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For each K, independently draw Np random pairs (x ,i , ,i ), i = 1, , Np , from the particle spanned empirical measure, using (911): px , | +h , k (dx, | , 1)
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This yields, for each K, a set Np {x ,i , ,i , ,i }i=1 with ,i := ( )/Np
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If k < m, then repeat steps 1 3 for k := k + 1 and N := Np Remark 1 The key extensions of HHIPS over IPS for an SHS [8,9] are: (i) embedding of an aggregation mode process; (ii) particles are maintained per aggregation mode;
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(iii) importance switching of aggregation mode is used for the conditional prediction of SHS particles; (iv) hierarchical interaction is used for the resampling of particles that reached Qk , k = 1, , m 1
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We consider a speci c free ight operational concept that has been developed within a recent European research project [37] In the free ight air traf c example, the airspace is an en-route airspace without xed routes and without support by air traf c control All aircraft ying in this airspace are assumed to be properly equipped and enabled for free ight: the pilots can try to optimize their trajectory, due to the greater freedom to choose path and ight level The pilots are only limited by their responsibility to maintain airborne separation, in which they are assisted by the so-called Airborne Separation Assistance System (ASAS) This system processes the information ows from the data-communication links between aircraft, the navigation systems and the aircraft guidance and control systems ASAS detects con icts, determines con ict resolution maneuvers and presents the relevant information to the aircrew The number of agents involved in the free ight operation is huge and ranges from the Control Flow Management Unit to ight attendants In the setting chosen for an initial risk assessment, the following agents are taken into account: one ying pilot in each aircraft; one non- ying pilot in each aircraft; various systems and entities per aircraft, such as the aircraft position evolution and the con ict management support systems; some global systems and entities, such as the communication frequencies and a satellite system The approach taken in developing the speci c free ight concept of operation [37] is to avoid much information exchange between aircraft and to avoid dedicated decision-making by arti cial intelligent machines Although the con ict detection and resolution approach developed for this free ight concept has its roots in the modi ed potential eld approach [30], it has some signi cant deviations from this The main deviation is that con ict resolution is intentionally designed not to take the potential eld of all aircraft into account The resulting design can be summarized as follows: All aircraft are supposed to be equipped with Automatic Dependent Surveillance-Broadcast (ADS-B), which is a system that periodically broadcasts own aircraft state information, and continuously receives the state information messages broadcasted by aircraft that y within broadcasting range ( 100 Nm)
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To comply with pilot preferences, con ict resolution algorithms are designed to solve multiple con icts one by one rather than in a fully concurrent way (see [30]) Con ict detection and resolution are state-based, that is, intent information, such as information at which point surrounding aircraft will change course or height, is assumed to be unknown The vertical separation minimum is 1000 ft and the horizontal separation minimum is 5 Nm A con ict is detected if these separation minima will be violated within 6 minutes The con ict resolution process consists of two phases During the rst phase, one of the aircraft crews should make a resolution maneuver If this does not work, then during the second phase, both crews should make a resolution maneuver Prior to the rst phase, the crew is warned when an ASAS alert is expected to occur if no preventive action would be implemented on time; this prediction is done by a system referred to as Predictive ASAS (P-ASAS) Con ict co-ordination does not take place explicitly, that is, there is no communication on when and how a resolution maneuver will be executed All aircraft are supposed to use the same resolution algorithm, and all crew are assumed to use ASAS and to collaborate in line with the procedures Two con ict resolution maneuver options are presented: one in the vertical and one in the horizontal direction The pilot decides which option to execute ASAS related information is presented to the crew through a cockpit display of traf c information In order to use the HHIPS algorithm for the estimation of collision risk in this free ight operation, we need to develop an MC simulator of this operation, such that the simulated trajectories constitute realizations of a hybrid state strong Markov process Everdij and Blom [23,24,25,26] have developed a stochastically and dynamically colored Petri net (SDCPN) formalism that ensures the speci cation of a free ight MC simulation model which is of the appropriate class In [9] it is explained how the SDCPN formalism has been used to develop an MC simulation model of a particular free ight design The dimensionality of the resulting MC simulation model is very large, for example in simulating two aircraft there are about 1025 discrete mode combinations, and the Euclidean state may go up to R336 [9] For this very large stochastic hybrid system we want to estimate the probability of collision between aircraft This is practically infeasible using naive MC simulation
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