Guaranteed Polynesian Navigation

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Guaranteed Polynesian Navigation

Thibaut Nico, Luc Jaulin, Benoit Zerr

To cite this version:

Thibaut Nico, Luc Jaulin, Benoit Zerr. Guaranteed Polynesian Navigation. Summer Workshop on

Interval Methods (SWIM) 2019, Jul 2019, Paris, France. �hal-02304716�

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Guaranteed Polynesian Navigation

Thibaut Nico

¹

, Luc Jaulin

^∗²

, and Benoît Zerr

²

1

ECA, Lab-STICC UMR CNR 6285

2

ENSTA-Bretagne, Lab-STICC UMR CNRS 6285

Figure 1: Polynesian navifation

Keywords: Navigation; Intervals, Contrac- tors, No-lost zone

Introduction

The Polynesian navigation problem asks to move from islands to other islands without be- ing lost. The navigation should be performed without without GPS, compass and clocks.

The diculty of the navigation is illustrated by Figure 1: the ocean is huge, the islands are small, the boats are more or less uncertain.

Among the techniques used by Polynesian, the observation of the stars (see Figure 2) are useful to get the heading, but also to detect if the boat is on the route which leads us to the desired island. The approach we will fol- low to guarantee that we can reach an island from another island, uses guaranteed integra- tion [10], tube programming [9], [8], constraint programming [11], localization [3], contractors [2] and interval analysis [6][4].

∗Corresponding author.

Figure 2: Pair of stars technique: the boat is on the right route if the bottom star rises when the right star sets

Formalisation

The problem can be formalized as follows

• Given a set of geo-localized islands m

i

, i ≥ 0 .

• The ith coastal area is:

C

i

= {x | c

_i

(x) ≤ 0} .

• A robot has to move in this environment without being lost.

Figure 3 represents a set of 4 islands with the associated coastal zones C

1

, C

2

, C

3

, C

4

(painted blue).

We assume the following

• The coastal areas are small compare to the oshore area.

• In the coastal area, the robot knows its state.

• Oshore, the robot is blind and has an

open loop strategy, such as for instance

go North.

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50 100 150 200 250 300 350 400 450 20

40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400 420 440

y (m)

x (m)

Figure 3: Islands ans coastal zones

• The robot is described by blind state equations

x ˙ = f (x, u) , u (·) ∈ [u] (t) x (0) = x

0

where the input u(t) belongs to the un- certainty box [u] (t) .

We dene the set ow Φ : R × R

ⁿ

→ P ( R ) as:

Φ(t

₁

, x

₀

) = { a|∃u (·) ∈ [u] (t) , a = x(t

₁

), x ˙ = f(x, u), x (0) = x

0

} Given the set A (for instance a coastal area), the backward reach set [1] is dened by

Back ( A ) = { x | ∀ϕ ∈ Φ,

∃t ≥ 0, ϕ(t, x) ∈ A } Interval analysis is often used to compute backward reach sets in the case where the robot is nonlinear [5], [7]. We have

Back ( A ∪ B ) ⊃ Back ( A ) ∪ Back ( B ) . This is the Archipelago eect which tells us that nding an Archipelago (A ∪ B) is easier than nding individual islands, as illustrated by Figures 4 and 5.

100 150 200 250 300 350 400 450

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340

y (m)

x (m)

Figure 4: Back ( A ) ∪ Back ( B )

100 150 200 250 300 350 400 450

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380

y (m)

x (m)

Figure 5: Back ( A ∪ B )

Moving between coastal zones

Assume that we have m coastal sets C

1

, C

2

, . . . , i ∈ {1, 2, . . . } and open loop con- trol strategies u

j

, j ∈ {1, 2, . . . } or equiva- lently, we have set ows Φ

j

(t, x

0

). Moreover, we assume that the control strategy cannot change oshore. As a consequence,

• From C

1

we can reach C

2

with the j th control strategy if C

1

∩ Back (j, C

2

) 6= ∅.

• From C

1

we can reach C

2

with at least one control strategy if C

1

∩ S

j

Back (j, C

2

) 6= ∅.

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Figure 6: Reach an island from another island using a 'Go-East' strategy

Figure 7: Reachability graph

• From C

1

we can reach C

2

∪ C

3

with at least one control strategy if C

1

∩ S

j

Back (j, C

2

∪ C

3

) 6= ∅.

Therefore, we dene the reachability relation , → as:

• C

a

, → C

b

if from C

a

we can reach C

b

with at least one control strategy j .

• , → is the smallest transitive relation which satises

∀k ∈ K , C

ik

, → C

b

∃j, C

a

∩ Back (j, S

k∈K

C

ik

) 6= ∅

⇒ C

a

, → C

b

Consider for instance, the hyper-graph of Fig- ure 6 where the relation A →

^j

B , C means that from A the robot can reach either B or C using the jth strategy. For instance, in our graph

C

1

∩ Back (1, C

3

∪ C

4

) 6= ∅ ⇒ C

1

→

1

( C

3

, C

4

) Thus, the associated reachability graph (corresponding to , → ) is given by Figure 7.

In a similar way, we can also dene the for- ward reach set as illustrated by Figure 8.

100 150 200 250 300 350 400 450

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360

y (m)

x (m)

Figure 8: Forward reach set from a given is- land

No lost zone

We dene the no-lost zone as the set S of all states that we may visit from a coastal area without being lost with the available control strategies. Dene the index set associated with the strategy I

_j

as

I

_j

= {k| C

k

∩ Back (j, [

i6=k

C

i

) 6= ∅}.

If we start from C

k

, k ∈ I

_j

, then we will reach at least another coastal area with the control strategy j . We have

x ∈ Back (j, S

i

C

i

)

x ∈ Forw(j, C

k

), k ∈ I

_j

⇒ x ∈ S Thus

S ⊂ [

j

[

k∈I_j

Forw (j, C

k

) ∩ Back (j, [

i

C

i

).

This property will allow us to have an inner

approximation of the no-lost zone, which is

the main contribution of this paper. This is il-

lustrated by Figure 9 with 8 strategies: North,

East, South, West, North-East, East-South,

South-West, West-North.

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50 100 150 200 250 300 350 400 450 500 0

20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400 420 440

y (m)

x (m)

Figure 9: No-lost zone associated with the 5 islands

Aknowledgement

This work has been supported by the French Government Defense procurement and tech- nology agency (DGA).It also beneted from the support of the project CONTREDO of the French National Research Agency (ANR).

References

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