Problem Statement
Possible Methods
Interval Method
Guaranteed Assessment of the Area Explored
by an AUV
Benoît Desrochers, Luc Jaulin
DGA Techniques Navales, Ensta Bretagne
October 12, 2016
Benoît Desrochers, Luc Jaulin
Explored
1 / 20
Problem Statement
Possible Methods
Interval Method
Outlines
1
Problem Statement
2
Possible Methods
3
Interval Method
Benoît Desrochers, Luc Jaulin
Mission requirements
Explored
2 / 20
Problem Statement
Possible Methods
Interval Method
Mission requirements
Problem Statement
Scenario with SSS
long mission
position drift
unknown environment
mission re-planning
no communication
Benoît Desrochers, Luc Jaulin
Explored
3 / 20
Problem Statement
Possible Methods
Interval Method
Mission requirements
Problem Statement
Scenario with SSS
long mission
position drift
MUD
unknown environment
mission re-planning
no communication
Benoît Desrochers, Luc Jaulin
Obstacles
Explored
3 / 20
Problem Statement
Possible Methods
Interval Method
Mission requirements
Problem Statement
Scenario with SSS
long mission
?
position drift
unknown environment
mission re-planning
no communication
Issues
area explored ?
data quality ?
number of view ?
Benoît Desrochers, Luc Jaulin
Explored
3 / 20
Side scan sonar
Problem Statement
Possible Methods
Interval Method
Mission requirements
Hypothesis
side scan sonar with gapller
1
R
Condent
0.9
0.8
0.7
0.6
0.5
Range (m)
Sonar performance
Benoît Desrochers, Luc Jaulin
Explored
4 / 20
Side scan sonar
Problem Statement
Possible Methods
Interval Method
Mission requirements
Hypothesis
side scan sonar with gapller
1
R
Condent
0.9
0.8
0.7
0.6
0.5
Range (m)
Sonar performance
Benoît Desrochers, Luc Jaulin
Explored
4 / 20
Side scan sonar
Problem Statement
Possible Methods
Interval Method
Mission requirements
Hypothesis
side scan sonar with gapller
eective range depends of environmental parameters
1
R
L
Condent
0.9
0.8
0.7
mud
sand
0.6
0.5
Range (m)
Sonar performance
Benoît Desrochers, Luc Jaulin
Explored
4 / 20
Problem Statement
Possible Methods
Interval Method
Outlines
1
Problem Statement
2
Possible Methods
3
Interval Method
Benoît Desrochers, Luc Jaulin
Geometrical
Probabilistic
Explored
5 / 20
Possible Methods
Problem Statement
Possible Methods
Interval Method
Geometrical
Probabilistic
Geometrical approach
Geometrical footprint of sensors are used to take into account
position drift / eective range
Remarks
geometrical assessment
need a priori knowledge
L
easy to use
σ
not resilient
Benoît Desrochers, Luc Jaulin
Explored
6 / 20
Possible methods
Problem Statement
Possible Methods
Interval Method
Geometrical
Probabilistic
Probabilistic (see Liam Paull [PSL14]
) Use probabilistic framework to build an occupancy grid [PSL14]
1
Condent
0.9
0.8
Remarks
Tackles with the "mean" case
0.7
0.6
0.5
Range (m)
Benoît Desrochers, Luc Jaulin
Threshold to make decision ?
Explored
7 / 20
Possible methods
Problem Statement
Possible Methods
Interval Method
Geometrical
Probabilistic
Probabilistic (see Liam Paull [PSL14]
) Use probabilistic framework to build an occupancy grid [PSL14]
1
Condent
0.9
0.8
Remarks
Tackles with the "mean" case
0.7
0.6
0.5
Range (m)
Benoît Desrochers, Luc Jaulin
Threshold to make decision ?
Explored
7 / 20
Possible methods
Problem Statement
Possible Methods
Interval Method
Geometrical
Probabilistic
Sum up
Main issues
manual parameters need to be set
diculties to work with multiple views
no guaranteed
Proposed approach
guaranteed (based on interval arithmetics)
simple/relevant parameters
can be used online
deals with multiple views
Benoît Desrochers, Luc Jaulin
Explored
8 / 20
Problem Statement
Possible Methods
Interval Method
Outlines
1
Problem Statement
2
Possible Methods
3
Interval Method
Benoît Desrochers, Luc Jaulin
Bounded error context
Description
Multiple views
Explored
9 / 20
Problem Statement
Possible Methods
Interval Method
Bounded error context
AUV trajectory
Bounded error context
Description
Multiple views
Range of the SSS
1
Condent
0.9
0.8
0.7
0.6
x(.) ∈ [x](.)
Benoît Desrochers, Luc Jaulin
0.5
Explored
Range (m)
10 / 20
Problem Statement
Possible Methods
Interval Method
Bounded error context
AUV trajectory
Bounded error context
Description
Multiple views
Range of the SSS
1
Condent
0.9
0.8
0.7
0.6
0.5
x(.) ∈ [x](.)
Benoît Desrochers, Luc Jaulin
Explored
Range (m)
10 / 20
Problem Statement
Possible Methods
Interval Method
Bounded error context
AUV trajectory
Bounded error context
Description
Multiple views
Range of the SSS
1
Condent
0.9
0.8
0.7
0.6
0.5
x(.) ∈ [x](.)
Benoît Desrochers, Luc Jaulin
Explored
L
Range (m)
10 / 20
Problem Statement
Possible Methods
Interval Method
Thick set inversion problem
Bounded error context
Description
Multiple views
Consider :
an uncertain trajectory x(.) ∈ [x](.)
ϕ : R × R −→ R the
p
n
visibility function
(continuous)
ψ : R × R −→ R models the scope of the sensor
p
Given
n
x ∈ [x](.), we aim at nding the set:
Z(x ) = {z ∈ R |ϕ(z , x ) = 0 and ψ(z , x ) ≤ 0}.
q
Benoît Desrochers, Luc Jaulin
Explored
11 / 20
Problem Statement
Possible Methods
Interval Method
Thick set inversion problem
Example SSS
ϕ(z, x) =
x
u
Benoît Desrochers, Luc Jaulin
Bounded error context
Description
Multiple views
det
(u, z − x)
z
Explored
12 / 20
Problem Statement
Possible Methods
Interval Method
Thick set inversion problem
Example SSS
ϕ(z, x) =
x
u
θ
Benoît Desrochers, Luc Jaulin
Bounded error context
Description
Multiple views
det
(u, z − x)
z
Explored
12 / 20
Problem Statement
Possible Methods
Interval Method
Thick set inversion problem
Example SSS
ϕ(z, x) =
x
Benoît Desrochers, Luc Jaulin
Bounded error context
Description
Multiple views
det
(u, z − x)
u
θ
z
Explored
12 / 20
Problem Statement
Possible Methods
Interval Method
Thick set inversion problem
Example SSS
ϕ(z, x) =
det
Bounded error context
Description
Multiple views
(u, z − x)
ψ(z, x) = || < u, (z − x) > || − L
x
u
Benoît Desrochers, Luc Jaulin
z
Explored
12 / 20
Problem Statement
Possible Methods
Interval Method
Thick set inversion problem
Bounded error context
Description
Multiple views
Given:
an uncertain trajectory x(.) ∈ [x](.)
ϕ : R × R −→ R the
p
n
visibility function
(continuous)
ψ : R × R −→ R models the scope of the sensor
(continuous)
p
Given
n
x ∈ [x](.), we aim at nding the set:
Z(x ) = {z ∈ R |ϕ(z , x ) = 0 and ψ(z , x ) ≤ 0}.
q
Benoît Desrochers, Luc Jaulin
Explored
13 / 20
Problem Statement
Possible Methods
Interval Method
Thick set inversion problem
Bounded error context
Description
Multiple views
Given:
an uncertain trajectory x(.) ∈ [x](.)
ϕ : R × R −→ R the visibility function (continuous)
ψ : R × R −→ R models the scope of the sensor
(continuous)
Given x ∈ [x](.), we aim at nding the set:
p
n
p
n
Z(x ) = {z ∈ R |ϕ(z , x ) = 0 and ψ(z , x ) ≤ 0}.
q
Using interval analysis, can be enclosed by [DJZ13, DJ16]:
Z− ⊂ Z ⊂ Z+ .
\
Z− =
x
[
[
Z(x (t )) and Z+ =
(.)∈[x ](.) t ≥0
Benoît Desrochers, Luc Jaulin
x
Explored
[
Z(x (t )).
(.)∈[x ](.) t ≥0
13 / 20
Example
Problem Statement
Possible Methods
Interval Method
Benoît Desrochers, Luc Jaulin
Bounded error context
Description
Multiple views
Explored
14 / 20
Example
Problem Statement
Possible Methods
Interval Method
Benoît Desrochers, Luc Jaulin
Bounded error context
Description
Multiple views
Explored
14 / 20
Example
Problem Statement
Possible Methods
Interval Method
Benoît Desrochers, Luc Jaulin
Bounded error context
Description
Multiple views
Explored
14 / 20
Bigger example
Problem Statement
Possible Methods
Interval Method
5 hours of mission
error model δx = 0.2 ∗
Bounded error context
Description
Multiple views
√
t
track to track dist. = 150 m
eective range 120 m
Daurade
Benoît Desrochers, Luc Jaulin
Explored
15 / 20
Problem Statement
Possible Methods
Interval Method
Bounded error context
Description
Multiple views
Relaxed Intersection
Denition
Given m subsets X1 , . . . , X of R .
T
The q -relaxed intersection, denoted by X{ } = { } X , is the set
of all x ∈ R which belong to all X 's, except q at most.
n
m
q
q
i
n
i
Benoît Desrochers, Luc Jaulin
Explored
16 / 20
Problem Statement
Possible Methods
Interval Method
Relaxed Intersection
Bounded error context
Description
Multiple views
Examples
Given 3 disks,
JX{0} K
JX{1} K
Benoît Desrochers, Luc Jaulin
Explored
JX{2} K
17 / 20
Illustration
Problem Statement
Possible Methods
Interval Method
Benoît Desrochers, Luc Jaulin
Bounded error context
Description
Multiple views
Explored
18 / 20
Illustration
Problem Statement
Possible Methods
Interval Method
Benoît Desrochers, Luc Jaulin
Bounded error context
Description
Multiple views
Explored
18 / 20
Sum up
Problem Statement
Possible Methods
Interval Method
Bounded error context
Description
Multiple views
Interval methods are powerful to tackle with uncertainty
The proposed interval method
real time computing
simple hypothesis
continuous representation of the space
useful for small AUV with strong pose uncertainty
Benoît Desrochers, Luc Jaulin
Explored
19 / 20
Sum up
Problem Statement
Possible Methods
Interval Method
Bounded error context
Description
Multiple views
Interval methods are powerful to tackle with uncertainty
The proposed interval method
real time computing
simple hypothesis
continuous representation of the space
useful for small AUV with strong pose uncertainty
Thank you !
Benoît Desrochers, Luc Jaulin
Explored
19 / 20
References
Problem Statement
Possible Methods
Interval Method
Bounded error context
Description
Multiple views
B. Desrochers and L. Jaulin.
Computing a guaranteed approximation of the zone explored by
a robot.
IEEE Transactions on Automatic Control, PP(99):11, 2016.
V. Drevelle, L. Jaulin, and B. Zerr.
Guaranteed characterization of the explored space of a mobile
robot by using subpavings.
In Proc. Symp. Nonlinear Control Systems (NOLCOS'13),
Toulouse, 2013.
L. Paull, M. Seto, and H. Li.
Area coverage planning that accounts for pose uncertainty with
an auv seabed surveying application.
In Robotics and Automation (ICRA), 2014 IEEE International
Conference on, pages 65926599, May 2014.
Benoît Desrochers, Luc Jaulin
Explored
20 / 20