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An algorithmic approach to air path computation

Devan SOHIER

LDCI-EPHE (Paris)

23/11/04

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Outline

IntroductionSituationModelingMarkov decision processesConclusion and perspectives

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Outline

IntroductionSituationModelingMarkov decision processesConclusion and perspectives

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Problem

Find safe trajectories for all aircrafts in a given portion of the airspace

Taking into account stochastic events:Temporary flyover interdiction (due to

meterological conditions or some other reason)

Deviation…

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Levels of ATC

ATC can be divided in several levels:Strategic level for mid-term planning of

flights:many aircraftsmeteorological uncertainties

Tactical level for short-term managementfew aircrafts (2 or 3)uncertainties about the location (deviation)

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Outline

IntroductionSituationModelingSolutionConclusion and perspectives

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Aircrafts crossing

Two aircrafts x and ygo from xd and yd to xf and yf

risks of conflictFind a « good » trajectory for each of

them (safe, and cheap)

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Aircrafts crossing

Minimize:

∫x’2+y’2dt Under the safety constraint:

d(x,y)>ds

and some constraints on speed! We work on (x, y)R6

The trajectory of (x, y) is composed of segments of straight lines and arcs of ellipses in R6

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Stochastic aircrafts crossing

A stochastic deviation (d1, d2) is added to the model:minimize:

E[∫(x+d1)’2+(y+d2)’2dt]under the constraint:

d(x+d1,y+d2)>ds

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Problems

Continuous modeling of the deviation:difficult to determinedifficult to exploit (a continuous time

markovian modeling cannot be adequate)Moreover will it provide useful

information?

discretization

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Outline

IntroductionSituationModelingMarkov decision processesConclusion and perspectives

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Our Modeling

Existing modelings use:Continuous spaceContinuous time

We propose a discrete modeling more adequate to programming

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Bricks

Discretization of the airspace :Bricks (parallelepipeds)Size = safety distances

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Modeling of the airspace

To improve the modeling:Use of a honeycomb pavingDiscrete time

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Voronoi paving

Introduction of dynamic safety distances by the use of a Voronoi paving

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The graph

Allowed movements are modeled by a graph

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Statistics

Markov (resp. semi-markovian) processes are a simple, general and well-known modeling

All the information is contained in the most recent observation(s)

The deviation evolves in a memoryless way: the deviation at time t+1 only depends on the deviation at time t (resp. t, t-1, …, t-k)

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Statistics

Preliminary Markov tests on the deviationhighlights a different behaviour of

transversal and longitudinal deviationssemi-Markovian with a dependence to

history of about 5

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Outline

IntroductionSituationModelingMarkov decision processesConclusion and perspectives

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Static vs. dynamic

Static solutionsWorst-case analysisLoss of airspace

DynamicityAdapt the solution to the current situationUse all the available information

But dynamicity requires more computing power

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Dynamic programming

An optimal path (xt)t>0 is such that for all t0, (xt)t>t0 is also optimal starting from the situation xt0

Continuous time difficult to applyThrough discretization we obtain an

adequate framework

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Markov Decision Process

Dynamic programming with a Markov « opponent »

Find rules giving the decision to make in each situation, taking into account the probabilities of evolution under constraints

Safe: in each safe situation, a safe reaction is proposed

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Markov Decision Process

We define for each deviation d and situation s:

dk,d(s, g)=min{d(s,s’)+d2 pd,d2 dk-1,d2(s’, g)/ss’}

Nextk,d(s)=argmin{d(s,s’)+d2 pd,d2 dk-1,d2(s’,g)/ss’}

with g=(xf, yf) the final situation, for all k: dk, d(s1, s2)= if s1+d is forbidden and dk, d(s, s)=0

When these quantities do not evolve any longer, we obtain the optimization rules.

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Complexity

Complexity of this MDP grows with the size of the history (5 in this case) of the Markov chain

Much more efficient than the computation of exact optimal solutions

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Outline

IntroductionSituationModelingMarkov decision processesConclusion and perspectives

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Conclusions

Dynamic computation of air trajectories may save much airspace without decreasing the safety

Markovian (memoryless) discrete modelings provide an efficient and adequate framework allowing computer programming of the solution

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Works in Progress

Works in collaboration with L. El Ghaoui (Berkeley), A. d’Aspremont (Princeton) on the strategic level

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Perspectives

Statistical validation of the modelingUse of continuous modeling and

decision rules, and discretization of the solution

Use of pretopological tools to refine the notion of conflict

Decentralization of the decision by the use of negociations protocols

Introduction of some equity constraints