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Model predictive control approach to global air collision avoidance K. Bousson Avionics and Control Laboratory, Department of Aerospace Sciences, University of Beira Interior, Covilha˜, Portugal Abstract Purpose – Most of the existing approaches for flight collision avoidance are concerned with local traffic alone for which the separation is based on the pairwise analysis of aircraft trajectory trends, which is not efficient with regard to some flight path requirements along waypoints. The purpose of this paper is to deal with the global collision avoidance problem which aims at separating aircraft taking into consideration the global traffic in a given area instead of considering them pairwise. It aims to model the concept of global collision avoidance and propose a validated algorithm for the purpose in the framework of free-flight. Design/methodology/approach – The collision avoidance procedure computes online the appropriate speed and heading for each aircraft, at each sampling time-instant, to generate a collision-free flight trajectory along scheduled waypoints. The method accounts for automatic assignment of priority indexes that are updated from one control time horizon to the next. The paper considers here the case of aircraft flying at the same altitude, but the proposed method is easily extendable to the general 3D situation. Owing to the predictive features that are inherent to collision avoidance, the collision-free trajectories are generated using model predictive control approach. A simulation example is presented in the end and its results show the suitability of the proposed approach. Findings – Since the model predictive control approach is used, the collision avoidance procedure is anticipative; therefore, the avoidance capability depends only on the prediction horizon rather than on the control horizon. Practical implications – The computation of the trajectory guidance information (speed and heading) at each time-step is fast, therefore the proposed method is well suited for online processing requirements in real world applications. Originality/value – The paper provides a flexible modelling framework and a computationally effective algorithm, both based on model predictive control concepts, to cope with global collision avoidance for flight safety enhancement in the framework of free-flight. Keywords Air traffic control, Collisions, Air safety Paper type Research paper

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controllers is called free-flight. Current advances in satellitebased air navigation systems (EGNOS, GPS, Galileo, etc.) have led air traffic research laboratories to consider the free-flight concept as a major component for the future air traffic control system (Bousson, 2003; Hu et al., 2002, 2000; Menon and Sweriduk, 1999; Yang and Kuchar, 1997; Paielli and Erzberger, 1999). Meanwhile, enabling aircraft to evolve freely in space will result in the complexity of air traffic control and flight safety. There are two groups of approaches about air collision avoidance: the local and the global separation approaches. The local approach consists in separating aircraft based on the air traffic trends in the neighbourhood of a given aircraft, that is, aircraft are separated by pairs. The global approach aims at separating aircraft taking into consideration the overall traffic trends in a given control area so that the separation of an aircraft A from B may not induce a conflict between aircraft A and another aircraft C. The traffic alert and collision avoidance system (TCAS) is an operational airborne collision avoidance system that only ensures local separation (Brooker, 2005a, b). TCAS provides ground-independent protection from mid-air collisions as a backup to conventional air traffic control system. Effort has also been made in processing air traffic control in terminal

Aircraft Engineering and Aerospace Technology: An International Journal 80/6 (2008) 605– 612 q Emerald Group Publishing Limited [ISSN 1748-8842] [DOI 10.1108/00022660810911545]

This research was conducted in the Aeronautics and Astronautics Research Center (AeroG) of the University of Beira Interior at Covilha˜, Portugal, and supported by the Portuguese Foundation for Sciences and Technology.

1. Introduction In current air traffic control concepts, the freedom to fly preferred trajectories is constrained on the one hand by the fact that traffic flow control is the responsibility of air traffic controllers, and on the other hand, aircraft must fly along predetermined airways due to the location of ground-based radio navigation systems and other environmental considerations. This is obviously not optimal since it impedes aircraft to take advantage, for instance, of flight time and fuel consumption requirements or even of favourable winds. Owing to the increasing air traffic density and to the relatively limited number of airways, which results in increasing flight delays and collision risks, the future solution for safe and optimal airspace use would be to design a new organization for air traffic control (Cavcar and Cavcar, 2004; Debbache, 2001), or to give the capability to aircraft to evolve not only in pre-specified airways but also out of these airways. The capability of an aircraft to choose freely its own trajectory, to fly along it and at the same time being able to ensure its safety independently of air traffic

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areas too, such as the Center-TRACON automation system (Alcabin and Erzberger, 1985; Davis et al., 1991; Erzberger et al., 1993) which serves as a decision-support tool for ground controller in an effort to reduce air traffic control workload and optimize capacity close to highly congested airports. Other research activities on local separation have been dedicated to free-flight in academic institutions. Durand et al. (1995) use genetic algorithms to solve aircraft conflicts in which manoeuvres were used to construct the new trajectories. Paielli and Erzberger (1999) use probability calculus for predicting conflict between aircraft along a predicted trajectory. A dual cooperative and non-cooperative conflict resolution architecture for multi-agent hybrid systems with emphasis on air traffic management systems is presented by Tomlin et al. (1998), and a prototype alerting system for a conceptual freeflight environment is discussed in Paielli and Erzberger (1999) where the alerting logic is based on a probabilistic model of aircraft sensor and trajectory uncertainties. In Campos and Marques (2002), safety metrics are proposed for aircraft separation based on Gaussian and Laplacian distributions, as well as a generalized exponential family of probability distributions stemming from a mixture of the Gaussian and Laplacian distributions. All the probability-based collision prediction methods mentioned above assume the root mean squares position error to be identical for all the aircraft involved in the conflict situation. Meanwhile, since the root mean squares position error may be the consequence of atmospheric disturbances or flight path deviations related to airplane stability, Campos (2001) analyzed the case of dissimilar position errors. As to global collision avoidance, Menon and Sweriduk (1999) state the global collision avoidance problem as a trajectory optimization problem. Their work assumes tacitly all the aircraft subject to separation in a given control area to be initially known, and the nominal trajectory to be prespecified for each of them. The avoidance strategy consists in controlling each aircraft along its nominal trajectory minimizing deviations such that collision may be avoided. The avoidance trajectories are computed off-line, therefore, that approach cannot be applied to cases when unforeseen aircraft may enter the control area. In Bousson (2001), the global avoidance problem in terminal areas is dealt with as an online parameter optimization problem (Bousson and Correia, 2006) which accounts for priority between aircraft, as well as for the variability of the number of aircraft in the considered terminal area, contrarily to the work by Menon and Sweriduk quoted above. As mentioned above, most of the existing research activities in collision avoidance are mainly based on pairwise considerations in which the collision avoidance problem is dealt with considering pairs of aircraft in a given area. That approach is clearly inefficient for high-traffic densities since the resolution advisories take into account only the local traffic, but not the overall traffic in the area. Indeed, the local resolution of conflicts may induce other conflicts soon after since the global traffic has not been accounted for. The present work deals with the global collision avoidance problem in the horizontal plane in the framework of free-flight and builds on previous work by Bousson (2003, 2001) that rather focused on terminal area collision avoidance problems. The main contribution of the proposed method is to devise

collision avoidance model for free-flight based on simultaneous separation of all the aircraft in a given control area, using model predictive control concepts (Mayne and Michalska, 1990), which means that the collision avoidance process will rather take into account the global traffic in the area instead of considering aircraft only by pairs. One of the main interests of such a method is that the collision avoidance process in high-traffic density will be more effective for a much more flexible traffic flow.

2. Problem statement Let D be a specified air traffic control sector, and n(t) be the number of aircraft in D at time t. Assume IðtÞ ¼ {1; 2; . . . ; nðtÞ} to be the index set of all the aircraft in D at time t, and each aircraft to be identified by its indexing number i [ I(t). It comes from that assumption that the number of aircraft in the control sector may vary across time. Navigation requires to be given a set of specified waypoints for each aircraft. However, the flight control deals with the way to optimize some given criterion from a waypoint to the next. Therefore, collision avoidance copes with space-time trajectory optimization and control between two consecutive waypoints for each aircraft with respect to other possibly conflicting aircraft. Assume P f1 ; . . . ; P fnðtÞ to be the next waypoints, in sector D, of aircraft 1, . . ., n(t), respectively, at any given time t. Waypoints are checkpoints from the standpoint of navigation. As such, any such a point can be considered as a temporary destination point for the corresponding aircraft. The problem to be solved is that of driving each aircraft i from its current position to its next waypoint P fi such that its trajectory up to P fi remains fully in sector D and that the aircraft does not collide with any other aircraft.

3. Global collision avoidance modelling Classical modelling formalisms for tackling the collision avoidance problem are based on pairwise considerations of aircraft (Clements, 2002), that is, if they are n aircraft in a given control area, the collision avoidance policy examines independently q ¼ nðn 2 1Þ=2 pairs of aircraft for potential conflict cases. This is a local collision avoidance that does not account for the global traffic. The problem is that, since pairs of aircraft are independently checked for conflict, the aircraft separation procedure may not be efficient in term of meeting flight path preferences in the sense that some of the aircraft may significantly deviate from their chosen waypoint track set in their flight plans, or such a pairwise strategy may even induce other conflicts. We may think of a collision avoidance policy in which the separation strategy takes simultaneously into account all the aircraft in a given air traffic control sector and elaborates a global guidance which is efficient in the sense that any separation solution is void of potentially induced conflicts in the near-future and at the same time minimizes the flight path from one waypoint to the next. We propose such a global collision avoidance model in the current section, and an algorithm to deal with it in the next. 3.1 Flight model As far as horizontal collision avoidance is concerned, the aircraft are assumed to fly at the same flight altitude. Therefore, we consider the navigation equations in the 606



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horizontal plane. In that case, the flight model for each aircraft i may then be modelled as follows:

y ik ; yi ðt k Þ, the subscript f denotes for the final (terminal) value of the corresponding variable. Then, we define the priority index of aircraft i at tk as:

x_ i ¼ vi cos xi ; y_ i ¼ vi sin xi

v_ i ¼ ai ; x_i ¼ bi

ð1Þ

d 22 lik ¼ 100 Pnkik 22 j¼1 d jk

where xi and yi are, respectively, the longitudinal and lateral displacements of aircraft i with respect to the Cartesian coordinate system tied to the earth. Variables vi and xi are, respectively, the aerodynamic speed and heading angle of aircraft i, the heading angle being measured counter clockwise from the x-axis. Quantities ai and bi are the speed and heading rates, respectively. Owing to structural limitations, to actuator saturations and to the fact that each aircraft should remain in the same horizontal plane, bound constraints may be imposed on the speeds, the speed rates and the heading rates: 0 , vimin # vi # vimax ;

jai j # aimax ;

jbi j # bimax :

2 with: d 22 ik ¼ 1=d ik . It can be noticed that the closer an aircraft is to its destination point, the higher is its priority index.

3.4 Model predictive control formulation Two main problems have to be solved here: the collision avoidance problem that consists in guaranteeing minimum safety distance between pairs of aircraft, and the navigation problem that consists in driving each aircraft to its destination point. Satisfying equation (6) solves the collision avoidance problem, which is not enough. To solve the navigation problem, one has to continuously minimize the distance between each aircraft to its destination point, on a sequence of contiguous time intervals of pre-specified constant length t. That means the controls have to guarantee collision-free navigation and at the same time drive each aircraft to its destination point. The strategy to deal with both collision avoidance and navigation to specified destinations must be predictive so that potential near future collision occurrences can be foreseen and prevented properly. Such a strategy may be cast into the framework of predictive control (Mayne and Michalska, 1990). Indeed, predictive control allows one to obtain online feedback controls that drive a system from a specified state to a final one while minimizing a given cost function J over a given prediction horizon of length T, eventually subject to constraints on the state and control vectors. Consider a sequence of sampling time-instants t 0 , t 1 , · · · , t n , · · ·, with t kþ1 ¼ t k þ t for k ¼ 1, 2, . . . and 0 , t , T. In fact, the prediction horizon T is much larger than t. Then, the model predictive control algorithm is as follows: 1 set initially k ¼ 0; 2 measure the current state Xk of the system to be controlled; 3 find an open-loop control U^ that minimizes the cost function J on the current time-interval ½t k ; t k þ T ; where tk is the current time; 4 apply the optimal control U^ to the system on subinterval ½t k ; t k þ t; and 5 set k ¼ k þ 1, and repeat the procedure from Step 1 above.

ð2Þ

In the sequel, we denote, respectively, the position Pi and the control vector Ui of aircraft i by: P i ¼ ½xi yi T U i ¼ ½ai bi T :

ð3Þ

The initial conditions of each aircraft are assumed to be known as: P0i ¼ P i t 0i ð4Þ where: t 0i is the time from which aircraft i initiates its flight in the corresponding area. 3.2 Safety net It is natural that each aircraft should keep a safety distance from any other aircraft for the sake of collision avoidance. A circular safety envelope of radius d . 0, centred at the geometrical centre of each aircraft, may be used as the safety net for aircraft separation, and a proximity measure dij between two different aircraft i and j may be defined as: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dij ¼ Dx2ij þ Dy2ij 2 d; i – j ð5Þ where Dxij ¼ xi 2 xj , and Dyij ¼ yi 2 yj . As may be seen, dij is non-negative when aircraft i (respectively aircraft j) is outside the safety envelope of aircraft j (respectively aircraft i), and negative otherwise. Therefore, the following inequality expresses the non-intrusion constraint that should be met to avoid collision:

dij . 0; for each i; j [ IðtÞ with i , j:

ð7Þ

ð6Þ

Therefore, taking the total distance to go for all the aircraft as a cost function to be minimized, and equation (1) and inequalities (6) as constraints, one can guess that the collision avoidance problem may be formulated as a model predictive control as done in the sequel. One of the most interesting aspects of model predictive control is that it is actually a feedback control method since the control law is dependent on the state of the system. As to the formulation of the collision avoidance problem in the setting of model predictive control, the controls may be computed on p consecutive temporal intervals ( p $ 2) of ð1Þ equal length t from time tk, that is, controls U i ; ð2Þ ð pÞ U i ; . . . ; U i are computed on intervals:

3.3 Priority index In this section and in the next, the collision avoidance problem will be analyzed according to a discrete time concept, to enable the implementation of the method that will be presented. Consider S ¼ {1; 2; . . . ; nk } the set (or the indexing set) of the aircraft in sector D at a given discrete time tk, where nk ; n(tk). Let: 2 2 d 2ik ; d 2i ðt k Þ ¼ x ik 2 x fi þ y ik 2 y fi be the distance between the position of aircraft i at time tk and its destination point (next waypoint), with x ik ; xi ðt k Þ;

½t k ; t k þ t ; ½t k ; t k þ 2t ; . . . ; ½t k ; t k þ pt 607



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Finally, the collision avoidance problem can be formulated as the following constrained optimization problem (P) at each discrete time tk. Problem (P): . minimize function J defined in equation (11); . subject to constraints defined in equations (2) and (6) at sampling time-instants t k þ t; t k þ 2t; . . . ; t k þ pt; and . where ðx*ik y*ik Þ is the actual state vector (here the planar position) of aircraft i at time tk.

respectively for each aircraft i, then the control Ui is applied to aircraft i on interval ½t k ; t k þ t (let us notice that the ð jÞ ð jÞ ð jÞ components of each vector Ui are vi and xi ). From ð1Þ time t kþ1 ¼ t k þ t, another set of p controls Ui ; ð2Þ ð pÞ Ui ; . . . ; Ui are computed for each aircraft, and the newly ð1Þ computed control Ui is applied to aircraft i on interval ½t kþ1 ; t kþ1 þ t , and so on . . . Driving each aircraft to its destination point requires that the distance to go be minimized at each step mainly when there is no obstacle on the way. This fact suggests the following cost function to be minimized at each discrete time tk: ð1Þ ð pÞ ð pÞ J U1 ; . . . ; U1 ; . . . ; Uð1Þ nk ; . . . ; Unk ¼

p X nk X

P i ðt k þ stÞ 2 P fi

T

Q ik P i ðt k þ stÞ 2 P fi :

4. Global collision avoidance algorithm The general algorithm which is proposed for the global collision avoidance is stated hereafter: 1 Initialization: k :¼ 0; t k :¼ 0; let S be the set of aircraft in control sector D, and Pi,k be the initial state vector of each aircraft i. Set a control period t and a value p for the control horizon. Set ^tk ¼ t k 2 1 , t k , (1 . 0). 2 Repeat until set S is empty: . at time ^tk , compute the solution: ð1Þ ð pÞ ð pÞ U1 ; . . . ; U1 ; . . . ; Uð1Þ nk ; . . . ; Unk

ð8Þ

s¼1 i¼1

The subscript i denotes aircraft I, matrix Q ik is positive definite, vector Pi(tk þ st) is the prediction of the position of aircraft i at time t k þ st; ðs ¼ 1; . . . ; pÞ, P fi is its destination ð jÞ point. Control vectors Ui are unknown constant arguments of the objective function J that has to be minimized. In the present paper, Q ik is taken as a diagonal matrix with positive diagonal entries equal to the priority index of aircraft i at time tk. Therefore: " # lik 0 : ð9Þ Q ik ¼ 0 lik

of the optimization problem (P) defined above as the solution of the collision avoidance problem on temporal intervals: ½t k ; t k þ t ; ½t k ; t k þ 2t ; . . . ; ½t k ; t k þ pt ; .

.

It has been mentioned earlier that the closer an aircraft to its destination point, the higher its priority index lik. Indeed, to drive each aircraft to its destination point, it is natural to give higher priorities to aircraft which are closer to their destination points than the others which are farther. Owing to equation (9), the cost function in equation (8) may be written as: ð1Þ ð pÞ ð pÞ J U1 ; . . . ; U1 ; . . . ; Uð1Þ nk ; . . . ; Unk ¼

p X nk X s¼1 i¼1

2 li;k P i ðt k þ stÞ 2 P fi

.

.

While the controls are assumed to be valid for a time horizon of length pt ( p $ 2) as computed in Step 2.1, they are updated, for safety reason, according to a time period of length t, and so is the set S of aircraft in sector D (Steps 2.4 and 2.5). The actual control horizon is t, whilst T ¼ pt is the prediction horizon. The actual control is updated according to a period less lengthy than the prediction horizon T so that new control can be computed before the prediction horizon elapses. Controls are computed at time ^tk ¼ t k þ r · t , t k þ t ¼ t kþ1 so that they can be already available and used on the following control time period starting at tkþ 1. Updating the set S requires to list the incoming aircraft as well in sector D and to unlist the aircraft which have already reached their destination points in D.

ð10Þ

2

where k · k2 denotes the Euclidean norm. Using the Euler integration scheme, we get explicitly the expression of the cost function: ð1Þ ð pÞ ð pÞ J U1 ; . . . ; U1 ; . . . ; Uð1Þ nk ; . . . ; Unk ( !2 p nk X X ðsÞ ðsÞ f * lik xik þ t · vik cos xik 2 x i ¼ ð11Þ s¼1 i¼1 !2 ) p X ðsÞ ðsÞ vik sin xik 2 y fi þ y*ik þ t ·

5. Application example The conflict scenario was generated by including four aircraft numbered from 1 to 4. To validate the method presented above, we assume that no other aircraft comes in the control area during the conflict avoidance procedure.

s¼1

with: ðsÞ

ðs21Þ

vik ¼ vik

ðs21Þ

þ taik ðsÞ

;

ðsÞ

ðs21Þ

xik ¼ xik

ðs21Þ

þ tbik

ð1Þ

apply the control Ui to aircraft i on time interval ½t k ; t k þ t ; set ^tk ¼ t k þ r · t, with 0.5 , r , 1, k: ¼ k þ 1, and tk ¼ tk2 1 þ t ; list in S any aircraft i in control area D such that rij # 2d for some other aircraft j, and unlist any aircraft i for which rij . 2d for any other aircraft j; and update the set S of aircraft in control area D at ^tk .

ð12Þ

5.1 Data The coordinate positions xi and yi are in nautical mile, the velocities in knot, the headings in degree. The initial and terminal conditions for the simulation are, respectively, in Tables I and II. The speed constraints are defined in Table III.

ðs21Þ

where the speeds vik , the speed rates aik , and the heading ðs21Þ rates bik relate to aircraft i at time tk and must satisfy (equation (2)). 608



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Figure 1 Collision-free trajectories

Table I Initial conditions

1 2 3 4

yi (nm)

Speed (knot)

Heading (degree)

30 100 300 350

100 400 150 460

257 262 210 220

25 285 130 235

500 B

450

(4)

C

400 350 y (nm)

Aircraft Aircraft Aircraft Aircraft

x i (nm)

(2)

300 250

(3)

200 150


1 2 3 4

xf (nm)

yf (nm)

350 300 100 30

460 150 400 100

50

1

2 3 4

1 2 3 4

vimin (knot)

vimax (knot)

200 200 200 200

290 290 290 290

(1)

A 0

50

100

150

200 250 x (nm)

300

350

400

Figure 2 shows the speeds of the four aircraft. It can be noticed that Aircrafts 1 and 2 keep their speeds constant, but Aircrafts 3 and 4 vary theirs to meet the minimal separation constraints. This is explained by the fact that higher priorities are given to aircraft that are closer to their arrival points, and aircraft with higher priorities perform less manoeuvres than aircraft with lower priorities. Indeed, since the initial speeds of Aircrafts 1 and 2 are greater than those of Aircrafts 4 and 3, respectively, the priority index of Aircraft 1 is higher than that of Aircraft 4, and so is the priority index of Aircraft 2 compared with that of Aircraft 3. Therefore, Aircrafts 1 and 2 maintain their speeds, whereas Aircrafts 4 and 3 change theirs to meet the separation constraints. Figure 3 shows the heading of the four aircraft. Here, only the heading of Aircraft 3 remains constant. Finally, Figure 4 shows the proximity measures (dij defined in equation (5)) of all the pairs of aircraft. As can be seen, all the proximity measures are positive, which means that the minimum separation distance d is fulfilled for any pair of aircraft.

Table III Speed constraints


D

100

Table II Terminal conditions

Rates constraints for all the aircraft we consider the speed rates ai [ [2 2, 2] (m/s2), and the heading rates bi [ [215, 15] degree/s. Minimum safety distance: d ¼ 5 nautical miles. Prediction horizon: T ¼ 120 s. Control horizon: t ¼ 10 s, and r ¼ 0.9.

One may deduce from the above values of the prediction and control horizons that the number of time steps within a prediction horizon is p ¼ 12.

6. Conclusion 5.2 Simulation results The choice of the initial and terminal conditions as stated in Tables I and II, respectively, is such that Aircraft 1 is in conflict with Aircraft 4, and Aircraft 2 conflicts with Aircraft 3. This is so because the departing point of Aircraft 1 is the arrival point of Aircraft 4, and vice-versa; and the departing point of Aircraft 2 is the arrival point of Aircraft 3, and viceversa. The objective of the simulation is to validate the collision avoidance method presented earlier. Indeed, Figure 1 shows the trajectories of the four aircraft. The trajectories are labelled by the corresponding aircraft numberings, that is, number 1 (in Figure 1) denotes the trajectory of Aircraft 1, number 2 denotes the trajectory of Aircraft 2, and so on . . . As can be seen the trajectories of Aircrafts 1 and 4 do not meet, and nor do the trajectories of Aircrafts 2 and 3. Trajectory separations are performed for all four aircraft as well, ensuring a collision free navigation.

The present paper copes with global collision avoidance in the horizontal plane in the context of free-flight. A model and an effective algorithm for the collision avoidance problem are presented from the standpoint of model predictive control. The method accounts for automatic assignment of priority indexes which are updated from one control time horizon to the next. The model and the algorithm that are devised for the purpose are simple and well suited for online processing requirements in real world applications. A simulation example is presented to demonstrate the suitability of the proposed approach. Future research work will deal with the extension of the model presented in this paper so that it can be used for 3D global collision avoidance involving trajectory prediction uncertainties due to atmospheric disturbances and measurement errors. 609



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Figure 2 Aircraft speeds 262 Speed of aircraft 2 (knot)

Speed of aircraft 1 (knot)

258

256

254

252

250

0

2,000

4,000

6,000

256 254 252 0

1,000

2,000

3,000

4,000

5,000

0

2,000

4,000 6,000 Time (s)

8,000

10,000

222.5 Speed of aircraft 4 (knot)

Speed of aircraft 3 (knot)

258

250

8,000

212.5 212 211.5 211 210.5 210 209.5

260

0

2,000

4,000

222 221.5 221 220.5 220 219.5

6,000

Time (s)

140

Heading of aircraft 2 (degree)


80


Figure 3 Aircraft headings

70 60 50 40 30 0

2,000

4,000

6,000

8,000 Heading of aircraft 4 (degree)

20

120 100 80 60 40 20

0

2,000

4,000

6,000

Time (s)

610

350 340 330 320 310 300 290 280

0

1,000

2,000

3,000

4,000

5,000

0

2,000

4,000 6,000 Time (s)

8,000

10,000

340 320 300 280 260 240 220 200



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Figure 4 Proximity measures 300

300

δ13 (nm)

δ12 (nm)

400

200 100 0

0

1,000

2,000

3,000

4,000

δ14 (nm)

300 200 100

2,000

4,000

6,000

0

1,000

2,000

3,000

4,000

400 200 0

5,000

300

400

250

300

200 150 100

0

600

δ34 (nm)

δ23 (nm) δ24 (nm)

100 0

5,000

400

0

200

0

1,000

2,000 3,000 Time (s)

4,000

5,000

References

0

2,000

4,000

6,000

8,000

200 100 0

0

2,000 4,000 Time (s)

6,000

Clements, J.C. (2002), “Optimal simultaneous pairwise conflict resolution maneuvers in air traffic management”, Journal of Guidance, Control, and Dynamics, Vol. 25 No. 4, pp. 815-8. Davis, T.J., Erzberger, H., Green, S.M. and Nedell, W. (1991), “Design and evaluation of an air traffic control final approach spacing tool”, Journal of Guidance, Control, and Dynamics, Vol. 14 No. 4, pp. 848-54. Debbache, N.E. (2001), “Toward a new organization for air traffic control”, Aircraft Engineering & Aerospace Technology, Vol. 73 No. 6, pp. 561-7. Durand, N., Alliot, J-M. and Chansou, O. (1995), “Optimal resolution of en route conflicts”, Air Traffic Control Quarterly, Vol. 3 No. 3, pp. 139-61. Erzberger, H., Davis, T.J. and Green, S. (1993), “Design of Center-TRACON automation system”, AGARD CP-538, pp. 11.1-11.12. Hu, J., Prandini, M. and Sastry, S. (2000), “Optimal maneuver for multiple aircraft conflict resolution. A braid point of view”, Proceedings of the 39th IEEE Conference on Decision and Control, Vol. 4, IEEE Publ., Piscataway, NJ, pp. 4164-9. Hu, J., Prandini, M. and Sastry, S. (2002), “Optimal coordinated maneuvers for three-dimensional aircraft conflict resolution”, Journal of Guidance, Control, and Dynamics, Vol. 25 No. 5, pp. 888-900. Mayne, D.Q. and Michalska, H. (1990), “Receding horizon control of nonlinear systems”, IEEE Transactions on Automatic Control, Vol. 35, pp. 814-24. Menon, P.K. and Sweriduk, G.D. (1999), “Optimal strategies for free-flight air traffic conflict resolution”, Journal of Guidance, Control and Dynamics, Vol. 22 No. 2, pp. 202-11.

Alcabin, M. and Erzberger, H. (1985), “Simulation of timecontrol procedures for terminal area flow management”, Proceedings of the American Control Conference, Boston, USA. Bousson, K. (2001), “Automatic guidance of aircraft for collision avoidance in terminal areas”, Russian-American Journal of Actual Problems of Aviation and Aerospace Systems, Vol. 6 Nos 2/12, pp. 49-57. Bousson, K. (2003), “Waypoint-constrained free-flight collision avoidance”, paper No. 2003-01-2996, Proceedings of the SAE Advances in Aviation Safety Conference (ACE), Montreál, Que´bec, Canada. Bousson, K. and Correia, S.D. (2006), “Optimization algorithm based on densification and dynamic canonical descent”, Journal of Computational and Applied Mathematics, Vol. 191, pp. 269-79. Brooker, P. (2005a), “STCA, TCAS, airproxes and collision risk”, The Journal of Navigation, Vol. 58, pp. 389-404. Brooker, P. (2005b), “Airborne collision avoidance systems and air traffic management safety”, The Journal of Navigation, Vol. 58, pp. 1-16. Campos, L.M.B.C. (2001), “Probability of collision of aircraft with dissimilar position errors”, Journal of Aircraft, Vol. 38 No. 4, pp. 593-9. Campos, L.M.B.C. and Marques, J.M.G. (2002), “On safety metrics related to aircraft separation”, The Journal of Navigation, Vol. 55, pp. 39-63. Cavcar, A. and Cavcar, M. (2004), “Impact of aircraft performance differences on fuel consumption of aircraft in air traffic management environment”, Aircraft Engineering & Aerospace Technology, Vol. 76 No. 5, pp. 502-15. 611



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Volume 80 · Number 6 · 2008 · 605 –612

About the author

Paielli, R.A. and Erzberger, H. (1999), “Conflict probability estimation generalized to non-level flight”, Air Traffic Control Quarterly, Vol. 7 No. 3, pp. 195-222. Tomlin, C., Pappas George, J. and Sastry, S. (1998), “Conflict resolution for air traffic management: a case study in multi-agent hybrid systems”, IEEE Transactions on Automatic Control, Vol. 43 No. 6, pp. 509-21. Yang, L.C. and Kuchar, J.K. (1997), “Prototype conflict alerting system for free flight”, Journal of Guidance, Control and Dynamics, Vol. 20 No. 4, pp. 768-73.

K. Bousson is a Professor in the Department of Aerospace Sciences at the University of Beira Interior in Covilha˜, Portugal, since February 1995. Formerly, he was a Researcher at the LAAS Laboratory of the French National Council for Scientific Research (CNRS) in Toulouse, France, from 1993 to 1995. His current research activities concern flight dynamics and control systems, trajectory optimization and control, navigation guidance, and state/parameter estimation. K. Bousson can be contacted at: [email protected]

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