Motivation Formalization Test-case
Interval State Estimation With Data Association
B. Desrochers, L. Jaulin
Rostock, SWIM 2018, July 25-27
La Cordelière
B. Desrochers, L. Jaulin Interval State Estimation With Data Association
Motivation Formalization Test-case
Reconstitution de la bataille youtu.be/yP4cM1UGrqY
B. Desrochers, L. Jaulin Interval State Estimation With Data Association
Motivation Formalization Test-case
Magnetism
Romain Schwab
B. Desrochers, L. Jaulin Interval State Estimation With Data Association
Motivation Formalization Test-case
Boatbot
B. Desrochers, L. Jaulin Interval State Estimation With Data Association
Motivation Formalization Test-case
B. Desrochers, L. Jaulin Interval State Estimation With Data Association
Motivation Formalization Test-case
Robots
B. Desrochers, L. Jaulin Interval State Estimation With Data Association
Motivation Formalization Test-case
Folaga with Kopadia
youtu.be/VqXG9zO_q1A
B. Desrochers, L. Jaulin Interval State Estimation With Data Association
Motivation Formalization Test-case
Formalization
B. Desrochers, L. Jaulin Interval State Estimation With Data Association
Motivation Formalization Test-case
State estimation problem
x ˙ (t ) = f ( x (t )), u (t)) (evolution equation)
g(x(t i )) ∈ [y](t i ) (observation constraint)
x ( 0 ) ∈ X 0 ( initial state )
Implicit form
x ˙ (t ) = f ( x (t)), u (t)) g(x(t i ),y(t i )) = 0, y (t i ) ∈ [ y ](t i ) x ( 0 ) ∈ X 0
B. Desrochers, L. Jaulin Interval State Estimation With Data Association
Motivation Formalization Test-case
Example. Assume that we have a robot at position (x 1 ,x 2 ) with a
heading x 3 . It measures a landmark located at (4,5).
We have
g ( x , y ) =
x 1 + y 1 · cos (x 3 + y 2 ) − 4 x 2 + y 1 · sin (x 3 + y 2 ) − 5
B. Desrochers, L. Jaulin Interval State Estimation With Data Association
Motivation Formalization Test-case
If now, the robot only measures the distance to the landmark, we get
g ( x ,y) = (x 1 − 4 ) 2 + (x 2 − 5 ) 2 − y 2 .
In the more general case, the observation constraint is g(x(t i ),y(t i ), m(t i )) = 0
y (t i ) ∈ [ y ](t i ) m (t i ) ∈ [ m ](t i )
B. Desrochers, L. Jaulin Interval State Estimation With Data Association
Motivation Formalization Test-case
In our context, the data are not associated. The observation constraint has the form
g ( x (t i ), y (t i ), m (t i )) = 0 m (t 1 ) ∈ [ m 1 ] ∨ · · · ∨ m (t ` ) ∈ [ m ` ] or equivalently
g ( x (t i ), y (t i ), m (t i )) = 0
m(t i ) ∈ M = [m 1 ] ∪ · · · ∪ [m ` ]
x ˙ (t) = f ( x (t)), u (t)) g ( x (t i ), y (t i ), m (t i )) = 0 y (t i ) ∈ [ y ](t i )
m (t i ) ∈ M x(0) ∈ X 0
B. Desrochers, L. Jaulin Interval State Estimation With Data Association
Motivation Formalization Test-case
Contractors
The operator C : IR n → IR n is a contractor for f ( x ) = 0 , if
C ([ x ]) ⊂ [ x ] (contractance)
x ∈ [ x ] and f ( x ) = 0 ⇒ x ∈ C ([ x ]) (consistence)
B. Desrochers, L. Jaulin Interval State Estimation With Data Association
Motivation Formalization Test-case
Building contractors Consider the equation
x 1 + x 2 − x 3 = 0
with x 1 ∈ [x 1 ] , x 2 ∈ [x 2 ] , x 3 ∈ [x 3 ].
We have
x 3 = x 1 + x 2 ⇒ x 3 ∈ [x 3 ] ∩ ([x 1 ] + [x 2 ]) x 1 = x 3 − x 2 ⇒ x 1 ∈ [x 1 ] ∩ ([x 3 ] − [x 2 ]) x 2 = x 3 − x 1 ⇒ x 2 ∈ [x 2 ] ∩ ([x 3 ] − [x 1 ])
B. Desrochers, L. Jaulin Interval State Estimation With Data Association
Motivation Formalization Test-case
The contractor associated with x 1 + x 2 = x 3 is thus
C
[x 1 ] [x 2 ] [x 3 ]
=
[x 1 ] ∩ ([x 3 ] − [x 2 ]) [x 2 ] ∩ ([x 3 ] − [x 1 ]) [x 3 ] ∩ ([x 1 ] + [x 2 ])
Our constraint :m ∈ M
B. Desrochers, L. Jaulin Interval State Estimation With Data Association
Motivation Formalization Test-case
B. Desrochers, L. Jaulin Interval State Estimation With Data Association
Motivation Formalization Test-case
For eciency, a balanced quadtree is created rst
The contractor has now a logarithmic complexity
B. Desrochers, L. Jaulin Interval State Estimation With Data Association
Motivation Formalization Test-case
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B. Desrochers, L. Jaulin Interval State Estimation With Data Association
Motivation Formalization Test-case
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