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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.: A A A A A A A A A A A A A A A A A. 2. d 1. 1. d 2. d 3. 3. 1.

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Undirected Formations

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  1. 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

  2. 2 d1 1 d2 d3 3

  3. 1 3.989 4 6.5 6.509 2 3 3.003 3

  4. period = 270 seconds

  5. 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) 2B. Then e(t) ! 0 as fast as e-¸ t! 0.

  6. There exists an open ball Babout e = 0 in R3 and a vector q(¹) 2 R3 depending continuously on ¹such 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

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

  8. Suppose z is not in N. Then z1 andz2 are linearly independent. Suppose error system is in an equilibrium state e = q(¹) and . . Then either

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

  10. 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 zi2R2 not true in R3 So each xi(t) is also sinusoidal at frequency !

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