ICAPS05 WS6  icaps 2005
W 1
TOW
ENU1
ENU2
ENU3
ENU4
W i Wj
W 2
W 5
W 3
ENO11
ENO12
ENO21
ENO22
W i is in S(W i )
W 4
EXO11
EXO12
EXO21
EXO22
W 6
W7
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:
TW1
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 takeoff 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) −
o∈Eo
W 8
EXU1
EXU2
EXU3
EXU4
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
form:
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 .
W e
LW1
LW2
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
by:
˜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
(nj−1,nj,aj)∈Eem
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