Distributed localization and control of a group of underwater robots - - PDF document

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Distributed localization and control of a group of underwater robots - - PDF document

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Distributed localization and control of a group of underwater robots using contractor programming L. Jaulin, S. Rohou, J. Nicola, M. Saad, F. Le Bars and B. Zerr. SWIM15 Prague, June 9-11, 2015 ENSTA Bretagne, OSM, LabSTICC. Video of the

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Distributed localization and control

  • f a group of underwater robots

using contractor programming

  • L. Jaulin, S. Rohou, J. Nicola,
  • M. Saad, F. Le Bars and B. Zerr.
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SWIM’15 Prague, June 9-11, 2015 ENSTA Bretagne, OSM, LabSTICC. Video of the presentation https://youtu.be/q6F7WDCcf2A

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1 Scout project

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Goal : (i) coordination of underwater robots ; (ii) collabo- rative behavior.

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Supervisors: L. Jaulin, C. Aubry, S. Rohou, B. Zerr, J. Nicola, F. Le bars Compagny: RTsys (P. Raude) Students: G. Ricciardelli, L. Devigne, C. Guillemot, S. Pommier, T. Viravau, T. Le Mezo, B. Sultan, B. Moura,

  • M. Fadlane, A. Bellaiche, T. Blanchard, U. Da rocha, G.

Pinto, K. Machado.

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2 Controller

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3 Localization problem

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Range only Based on interval analysis Robust with respect to outliers Distributed computation Low rate communication We propose here to use a contractor programming ap- proach

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4 Matrices and contractors

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linear application → matrices L :

  • α = 2a + 3h

γ = h − 5a → A =

  • 2

3 1 −5

  • We have a matrix algebra and Matlab.

We have: var(L) = {a, h}, covar(L) = {α, γ} . But we cannot write: var(A) = {a, h}, covar(A) = {α, γ}.

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constraint → contractor a · b = z →

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Contractor fusion

  • a · b = z

→ C1 b + c = d → C2 Since b occurs in both constraints, we fuse the two con- tractors as: C = C1 × C2⌋(2,1) = C1|C2 (for short)

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5 Localization with contractors

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xj

h+1 = f(xj h, uj h), h ∈ {k − ¯

h, . . . , k} The observer: Ck,j

x

=

h∈{k−¯ h,...,k} Cj

x(h)

var(Ck,j

x ) =var(Ck,j x(h)) =

  • xj

k−¯ h, . . . , xj k, xj k+1

  • .
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zj

h = h(xj h)

Observer RSO: Ck,j

x,z = Ck,j x ∩ {q1}

h∈{k−¯ h,...,k}

  • Ck,j

x |Ch,j

z

  • .
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To get an outer approximation of set(Ck,j

x,z), we need a

paver. We can also obtain an inner approximation using separators [SMART 2015].

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t = 0

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t = 0.1

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t = 0.2

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t = 0.3

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t = 0.4

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t = 0.5

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t = 0.6

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t = 0.7

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t = 0.8

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t = 0.9

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t = 1.0

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t = 1.1

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t = 1.2

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t = 1.3

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t = 1.4

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6 Distributed localization

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yj,ℓ

h = g(xj h, xℓ h)

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Observer: Ck,j

x,z

= Ck,j

x

∩

{q1}

h∈{k−¯ h,...,k}

  • Ck,j

x |Ch,j

z

  • ∩

{q2}

h∈{k−¯ h,...,k}

  • Ck,j

x |Ch,j,ℓ

y

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7 Singularity

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8 Test case

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QT/C++ code available at http://www.ensta-bretagne.fr/jaulin/easibex.html

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9 Tests

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