Characterization of trajectories using constraint programming and - PowerPoint PPT Presentation

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

Capture set Maze Braketting ← − T Applications Capture set T. Le Mézo, L. Jaulin , B. Zerr Characterization of trajectories using constraint programming

Capture set Maze Braketting ← − T Applications We consider a state equation ˙ x = f ( x ) . Example : The Van der Pol system � ˙ = x 1 x 2 � 1 − x 2 � ˙ = · x 2 − x 1 x 2 1 T. Le Mézo, L. Jaulin , B. Zerr Characterization of trajectories using constraint programming

Capture set Maze Braketting ← − T Applications T. Le Mézo, L. Jaulin , B. Zerr Characterization of trajectories using constraint programming

Capture set Maze Braketting ← − T Applications Let ϕ be the flow map. The capture set of the target T ⊂ R n is: ← − T = { x 0 | ∃ t ≥ 0 , ϕ ( t , x 0 ) ∈ T } . T. Le Mézo, L. Jaulin , B. Zerr Characterization of trajectories using constraint programming

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

Capture set Maze Braketting ← − T Applications Graph analogy T. Le Mézo, L. Jaulin , B. Zerr Characterization of trajectories using constraint programming

Capture set Maze Braketting ← − T Applications A deterministic graph G 1 with a target T (red), a dead path (blue). T. Le Mézo, L. Jaulin , B. Zerr Characterization of trajectories using constraint programming

Capture set Maze Braketting ← − T Applications It can be approximated by a non deterministic graph G 2 : T. Le Mézo, L. Jaulin , B. Zerr Characterization of trajectories using constraint programming

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

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

Capture set Maze Braketting ← − T Applications Maze T. Le Mézo, L. Jaulin , B. Zerr Characterization of trajectories using constraint programming

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 R n . A maze is a domain which encloses a path. T. Le Mézo, L. Jaulin , B. Zerr Characterization of trajectories using constraint programming

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

Capture set Maze Braketting ← − T Applications Mazes can be made more accurate by adding polygones. T. Le Mézo, L. Jaulin , B. Zerr Characterization of trajectories using constraint programming

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

Capture set Maze Braketting ← − T Applications 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

Capture set Maze Braketting ← − T Applications The set of mazes forms a lattice with respect to ⊂ . L a ⊂ L b means : the boxes of L a are subboxes of the boxes of L b . The polygones of L a are included in those of L b The doors of L a are thinner than those of L b . T. Le Mézo, L. Jaulin , B. Zerr Characterization of trajectories using constraint programming

Capture set Maze Braketting ← − T Applications 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

Capture set Maze Braketting ← − T Applications Inner approximation of ← − T T. Le Mézo, L. Jaulin , B. Zerr Characterization of trajectories using constraint programming

Capture set Maze Braketting ← − T Applications Main idea : Compute an outer approximation of the complementary of ← − T : ← − T = { x 0 | ∀ t ≥ 0 , ϕ ( t , x 0 ) / ∈ T } Thus, we search for a path that never reach T . T. Le Mézo, L. Jaulin , B. Zerr Characterization of trajectories using constraint programming

Capture set Maze Braketting ← − T Applications 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

Capture set Maze Braketting ← − T Applications 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

Capture set Maze Braketting ← − T Applications Inner propagation T. Le Mézo, L. Jaulin , B. Zerr Characterization of trajectories using constraint programming

Capture set Maze Braketting ← − T Applications T. Le Mézo, L. Jaulin , B. Zerr Characterization of trajectories using constraint programming

Capture set Maze Braketting ← − T Applications [ c ] [ b ] [ a ] [ e ] [ d ] [ f ] T. Le Mézo, L. Jaulin , B. Zerr Characterization of trajectories using constraint programming

Capture set Maze Braketting ← − T Applications T. Le Mézo, L. Jaulin , B. Zerr Characterization of trajectories using constraint programming

Capture set Maze Braketting ← − T Applications T. Le Mézo, L. Jaulin , B. Zerr Characterization of trajectories using constraint programming

Capture set Maze Braketting ← − T Applications T. Le Mézo, L. Jaulin , B. Zerr Characterization of trajectories using constraint programming

Capture set Maze Braketting ← − T Applications T. Le Mézo, L. Jaulin , B. Zerr Characterization of trajectories using constraint programming

Capture set Maze Braketting ← − T Applications T. Le Mézo, L. Jaulin , B. Zerr Characterization of trajectories using constraint programming

Capture set Maze Braketting ← − T Applications T. Le Mézo, L. Jaulin , B. Zerr Characterization of trajectories using constraint programming

Capture set Maze Braketting ← − T Applications Outer propagation T. Le Mézo, L. Jaulin , B. Zerr Characterization of trajectories using constraint programming

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

Capture set Maze Braketting ← − T Applications Car on the hill T. Le Mézo, L. Jaulin , B. Zerr Characterization of trajectories using constraint programming

Capture set Maze Braketting ← − T Applications � ˙ = x 1 x 2 � 11 � ˙ = 9 . 81sin 24 · sin x 1 + 0 . 6 · sin ( 1 . 1 · x 1 ) − 0 . 7 · x 2 x 2 T. Le Mézo, L. Jaulin , B. Zerr Characterization of trajectories using constraint programming

Capture set Maze Braketting ← − T Applications T. Le Mézo, L. Jaulin , B. Zerr Characterization of trajectories using constraint programming

Capture set Maze Braketting ← − T Applications Research box X 0 = [ − 1 , 13 ] × [ − 10 , 10 ] Blue: T out = X 0 ; Red: T in = [ 2 , 9 ] × [ − 1 , 1 ] T. Le Mézo, L. Jaulin , B. Zerr Characterization of trajectories using constraint programming

Capture set Maze Braketting ← − T Applications Combined with an outer propagation T. Le Mézo, L. Jaulin , B. Zerr Characterization of trajectories using constraint programming

Capture set Maze Braketting ← − T Applications Van der Pol system T. Le Mézo, L. Jaulin , B. Zerr Characterization of trajectories using constraint programming

Capture set Maze Braketting ← − T Applications Consider the system � ˙ = x 1 x 2 � 1 − x 2 � ˙ = · x 2 − x 1 x 2 1 and the box X 0 = [ − 4 , 4 ] × [ − 4 , 4 ] . T. Le Mézo, L. Jaulin , B. Zerr Characterization of trajectories using constraint programming

Capture set Maze Braketting ← − T Applications f → − f ; T = X 0 ∪ [ − 0 . 1 , 0 . 1 ] 2 . T. Le Mézo, L. Jaulin , B. Zerr Characterization of trajectories using constraint programming

Capture set Maze Braketting ← − T Applications f → − f ; T out = X 0 ; T in = [ 0 . 5 , 1 ] 2 . T. Le Mézo, L. Jaulin , B. Zerr Characterization of trajectories using constraint programming

Capture set Maze Braketting ← − T Applications Combined with an outer propagation T. Le Mézo, L. Jaulin , B. Zerr Characterization of trajectories using constraint programming