1. Undirected Formations
Supelec
EECI Graduate School in Control
Undirected Formations
A. S. Morse
Yale University
Gif – sur - Yvette
May 24, 2012
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2. d1 d2 d3 1 2 3

3. 1
3.989 4
6.509
6.5 2 3 3
3.003

4. period = 270 seconds

10. 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.

11. 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

12. 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

13. 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

14. 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

15. 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 !