Potential%20Fields%20for%20Maintaining%20Connectivity%20of%20Dynamic%20Graphs

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Potential Fields for Maintaining Connectivity of
Dynamic Graphs

• MEAM 620 Final Project
• Michael M. Zavlanos

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Problem Formulation

• n mobile agents in an obstacle free workspace,
  with single integrator dynamics dxi/dt ui.
• state dependent graph G(x) Nodes correspond to
  the agents and we draw an Edge between two nodes
  if their pairwise distance is smaller than some
  threshold R.
• Adjacency matrix A(x).
• Graph Laplacian L(x) D(x) A(x).
• ?1(L(x)) 0 with corresponding eigenvector 1.
• ?2(L(x)) gt 0 gt G(x) is connected.

3
Background

• Yoonsoo Kim and Mehran Mesbahi, On Maximizing
  the Second Smallest Eigenvalue of a State
  Dependent Graph Laplacian, IEEE Transactions on
  Automatic Control (to appear).
• Let P be a nx(n-1) projection matrix to the space
  perpendicular to the vector 1.
• L(x) positive semi-def. gt PTL(x)P positive
  semi-def.
• ?2(PTL(x)P) gt 0 ? PTL(x)P gt 0

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Potential Field Approach

• Eigenvalues of L(x) 0 ?1 lt ?2 lt lt ?n
• Eigenvalues of PTL(x)P 0 lt ?2 lt lt ?n
• ?2(PTL(x)P) gt 0 ? det(PTL(x)P) gt 0
• Control Law
• Connectivity modeled as an obstacle.

ui d/dxi ( 1/det(PTL(x)P) )
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Gradient of det(PTL(x)P)

• Let M(x) PTL(x)P. We can show that
• And hence