ICAPS05 WS6 - icaps 2005


ICAPS05 WS6 - icaps 2005

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W i Wj

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W 5

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W i is in S(W i )

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Figure 5: Finite state machine of the mission -

Objective 1: s(1) = 3, e(1) = 4, r(1) = 4;

Objective 2: s(2) = 5, e(2) = 6, r(2) = 7

Time and resources

The resources vector r ∈ R2 contains the probability of

being alive at current time (first component noted r1 ) and

the mass of the vehicle at current time (second component

noted r2 ). For a more complex description of the expression

of these terms, see (Chanthery, Barbier, & Farges 2004).

For each objective, the reward Ro is defined by the function

Go.po(ts(o)).r1 r(o) where Go is the maximum reward associated

to o, po(ts(o)) represents the quality of the observation

at time ts(o) and r1 r(o) represents the probability of being

alive at data transmission time. The cost function Re corresponds

to the costs of danger and consumption:


Re(re) = (r1 − re) ⊤ .C

where C is a vector of R 2 , whose first component is the

price of the aerial autonomous system (vehicle and payload

included), and whose second component is the price of the

fuel per mass unit. So, costs are decreasing with resources.

The planning goal is to find a sequence of states beginning

by the take-off waypoint, ending by the set of landing waypoints

of the mission and using the possible trajectories between

two sets of waypoints. The sequence has to minimize

the difference J between costs of danger and consumption

and rewards obtained for the data transmission while satisfying

the constraints on danger and fuel.

J = Re(re) −


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Go.po(t s(o)).r 1 r(o)

The constraint Ce(re) ≥ 0 expresses the fact that the vehicle

has enough chances to finish its mission. It has the following


re − rmin ≥ 0

Indeed, the probability of being alive at the end of the mis-

) under which

sion must be greater than a given limit (r1 min

the vehicle is considered as destroyed. The fuel being limited,

the mass of the vehicle cannot be lower than the mass

without fuel r2 min .

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Low level description

Different motion actions are possible to reach a node of N.

If there is no danger, the motion action is the straight line.

If there is a danger, it is possible to bypass the danger or

to cross it. During the treatment of an objective, the vehi-

cle can follow an outline or a trajectory for the area survey.

Bounds on ∆ ak+1

nk,nk+1 are computed by considering on the

one hand aerodynamic and propulsion characteristics of the

vehicle and on the other hand the traveled distance, the average

slope and the average height from the node nk to the

node nk+1 using action ak+1.

Some nodes of N have a time window. For the entrance

and exit points of the unsafe area, time windows correspond

to operational procedures to safely cross the frontier. For

each objective, time window indicates the times when the

observation is valid.

Resources consumption

Resources are consumable, so they decrease with time. Fuel

resource is decomposable. Let us simplify the notation

in ∆. The decrease of the fuel on the arc from

∆ai ni−1,ni

node ni−1 to node ni corresponding to the action ai is given


˜f 2 π(i) (ni−1, ni, ai, ∆) =


α(ni−1, ni, ai) 1

∆ 2 + β(ni−1, ni, ai).∆ 4

where α(ni−1, ni, ai) and β(ni−1, ni, ai) are computed by

considering the same parameters as for bounds on ∆ai ni−1,ni .

On the contrary, the probability of being alive is not decomposable.

It depends on the entire past path of the vehicle.

Indeed, the probability of being alive is the product, on all

the exposures to danger along the path, of the probability of

surviving the considered exposure. It is given by:

f 1 π(i) (n1, . . . , ni, a1, . . . , ai, ∆ a2

, . . .,∆ai ) =

n1,n2 ni−1,ni

1 − γm.pt(

∆.δ(nj−1, nj, aj))

m∈SM em∈Em


ICAPS 2005

where SM is the set of threats, Em the set of exposures

for threat m, γm the probability that the threat m actually

exists and is able to destroy the vehicle when it is detected,

Eem the set of arcs exposed to the threat during exposure

em, δ(nj−1, nj, aj) the ratio of the arc (nj−1, nj, aj) that is

exposed to the threat and pt the probability of being detected

in function of the time of exposure. The probability pt is

given on Figure 6.

The probability of being alive is not a linear function of

exposure duration.

Planning Algorithms

Algorithmic framework

The plan search is performed on the tree of possible actions.

Proposed algorithms are different from the ones of the literature:

for each developed node, the precise evaluation of

the criterion requires an optimization of the instants at each

Workshop on Planning under Uncertainty for Autonomous Systems 43

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