Simulation Based Aircraft Trajectories Design for Air Traffic Management

Transcription

1 Simulation Based Aircraft Trajectories Design for Air Traffic Management N. Kantas 1 A. Lecchini-Visintini 2 J. M. Maciejowski 1 1 Cambridge University Engineering Dept., UK 2 Dept. of Engineering, University of Leicester, UK Cambridge Statistics Initiative, 25 September 2009

2 Outline Air Traffic Management Introduction Problem Formulation Dynamical Model for Level flight Statistical Model for Wind Velocity Cost: Expected time of arrival at exit region Methodology Parameterise control policy Compute parameter using Sequential Monte Carlo Samplers for Simulated Anealling Simulations

3 Air Traffic Management Monitor air traffic at some designated area of airspace Aircraft must maintain safe separation (e.g. 9 km) Each aircraft wishes to reach some target exit zone sooner/using less fuel etc. ATM role is to solve problems Conflict Detection Conflict Resolution Long term plan: to handle more dense airspace by using automation Investigate Monte Carlo Algorithms (EPSRC,EC Projects: HYBRIDGE, ifly)

4 Simple Level Flight Model State Xt i :iscomposedofthexand y Cartesian coordinates of the aircraft. xt i = xt 1 i cos h i + t sin ht i vtasδ i + w t 1 (xt 1)δ i (1) v i tas: trueaircraftspeed h i t:heading(controlinput) δ: timediscretisationinterval w t (x) :is a random wind velocity at point x

5 Statistical Wind Modelling Example Wind Velocity modelled as in [Stroud et al 2001 JRSSB]: w t (x) =µ t (x; α t )+v t, v t iid N(0, Σt (x; β t )) with α t,β t being parameters, which are computed recursively from real forecast datasets (NOAA-RUC2).

6 Wind Velocity Snapshot

7 Control heading to counteract the wind true airspeed v tas B i e i wind velocity w 2 x i 3 h 2 x i 2 x i 1

8 Parameterisation of Control Inputs Divide the sequence of h1:t i to a sequence of H constant blocks (of varying dimensions): h1:t i = h i 1,...,h i T 1, h i 2,...,h i T 2,..., h i T H,...,hi H h 1 h 2 h H Block 1 Block 2 Block H Propose policy with a fixed heading for the last block: h i H is the angle to exit at the beginning of block H h i H = tan 1 ( ei (2) xt (H 1)p i (2) e i (1) xt (H 1)p i (2) (1)). Policy Parameter: θ i =[h i 1, hi 2,...,hi H 1 ] T,

9 Cost function For each aircraft the time of arrival T i at the random hitting time of target set B i. T i = inf t : X i t B i. t 1 Cast the Conflict Resolution problem for M aircraft in parallel as follows: min E x θ i i 0,θ i (T i ), i = 1,...,M. s.t. P x0,θ( Xk i X j k > d) 1 ε, i = j, k = 1,...,t where ε is some tolerance level.

10 Simulation based approach to compute θ Assuming E x0,θ i (T i ) <, estimatingθ i by maximising J x0 (θ i )= u(τ i )π(τ i )dτ i, (3) with u(τ) =τ τ, where τ is such that u > 0. Bayesian Optimal Design [Muller et al 2004, JASA]: sample from the joint posterior distribution γ n π n (θ i,τ1:γ i n ) p(θ i ) u(τj i )π(τj i ) (4) where{γ n } n 0 is a strictly increasing integer infinite sequence. Sample Θ i, T1:γ i n using a Sequential Monte Carlo [Del Moral et al 2004, JRSSB] Compute estimate of optimal θ i j=1

11 Simulations aircraft 1 aircraft 2 aircraft 3 aircraft 4 Latitude in km Expected cost approximation Longitude in km Iterations n 1000 ESS aircraft 1 aircraft 2 aircraft 3 aircraft 4 Separation between every aircraft pair in km Iterations n Time in mins

12 Possible Extensions and References Possible Future work: More complex dynamics (e.g. 3-D models with climb and descend) Compare with MCMC optimisation [Muller et al 2002] Generalise framework to include a wider class of policies, include feedback References: Del Moral P., Doucet A., Jasra A. (2006) Soc. B, vol. 68, no. 3, pp Sequential Monte Carlo Samplers, J. Royal Statist. Kantas N., Lecchini-Visintini A., Maciejowski J.M. (2009) Simulation Based Optimal Design of Aircraft Trajectories for Airtraffic Management, submitted. Muller P., Sanso B. G., De Iorio M. (2004). Optimal Bayesian design by Inhomogeneous Markov Chain Simulation. Journal of the American Statistical Association, pp Stroud, J., Mueller, P., and Sanso, B. (2001). Dynamic Models For Spatio-Temporal Data. Journal of the Royal Statistical Society, Series B, 63,