New trends in Air Traffic Complexity

Transcription

1 New trends in Air Traffic Complexity D. Delahaye and S. Puechmorel Applied Math Laboratory, ENAC March 8, 2010 D. Delahaye and S. Puechmorel (Applied Math Laboratory, New trends ENAC) in Air Traffic Complexity March 8, / 36

2 Agenda Why complexity metrics are needed? Dynamical system modeling of aircraft trajectories How a complexity map is built? Results D. Delahaye and S. Puechmorel (Applied Math Laboratory, New trends ENAC) in Air Traffic Complexity March 8, / 36

3 Agenda Why complexity metrics are needed? Dynamical system modeling of aircraft trajectories How a complexity map is built? Results D. Delahaye and S. Puechmorel (Applied Math Laboratory, New trends ENAC) in Air Traffic Complexity March 8, / 36

4 Why complexity metrics are needed? Airspace design Airspace comparison (US-Europe) Evaluation of Airspace Organisation Schemes Implementation of flexible use of airspace policies D. Delahaye and S. Puechmorel (Applied Math Laboratory, New trends ENAC) in Air Traffic Complexity March 8, / 36

5 Why complexity metrics are needed? Airspace design Airspace comparison (US-Europe) Evaluation of Airspace Organisation Schemes Implementation of flexible use of airspace policies 4D contract framework 4D Trajectory design. Forecasting of potentially hazardous traffic situations. Automated Conflict Solver enhancement (robustness of the solution). D. Delahaye and S. Puechmorel (Applied Math Laboratory, New trends ENAC) in Air Traffic Complexity March 8, / 36

6 Complexity vs Workload Workload Related to cognitive processes for human controllers. Easy/Hard forecasting of conflict occurence. Monitoring is a non negligible part of the workload. D. Delahaye and S. Puechmorel (Applied Math Laboratory, New trends ENAC) in Air Traffic Complexity March 8, / 36

7 Complexity vs Workload Workload Related to cognitive processes for human controllers. Easy/Hard forecasting of conflict occurence. Monitoring is a non negligible part of the workload. Complexity Related to traffic structure. Measure of intrinsic disorder of a set of trajectories. Increases with : Sensitivity to uncertainties. Interdependance of conflicts. D. Delahaye and S. Puechmorel (Applied Math Laboratory, New trends ENAC) in Air Traffic Complexity March 8, / 36

8 Intrinsic part of complexity Sensitivity Uncertainties on positions, wind speed, intents induce inaccuracy on trajectory prediction. Depending on the situation, prediction error can grow exponentially fast! In such a case, the situation is complex because nearly impossible to forecast. D. Delahaye and S. Puechmorel (Applied Math Laboratory, New trends ENAC) in Air Traffic Complexity March 8, / 36

9 Intrinsic part of complexity Sensitivity Uncertainties on positions, wind speed, intents induce inaccuracy on trajectory prediction. Depending on the situation, prediction error can grow exponentially fast! In such a case, the situation is complex because nearly impossible to forecast. Interdependance Solving a conflict may induce other ones. Even in conflict-less situations, interactions between trajectories can rise the perceived level of complexity. Complexity is related to mixing behaviour. D. Delahaye and S. Puechmorel (Applied Math Laboratory, New trends ENAC) in Air Traffic Complexity March 8, / 36

10 Sensitivity-interdependance No Sensitivity No conflict Easy situation Hight sensitivity Potential conflicts without interaction between solutions Hight sensitivity Potential conflicts with interactions between solutions Hard situation D. Delahaye and S. Puechmorel (Applied Math Laboratory, New trends ENAC) in Air Traffic Complexity March 8, / 36

11 Agenda Why complexity metrics are needed? Dynamical system modeling of aircraft trajectories How a complexity map is built? Results D. Delahaye and S. Puechmorel (Applied Math Laboratory, New trends ENAC) in Air Traffic Complexity March 8, / 36

12 Linear Dynamical System Modeling The key idea is to model the set of aircraft trajectories by a linear dynamical system which is defined by the following equation : X = A X + B where X is the state vector of the system : x X = y z Matrix A and vector B are the parameters of the model. D. Delahaye and S. Puechmorel (Applied Math Laboratory, New trends ENAC) in Air Traffic Complexity March 8, / 36

13 Regression of a Linear Dynamical System V 1 X 1 X 2 V 2 X 3 V 3 D. Delahaye and S. Puechmorel (Applied Math Laboratory, New trends ENAC) in Air Traffic Complexity March 8, / 36

14 Regression of a Linear Dynamical System D. Delahaye and S. Puechmorel (Applied Math Laboratory, New trends ENAC) in Air Traffic Complexity March 8, / 36

15 Regression of a Linear Dynamical System D. Delahaye and S. Puechmorel (Applied Math Laboratory, New trends ENAC) in Air Traffic Complexity March 8, / 36

16 Regression of a Linear Dynamical System Based on a set of observations (positions and speeds), one has to find a dynamical system which fits those observations.suppose that N obervations are given : Positions : x i X i = y i z i and speeds : V i = vx i vy i vz i D. Delahaye and S. Puechmorel (Applied Math Laboratory, New trends ENAC) in Air Traffic Complexity March 8, / 36

17 Regression of a Linear Dynamical System Based on a set of observations (positions and speeds), one has to find a dynamical system which fits those observations.suppose that N obervations are given : Positions : x i X i = y i z i and speeds : V i = vx i vy i vz i A LMS precedure is applied in order to extract the matrix A and the vector B. D. Delahaye and S. Puechmorel (Applied Math Laboratory, New trends ENAC) in Air Traffic Complexity March 8, / 36

18 Properties of the matrix A When real part of the eigenvalues of matrix A is positive, the system is in expansion mode and when they are negative, the system is in contraction mode. Furthermore, the imaginary part of such eigenvalues are related with curl intensity of the field. D. Delahaye and S. Puechmorel (Applied Math Laboratory, New trends ENAC) in Air Traffic Complexity March 8, / 36

19 Properties of the matrix A Imaginary axis full curl Organized Traffic convergence Real Axis divergence full curl D. Delahaye and S. Puechmorel (Applied Math Laboratory, New trends ENAC) in Air Traffic Complexity March 8, / 36

20 Linear Dynamical System Modeling : An example 1 v d v v 3 D. Delahaye and S. Puechmorel (Applied Math Laboratory, New trends ENAC) in Air Traffic Complexity March 8, / 36

21 Linear Dynamical System Modeling : An example Situation 1 Parallel trajectories (relative speed=0) Situation 2 Situation 3 Situation 4 Convergent Divergent Round about trajectories trajectories trajectories (relative speed=0) y y y y 2 v 2 2 d v v 2 v 1 v 90 x 1 v x v x v 1 1 x d v 3 3 v 3 3 v v Position of the eigenvalues of matrix A in the complex coordinate system eigenvalues Imaginary Imaginary Imaginary Imaginary Real 1 1 Real 1 1 Real 1 1 Real unit v/d D. Delahaye and S. Puechmorel (Applied Math Laboratory, New trends ENAC) in Air Traffic Complexity March 8, / 36

22 Linear Model limitations Give a global tendency of the traffic situation. Do not fit exactly with all traffic situations. Non Linear Extension D. Delahaye and S. Puechmorel (Applied Math Laboratory, New trends ENAC) in Air Traffic Complexity March 8, / 36

23 Non Linear Extension in Space X = f ( X ) Optimization problem f? such that : and i=n mine = V i f ( X i ) 2 i=1 min f ( x) 2 d x with f = R 3 2 f x x 2 2 f y x 2 2 f z x f x y f y y f z y f x z f y z f z z 2 D. Delahaye and S. Puechmorel (Applied Math Laboratory, New trends ENAC) in Air Traffic Complexity March 8, / 36

24 Non Linear Extension in Space Exact Solution (Amodei) with and f ( X N ) = Φ( X X i ). a i + A. X + B 2 6 Q = 4 i=1 Φ( X X i ) = Q( X X i 3 ) xx yy + 2 zz xx yy + 2 zz xx yy + 2 zz D. Delahaye and S. Puechmorel (Applied Math Laboratory, New trends ENAC) in Air Traffic Complexity March 8, / 36

25 Non Linear Extension in Space and time X = f ( X, t) Optimization problem We are looking for f such that : and i=n mine = k=k i=1 k=1 V i (t k ) f ( X i, t k ) 2 min f ( x) 2 + f R 3 t t 2 d xdt D. Delahaye and S. Puechmorel (Applied Math Laboratory, New trends ENAC) in Air Traffic Complexity March 8, / 36

26 Non Linear Extension in Space and time Exact Solution (Puechmorel and Delahaye) with f ( X, t) = N i=1 k=1 K Φ( X (t) X i (t k ), t t k ). a i,k + A. X + B Φ(r, t) = diag ( σ r. π.erf [ ]) r σ θ. t D. Delahaye and S. Puechmorel (Applied Math Laboratory, New trends ENAC) in Air Traffic Complexity March 8, / 36

27 Non Linear Extension in Space and time Gradient close form with Φ(r, t) x = (α β).x α = 2.σ r 2.π. 1.e 2 + θ. t β = Φ(r, t) r 2 r = x 2 + y 2 + z 2 r 2 σ 2.(2+θ. t ) D. Delahaye and S. Puechmorel (Applied Math Laboratory, New trends ENAC) in Air Traffic Complexity March 8, / 36

28 Agenda Why complexity metrics are needed? Dynamical system modeling of aircraft trajectories How a complexity map is built? Results D. Delahaye and S. Puechmorel (Applied Math Laboratory, New trends ENAC) in Air Traffic Complexity March 8, / 36

29 Characterization of sensitivity How fast two neighboring dynnamical system trajectories diverge? Let γ(t, s), t : [a, b], s V be a family of trajectories of the dynamical system in the neighborood V of a given point s 0. We assume that the nominal trajectory is t γ(t, s 0 ). A perturbed trajectory is t γ(t, s) with s V. Divergence to nominal trajectory with respect to time is thus γ(t, s 0 ) γ(t, s) = D(t, s). D. Delahaye and S. Puechmorel (Applied Math Laboratory, New trends ENAC) in Air Traffic Complexity March 8, / 36

30 Characterization of sensitivity How fast two neighboring dynnamical system trajectories diverge? Let γ(t, s), t : [a, b], s V be a family of trajectories of the dynamical system in the neighborood V of a given point s 0. We assume that the nominal trajectory is t γ(t, s 0 ). A perturbed trajectory is t γ(t, s) with s V. Divergence to nominal trajectory with respect to time is thus γ(t, s 0 ) γ(t, s) = D(t, s). Computing D(t, s) Main idea : when t γ(t, s) is the solution of a differential equation with initial condition γ(0, s) = s, it is possible to show that D itself satisfies a differential equation. D. Delahaye and S. Puechmorel (Applied Math Laboratory, New trends ENAC) in Air Traffic Complexity March 8, / 36

31 local behaviour of trajectories V s0 s D(t,s) γ (t,s ) 0 γ (t,s) D. Delahaye and S. Puechmorel (Applied Math Laboratory, New trends ENAC) in Air Traffic Complexity March 8, / 36

32 Local behaviour of trajectories The variational equation I The value of D(t, s) is the result of a cumulative process. Assume that γ(t, s) is defined to be a flow : γ(t, s) t = F (t, γ(t, s)) γ(0, s) = s with F a smooth vector field. Given a nonimal trajectory γ(t, s 0 ), then divergence of nearby trajectories can be found up to order one in s s 0 by solving : D(t, s) t = DF (t, γ(t, s 0 )).D(t, s) D(0, s) = s s 0 with DF the jacobian matrix of F (with respect to s). D. Delahaye and S. Puechmorel (Applied Math Laboratory, New trends ENAC) in Air Traffic Complexity March 8, / 36

33 Lyapunov exponents The variational equation II Since the previous equation is linear, it can be described by a matrix M(t) that obeys : dm(t) dt = DF (t, γ(t, s 0 )).M(t) M(0) = Id This equation is called the variational equation of the flow. The variational equation describes in some sense a linear dynamical system tangent to the original one. D. Delahaye and S. Puechmorel (Applied Math Laboratory, New trends ENAC) in Air Traffic Complexity March 8, / 36

34 Lyapunov exponents Lyapunov exponents Let U t (t)σ(t)v (t) = M(t) be the SVD decomposition of M(t). The Lyapunov exponents are mean values of the logarithms of the diagonal elements of Σ(t): κ(s) = 1 T T 0 log(σ ii (t))dt Σ ii (t) 1 D. Delahaye and S. Puechmorel (Applied Math Laboratory, New trends ENAC) in Air Traffic Complexity March 8, / 36

35 Summary Based on trajectories observations, a 4D non linear dynamical system model is buit. The associated 4D vector field is then computed on a cube of airspace. Lyapunov exponents are computed on each point of the cube in order to built a complexity map. D. Delahaye and S. Puechmorel (Applied Math Laboratory, New trends ENAC) in Air Traffic Complexity March 8, / 36

36 Interpretation of Lyapunov exponents Given an initial point, the Lyapunov exponents and the associated SVD decomposition provide us with a decomposition of space in principal directions and corresponding convergence/divergence rate. It is a localized version of the complexity based on linear systems. D. Delahaye and S. Puechmorel (Applied Math Laboratory, New trends ENAC) in Air Traffic Complexity March 8, / 36

37 Agenda Why complexity metrics are needed? Dynamical system modeling of aircraft trajectories How a complexity map is built? Results D. Delahaye and S. Puechmorel (Applied Math Laboratory, New trends ENAC) in Air Traffic Complexity March 8, / 36

38 Results D. Delahaye and S. Puechmorel (Applied Math Laboratory, New trends ENAC) in Air Traffic Complexity March 8, / 36

39 Results D. Delahaye and S. Puechmorel (Applied Math Laboratory, New trends ENAC) in Air Traffic Complexity March 8, / 36

40 Results D. Delahaye and S. Puechmorel (Applied Math Laboratory, New trends ENAC) in Air Traffic Complexity March 8, / 36

41 Conclusion An intrinsic trajectories complexity metric has been developped. This metric can be computed as a map. Give the areas of airspace where the traffic is organized and the ones where there is desorder D. Delahaye and S. Puechmorel (Applied Math Laboratory, New trends ENAC) in Air Traffic Complexity March 8, / 36

42 Current work Parallel computation by the mean of Local linear Model V i ( X i, t i ) = f ( X, t) + f ( X, t) (t i t) + t f ( X, t) + f ( X, t) X ( X i X )+ min a, b,c O( t i t + X i X ) = a + b(t i t) + C.( X i X ) + O( t i t + X i X ) N V i ( X i, t i ) a + b(t i t) + C.( X i X ) 2.ψ(t i t, X i X ) i=1 Robustness improvement by taking into account uncertainties on time position of aircraft on their trajectories D. Delahaye and S. Puechmorel (Applied Math Laboratory, New trends ENAC) in Air Traffic Complexity March 8, / 36