the result shows that there are at least one stable equilibrium, at least one unstable equilibrium, in initially collinear situations which will be trapped in collinearity all the time, and at most two other possible equilibria, which depends on the desired length of edges. A simulation supports the result.