Characterization of trajectories using constraint programming and - - PowerPoint PPT Presentation

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Characterization of trajectories using constraint programming and - - PowerPoint PPT Presentation

Jan 02, 2024 215 likes •674 views

Capture set Maze Braketting T Applications Characterization of trajectories using constraint programming and abstract interpretation T. Le Mzo, L. Jaulin , B. Zerr T. Le Mzo, L. Jaulin , B. Zerr Characterization of trajectories

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Capture set Maze Braketting ← − T Applications

Characterization of trajectories using constraint programming and abstract interpretation

  • T. Le Mézo, L. Jaulin, B. Zerr
  • T. Le Mézo, L. Jaulin, B. Zerr

Characterization of trajectories using constraint programming

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Capture set Maze Braketting ← − T Applications

Capture set

  • T. Le Mézo, L. Jaulin, B. Zerr

Characterization of trajectories using constraint programming

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Capture set Maze Braketting ← − T Applications

We consider a state equation ˙ x = f(x). Example: The Van der Pol system ˙ x1 = x2 ˙ x2 =

  • 1−x2

1

  • ·x2 −x1
  • T. Le Mézo, L. Jaulin, B. Zerr

Characterization of trajectories using constraint programming

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Capture set Maze Braketting ← − T Applications

  • T. Le Mézo, L. Jaulin, B. Zerr

Characterization of trajectories using constraint programming

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Capture set Maze Braketting ← − T Applications

Let ϕ be the flow map. The capture set of the target T ⊂ Rn is: ← − T = {x0 | ∃t ≥ 0,ϕ(t,x0) ∈ T}.

  • T. Le Mézo, L. Jaulin, B. Zerr

Characterization of trajectories using constraint programming

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Capture set Maze Braketting ← − T Applications

To each state, we associate a path.

  • T. Le Mézo, L. Jaulin, B. Zerr

Characterization of trajectories using constraint programming

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Capture set Maze Braketting ← − T Applications

Graph analogy

  • T. Le Mézo, L. Jaulin, B. Zerr

Characterization of trajectories using constraint programming

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A deterministic graph G1 with a target T (red), a dead path (blue).

  • T. Le Mézo, L. Jaulin, B. Zerr

Characterization of trajectories using constraint programming

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Capture set Maze Braketting ← − T Applications

It can be approximated by a non deterministic graph G2:

  • T. Le Mézo, L. Jaulin, B. Zerr

Characterization of trajectories using constraint programming

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Capture set Maze Braketting ← − T Applications

Using a backward method, we compute an interval containing ← − T .

  • T. Le Mézo, L. Jaulin, B. Zerr

Characterization of trajectories using constraint programming

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Capture set Maze Braketting ← − T Applications

Which corresponds to an interval of graphs: Our new approach: bracket ← − T , we search for paths not for states.

  • T. Le Mézo, L. Jaulin, B. Zerr

Characterization of trajectories using constraint programming

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Capture set Maze Braketting ← − T Applications

Maze

  • T. Le Mézo, L. Jaulin, B. Zerr

Characterization of trajectories using constraint programming

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Capture set Maze Braketting ← − T Applications

An interval is a domain which encloses a real number. A polygon is a domain which encloses a vector of Rn. A maze is a domain which encloses a path.

  • T. Le Mézo, L. Jaulin, B. Zerr

Characterization of trajectories using constraint programming

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Capture set Maze Braketting ← − T Applications

A maze is a set of paths.

  • T. Le Mézo, L. Jaulin, B. Zerr

Characterization of trajectories using constraint programming

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Mazes can be made more accurate by adding polygones.

  • T. Le Mézo, L. Jaulin, B. Zerr

Characterization of trajectories using constraint programming

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Capture set Maze Braketting ← − T Applications

Or using doors instead of a graph

  • T. Le Mézo, L. Jaulin, B. Zerr

Characterization of trajectories using constraint programming

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Here, a maze L is composed of A paving P A polygon for each box of P Doors between adjacent boxes

  • T. Le Mézo, L. Jaulin, B. Zerr

Characterization of trajectories using constraint programming

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Capture set Maze Braketting ← − T Applications

The set of mazes forms a lattice with respect to ⊂. La ⊂ Lb means : the boxes of La are subboxes of the boxes of Lb. The polygones of La are included in those of Lb The doors of La are thinner than those of Lb.

  • T. Le Mézo, L. Jaulin, B. Zerr

Characterization of trajectories using constraint programming

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The left maze contains less paths than the right maze. Note that yellow polygons are convex.

  • T. Le Mézo, L. Jaulin, B. Zerr

Characterization of trajectories using constraint programming

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Inner approximation of ← − T

  • T. Le Mézo, L. Jaulin, B. Zerr

Characterization of trajectories using constraint programming

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Main idea: Compute an outer approximation of the complementary of ← − T : ← − T = {x0 | ∀t ≥ 0,ϕ(t,x0) / ∈ T} Thus, we search for a path that never reach T.

  • T. Le Mézo, L. Jaulin, B. Zerr

Characterization of trajectories using constraint programming

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Target contractor. If a box [x] of P is included in T then remove [x] and close all doors entering in [x] .

  • T. Le Mézo, L. Jaulin, B. Zerr

Characterization of trajectories using constraint programming

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Flow contractor. For each box [x] of P, we contract the polygon using the constraint ˙ x = f(x).

  • T. Le Mézo, L. Jaulin, B. Zerr

Characterization of trajectories using constraint programming

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Inner propagation

  • T. Le Mézo, L. Jaulin, B. Zerr

Characterization of trajectories using constraint programming

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Capture set Maze Braketting ← − T Applications

[a] [b] [c] [d] [e] [f]

  • T. Le Mézo, L. Jaulin, B. Zerr

Characterization of trajectories using constraint programming

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Capture set Maze Braketting ← − T Applications

  • T. Le Mézo, L. Jaulin, B. Zerr

Characterization of trajectories using constraint programming

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  • T. Le Mézo, L. Jaulin, B. Zerr

Characterization of trajectories using constraint programming

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Capture set Maze Braketting ← − T Applications

  • T. Le Mézo, L. Jaulin, B. Zerr

Characterization of trajectories using constraint programming

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  • T. Le Mézo, L. Jaulin, B. Zerr

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  • T. Le Mézo, L. Jaulin, B. Zerr

Characterization of trajectories using constraint programming

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Outer propagation

  • T. Le Mézo, L. Jaulin, B. Zerr

Characterization of trajectories using constraint programming

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Capture set Maze Braketting ← − T Applications [a] [b] [c] [d] [e] [f]

An interpretation can be given only when the fixed point is reached.

  • T. Le Mézo, L. Jaulin, B. Zerr

Characterization of trajectories using constraint programming

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Car on the hill

  • T. Le Mézo, L. Jaulin, B. Zerr

Characterization of trajectories using constraint programming

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˙ x1 = x2 ˙ x2 = 9.81sin 11

24 ·sinx1 +0.6·sin(1.1·x1)

  • −0.7·x2
  • T. Le Mézo, L. Jaulin, B. Zerr

Characterization of trajectories using constraint programming

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Research box X0 = [−1,13]×[−10,10] Blue: Tout = X0; Red: Tin = [2,9]×[−1,1]

  • T. Le Mézo, L. Jaulin, B. Zerr

Characterization of trajectories using constraint programming

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Combined with an outer propagation

  • T. Le Mézo, L. Jaulin, B. Zerr

Characterization of trajectories using constraint programming

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Van der Pol system

  • T. Le Mézo, L. Jaulin, B. Zerr

Characterization of trajectories using constraint programming

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Consider the system ˙ x1 = x2 ˙ x2 =

  • 1−x2

1

  • ·x2 −x1

and the box X0 = [−4,4]×[−4,4].

  • T. Le Mézo, L. Jaulin, B. Zerr

Characterization of trajectories using constraint programming

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f → −f ; T = X0 ∪[−0.1,0.1]2.

  • T. Le Mézo, L. Jaulin, B. Zerr

Characterization of trajectories using constraint programming

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f → −f ; Tout = X0 ; Tin = [0.5,1]2.

  • T. Le Mézo, L. Jaulin, B. Zerr

Characterization of trajectories using constraint programming

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Capture set Maze Braketting ← − T Applications

Combined with an outer propagation

  • T. Le Mézo, L. Jaulin, B. Zerr

Characterization of trajectories using constraint programming

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Capture set Maze Braketting ← − T Applications

  • T. Le Mézo, L. Jaulin, B. Zerr

Characterization of trajectories using constraint programming