Analysis and Control of Acyclic Triangle Formations

The concept of congruent triangle formation is defined, which neglects the translation and rotation but focuses on the edge lengths and orientation of triangles. A set of three-variable, called TP-variables, is defined to describe the states of congruent triangles, which constitute the TP-space which is used to show the dynamic trajectories of shaping a desired triangle formation from initial setups. The TP-variables are also used to find out the equilibria of acyclic triangle formation under gradientlike control law in collinearity and to analyze the stability of the equilibria. Because there are two mirror acyclic triangle formations flipping over with each other which can not be distinguished by using gradient-like control law, and acyclic triangle formations will be trapped in collinearity if initially in collinear setups, a new control law, called orthogonal control law, is proposed. Under orthogonal control law the dynamical system approaches to the unique desired acyclic triangle formation in any initial condition except initially putting agent 1 and agent 2 in the same place. The correctness is proved and simulations support the claims.