Guaranteed Polynesian Navigation
Thibaut Nico1, Luc Jaulin ∗2, and Benoît Zerr2
1ECA, Lab-STICC UMR CNR 6285
2ENSTA-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 mi, i≥0.
• The ith coastal area is:
Ci={x|ci(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 C1,C2,C3,C4
(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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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) =x0
where the input u(t) belongs to the un- certainty box [u] (t).
We dene the set ow Φ : R×Rn → P(R) as:
Φ(t1,x0) ={ a|∃u(·)∈[u] (t),a=x(t1), x˙ =f(x,u),x(0) =x0 } Given the setA(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.
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y (m)
x (m)
Figure 4: Back(A)∪Back(B)
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x (m)
Figure 5: Back(A∪B)
Moving between coastal zones
Assume that we have m coastal sets C1,C2, . . . , i ∈ {1,2, . . .} and open loop con- trol strategies uj, j ∈ {1,2, . . .} or equiva- lently, we have set ows Φj(t,x0).Moreover, we assume that the control strategy cannot change oshore. As a consequence,
• From C1 we can reach C2 with the jth control strategy if C1∩Back(j,C2)6=∅.
• From C1 we can reach C2 with at least one control strategy if C1 ∩ S
jBack(j,C2)6=∅.
Figure 6: Reach an island from another island using a 'Go-East' strategy
Figure 7: Reachability graph
• From C1 we can reach C2 ∪ C3 with at least one control strategy if C1 ∩ S
jBack(j,C2∪C3) 6= ∅.
Therefore, we dene the reachability relation ,→ as:
• Ca,→Cbif fromCawe can reachCbwith at least one control strategy j.
• ,→ is the smallest transitive relation which satises
∀k∈K,Cik ,→Cb
∃j,Ca∩Back(j,S
k∈KCik)6= ∅
⇒ Ca,→Cb
Consider for instance, the hyper-graph of Fig- ure 6 where the relationA→j B,Cmeans that fromAthe robot can reach eitherBorCusing thejth strategy. For instance, in our graph
C1∩Back(1,C3∪C4)6= ∅ ⇒C1
→1 (C3,C4) 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.
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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 strategyIj as
Ij ={k|Ck∩Back(j,[
i6=k
Ci) 6= ∅}.
If we start fromCk,k∈ Ij, then we will reach at least another coastal area with the control strategy j. We have
x∈Back(j,S
iCi)
x∈Forw(j,Ck), k∈ Ij ⇒x∈S Thus
S⊂[
j
[
k∈Ij
Forw(j,Ck)∩Back(j,[
i
Ci).
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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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).
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