Undirected Formations Supelec EECI Graduate School in Control Undirected Formations A. S. Morse Yale University Gif – sur - Yvette May 24, 2012 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAAAAAAAA

d1 d2 d3 1 2 3

1 3.989 4 6.509 6.5 2 3 3 3.003

period = 270 seconds

Let q be any positive number such that Let e(t) be the solution to the unperturbed error system starting at some state e(0) 2 B. Then e(t) ! 0 as fast as e-¸ t ! 0.

There exists an open ball B about e = 0 in R3 and a vector q(¹) 2 R3 depending continuously on ¹ such that q(0) =0 and for every ¹ 2 B, q(¹) is an exponentially stable equilibrium of the perturbed error system Suppose error system is in an equilibrium state e = q(¹). Therefore the norm of each zi is constant. Therefore each z_i must be a constant or a linear combination of sinusoids. Suppose error system is in an equilibrium state e = q(¹) and . . Then either

Suppose error system is in an equilibrium state e = q(¹) and Suppose error system is in an equilibrium state e = q(¹) and . . Then either

Suppose z is not in N. Then z1 and z2 are linearly independent. Suppose error system is in an equilibrium state e = q(¹) and . . Then either

Suppose error system is in an equilibrium state e = q(¹) and Suppose error system is in an equilibrium state e = q(¹) and . . Then either

Main Results for Triangles Pick ¹ so that ¹1 + ¹2 + ¹3  0 and so that ||¹|| is small enough so that {x,G} is infinitesimally rigid for all x in the set Suppose the error system is in equilibrium at e = q(¹). Then z is not in N and each ||zi||2 = qi(¹) + di2 each zi is a linear combination of sinusoids each zi is nonconstant each zi 2 R2 not true in R3 So each xi(t) is also sinusoidal at frequency !