Automatic Collision Avoidance System based on Geometric ... - ICRAT

Automatic Collision Avoidance System based on Geometric Approach applied to Multiple Aircraft Paulo Machado

Kouamana Bousson

Department of Aerospace Sciences University of Beira Interior [email protected]

Department of Aerospace Sciences University of Beira Interior [email protected]

Abstract—A generalization of collision cone concept applied to real-time Collision Avoidance System (CAS) is proposed. Mostly due to the current trend of a wide range of applications which use unmanned air vehicles (UAV), the Collision Avoidance Systems take an important role either on airborne systems or even Air Traffic Collision Systems (ATC). Thus, a possible CAS shall lead with several aircraft even when solutions are required on short time period. Hence, the present work aims add to geometric resolution approach a generalized formulation as well as propose an algorithmic solution suitable for real-time implementations. The computational results involving up to ten aircraft on worst case scenario reveal the method effectiveness.

I. I NTRODUCTION Flight collision avoidance is a problem that adversely and significantly impacts major aspects of air traffic control, particularly in congestion-prone airspace. Therefore, many research activities have focused during the recent decades on making a conscious effort toward dealing with collision avoidance in terminal areas as well as in en-route control air spaces. It is encouraging that advanced air traffic management systems based on the Secondary Surveillance Radar (SSR) and the Automatic Dependent Surveillance-Broadcast (ADSB) are enabling more efficient use of existing air traffic flow network systems resulting in reduced traffic congestion, delays, fuel consumption and improved air safety. Furthermore, the SESAR Program that is underway will combine technological, economic and regulatory aspects and will use the Single European Sky (SES) legislation to synchronize the plans and actions of the different stakeholders and federate resources for the development and implementation of the required improvements throughout Europe mostly in airborne systems. Hence, a problem which unconditionally arise is how to avoid efficiently flight collisions. Today, that task is assigned mainly to air traffic controllers which based or not on computer systems try to avoid collisions. Of course, the trajectories chosen by air traffic controllers or even the computer aids are based on well established regulamentation. Several efforts and researches have been done by entities, organizations and academic institutions to enable collision-free air traffic giving rise to two groups of approaches about air collision avoidance: the local and the global separation approaches. The local approach [1] consists in separating aircraft based on the air traffic trends in the neighborhood of a given aircraft, that is,

aircraft are separated by pairs; that approach is clearly not efficient in highly congested areas since it does not account for more than two aircraft at the same time. The global approach [2], [3] aims at separating simultaneously more than two aircraft taking into consideration the overall traffic trends in a given control area so that the separation of an aircraft A from an aircraft B may not induce a conflict between aircraft A and another aircraft C. In the two main referred divisions, the collision avoidance problem falls into global or local approach, but that division does not solve the problem itself. Hence, the actual researches activities led to resolution methods which can be divided in three main categories namely, Probabilistic, Geometric and Optimization approaches. Studies based on conflict probability [4], [5] has been developed to find safety metrics to aircraft separation [6]. In those approaches there is not a direct solution which imply in the most cases the aid of alert systems [7], [8] using heuristics and procedures [9], [10] to solve the collision problem. The geometric approach is a technique widely used on collision avoidance problems between two aircraft [11], [12] and due to its intuitive formulation the resolution becomes easy to compute, further it also is used on path planning to avoid collisions with obstacles [13]. One of the must historical approaches belong to geometric formulation is the Proportional Navigation [14], [15], where the evasion aircraft is subject to a velocity variation proportional to Line-OfSide (LOS) angle. Another approach belong to geometric ones are those which make use of collision cone concepts [16], [17]. Collision Cone concept can be understood as a generalization of collision avoidance problem where evasion velocities vector are constant at a specific interval. Other technique that differs from Collision Cone techniques is the Dubins Paths [13], where a combination of line segments and constant-curvature segments allows built an evasion trajectory. Those main sets of techniques inside geometric approaches, after a suitable formulation, could be solved analytically if possible, or submitted to an optimization process if the closed form solution is not available. The last one, the Optimization approaches [2] are a more general point of view of collision avoidance problem. Both Probabilistic and Geometric approaches can at some point to be submitted to an optimization process or on other hand the

problem can be formulated from the beginning with the idea of optimize something. Hereupon, the idea developed in the present paper relies on the global approach applied to a multiple aircraft collision avoidance system (ACAS). The proposed approach uses a geometric concept stemming from cone-based assessment of collision risks. The underlying concept can be understood as a generalization of collision avoidance problem where evasion velocity vectors are kept constant during a specific timeperiod and pointing away from the collision cone, as present on previous study [18]. Beside that, it will be developed a method aimed for solve the dynamic constraints of aircraft, the idea is to use the predictive control philosophy [19] and a discretization technique based on pseudospectral methods [20] for ensure all aircraft fulfill the solutions found by the present method. The identification of a pattern on the generalized formulation has been implemented for preliminary simulation tests applied to UAVs that have revealed promising abilities of the method to handle simultaneously in real-time and successfully up to ten aircraft evolving in the Portuguese airspace. II. P ROBLEM S TATEMENT The geometric approaches applied to collision avoidance problem have been applied mostly to local avoidance scenarios on the other hand approaches based on Optimization techniques are used for global avoidance problems. Despite the use of geometric approaches on local collision avoidance problems those can be reduced to optimization problems and applied to global avoidance problems. The main problem of optimization techniques is essentially the computational time consumption which in general affects the feasibility at real scenarios. It is well known that geometric approaches have a simple formalism which normally relies at fast computations. Thus, the intention is start with the collision cone concept for local avoidance problems and augment its geometric expressiveness for multiple aircraft. However, there are studies which the desired expressiveness of mathematical modulation can be found using other approaches, indeed sometimes they are more expressive or more elegant. Therefore, it is also desired an associated computational method where the solutions are computed at efficient manner. So, in this work is developed an algorithmic solution which gives to the proposed approach the necessary computational efficiency to lead with global collision avoidance problems at real scenarios. III. C OLLISION C ONE C ONCEPT The formulation of Collision Cone concept it will now be built, first for two aircraft and expanded for multiple aircraft. It is also important to refer that the following concepts were developed based on some hypothesis. The first of them, we consider that each aircraft has a well defined priority, which implies that computed evasion maneuver must be took by aircraft with low priority. The second one, consider that during the evasion maneuver, aircraft with high priority do not change their velocities vectors.

A. Two Aircraft Formulation The main idea behind the collision avoidance problem is to avoid the distance between aircraft became dangerous for both. In that way, considering the case of two aircraft represented by points P0 and P1 with velocities V0 and V1 respectively, the distance between them is represented by the vector ~r. Then for collision avoidance problem the main requirement to be fulfill is, k~rk2 ≥ ∆

(1)

where ∆ is a value, which independently of aircraft size they never collide (on a general sense). The equation (1) per se does not solve the problem, indeed with only that equation the problem can not be formulated. Hence, it was considered aircraft with different priorities and assumed relative motions. Now, it will be introduced some definitions for a best understanding of next steps. Based in figure (1) the definition (1) comes naturally as well as the definition (2), they intend to clarify the starting point, i.e, the basic case of two aircraft interaction whose are abstracted at two points of mass. Definition 1 (Object to Object System). Consider the referential OXY Z where a point of mass with position P0 and velocity V0 coincide with origin of that referential. Suppose also there is other point of mass with position P1 , relatively to referential OXY Z with velocity V1 . The system formed by that two points of mass in the referred configuration, separated by the distance k~rk, with respective velocities, is called Object to Object System. Definition 2 (Line of Sight). The Line formed by the two objects on an Object to Object System, with distance k~rk, is called Line of Sight. Definition 3 (Collision). It is considered that two points of mass of an Object to Object System are in collision if k~rk < k~rCAD k, where k~rCAD k is the safety distance. As referred above the CAS is based on an idealized priority system, in current formulation that means that aircraft with the lowest priority has the positioning in the origin of referential OXY Z of Object to Object System. Defined the base system, the next step is to define more clearly the notion of collision on tree dimensions as well as the concept of Collision Cone, which will be an important concept to figure out the solution of a possible collision. The figure (1) represents geometrically these concepts which are translated for a mathematical formalism by means of definitions (3), (4) and (5). Definition 4 (Safety Sphere). The sphere built on Object to Object System where P1 is its center and the safety distance radius k~rCAD k, is called Safety Sphere. Definition 5 (Collision Cone). At Object to Object System the infinite cone with apex coincident with point of mass P0 and the straight lines tangent to a sphere with radius k~rCAD k,

Z

Px1 = Px01 + Vx1 t Py1 = Py01 + Vy1 t Pz1 = ⃗1 V

⃗rCAD P1

O. P0

+ Vz1 t

||~r||2 = ||~rCAD ||2

δCAD ⃗01 V

δ

(3)

where (2) and (3) shown the movement of P0 and P1 points after time t0 . ~ || = Now, it will be necessary to find the relative velocity ||V ~ ~ ~ ||V01 || = ||V0 − V1 || that system needs for attained,

.

⃗r

⃗0 V

Pz01

(4)

on time t. Known ||~r||2 = (Px1 −Px0 )2 +(Py1 −Py0 )2 +(Pz1 −Pz0 )2 , then,

β Y

||~r||2 = rx2 − 2trx Vx + Vx2 t2

α

+ ry2 − 2try Vy + Vy2 t2 + rz2 − 2trz Vz + Vz2 t2

X

= r2 − 2trV cos δ + V 2 t2 Fig. 1. Collision Cone

The right term of (4), can be switched by r sin δCAD . So, the equation (4) follows,

where k~rk > k~rCAD k, and center at the point of mass P1 is called the Collision Cone (figure (1)). Gathered the start conditions follows the exposition of some important laws. The lemma (1), which is the basis of theorem (1) shows, under some assumptions, the condition of noncollision. Since that assertions have been proved, by mean of theorem (1) it was tried to show the set of solutions of collision avoidance problem between two aircraft. It is also added an important conclusion to lemma (1) which shows the condition of interception between aircraft. ~01 the relative velocity vector between Lemma 1. Let’s be V the two points of mass of an Object to Object System with constant velocities during a time interval ∆t. For a time t0 ~01 where k~r(t0 )k > k~rCAD k, if the angle δ formed by vector V and the line of sight ~r is greater than the half aperture angle δCAD of a collision cone, then ∃∆t > 0 : ∀t ∈ [t0 , t0 + ∆t] , k~r(t)k > k~rCAD k. Proof. Let us consider an Object to Object System, where velocities of each point of mass are constant between a time interval ∆t. So, it can easily be realized the position of points of mass on time t = t0 + ∆t, where t0 is the initial time. Hence, the R3 positions can be be defined on the follow set of equations,

Px0 = Px00 + Vx0 t Py0 = Py00 + Vy0 t Pz 0 =

Pz00

+ Vz0 t

(5)

(2)

r2 − 2trV cos δ + V 2 t2 = r2 sin2 δCAD r  r 2 cos δV + V 2 = 0 cos2 δCAD − 2 ⇔ t t Solving in order to V we have,  p r V = cos δ ± cos2 δ − cos2 δCAD t where stay true if only if δ ≤ δCAD .

(6)

(7)

Theorem 1 (Set of Solutions). Consider an Object to Object System, where the point of mass P1 has a sphere of collision centered in it of radius k~rCAD k and k~rk > k~rCAD k. If the point of mass P1 has constant velocity, then the set of velocity ~ of point of mass P0 that produces a vector variations ∆V non-interception condition with P1 is given by, ∗ ∗ ~ ∈ R3 | V ~01 ~01 + ∆V ~ ; arccos(Vˆ01 Γ = {∆V =V · rˆ) > δCAD }

~01 is the relative velocity vector and δCAD is the where V ∗ half aperture angle of collision cone and Vˆ01 , rˆ are the normalized vectors. B. Collision Avoidance Problem Formulation So far, it was possible to develop a simple geometry that translates, by mean of intuitive manner, the collision avoidance problem between two aircraft into a mathematical formulation, also it was identified the set of solutions for the problem with two aircraft. Now, it will be tried to expand it to a number N ≥ 3 of aircraft. The idea is to expand the Object to Object System for an Object to Multi Object System (figure (2)) and try to find a set of solutions for the generalized collision avoidance problem.

Z

The theorem (2) assert where the solution can be found for the problem based on definition (6). Definition 6 (Object to Multi Object System). Considering the referential OXY Z where a point of mass with position P0 and velocity V0 coincides with origin of that referential. Suppose also there are others point of mass with position Pi , relatively to referential OXY Z, and velocity Vi , where i = 1, ..., N − 1 with N as the number of points of mass. The system formed by that N points of mass in the referred configuration, separated by the distance k~ri k, with respective velocities, is called Object to Multi Object System (figure (2)).

⃗rCAD1 ⃗1 V

.

δCAD1

P2

⃗0 V

⃗r2 δCAD2 δ2 β2

Theorem 2 (Generalized Set Of Solution). At Object to MultiObject System, the point of mass Pi has a sphere of collision centered in it of radius k~rCADi k and k~ri k > k~rCADi k. If the points of mass Pi have constant velocities, then the set of ~ of the point of mass P0 which variation velocity vector ∆V produces a non-interception with Pi is given by,

⃗r1

⃗2 V

⃗rCAD2

.P1

⃗02 V

δ1 ⃗01 V

O

β1

. P0

Y

α1

α2

X

Fig. 2. Collision Cones On Object Multi Object System

~ ∈ R3 | V ~0i∗ = V ~0i + ∆V ~ ; arccos(Vˆ0i∗ · rˆi ) > δCAD } Γ = {∆V i where i = 1, ..., N − 1 with N as the number of points of ∗ mass and Vˆ01 , rˆ are normalized vectors. ~0i = V ~0 − V ~i , then the velocity vector Proof. By definition V ∗ ~ V0 that yields a non-collision condition with Pi is given by, ~0∗ = V ~ 0 + ∆V ~ V

(8)

~ ∗ gets the next form, Thus, the new relative velocity vector V 0 ~0i∗ = V ~0∗ − V ~i V ~0i∗ = V ~ 0 + ∆V ~ −V ~i ⇔ V ~0i∗ = V ~0i + ∆V ~ ⇔ V

(9)

then, the sufficient conditions that guarantee a noninterception of P0 with Pi is, δi = arccos(Vˆ0i∗ · rˆi ) > δCADi

(10)

The theorem (2) takes an important role in this study, because can be known exactly where the guidance vector must be searched. Hence, the collision avoidance problem can be stated as, minimize ~ ∈Γ⊂R3 ∆V

~ k2 k∆V

(11)

~ k2 k∆V

subject to

~min ≤ V ~ + ∆V ~ ≤V ~max V

~ ∈Γ⊂R3 ∆V

Theorem 3. Considering an Object to Multi Object System, where the point of mass Pi has a sphere of collision centered in it of radius k~rCADi k and k~ri k > k~rCADi k, with i = 1, . . . , N −1 and N is the number of points of mass. If points of mass Pi have a constant velocity, then the formulation which solves the collision avoidance problem can be stated as, minimize

~ k2 k∆V

subject to

~min ≤ V ~ + ∆V ~ ≤V ~max V

~ ∈Γ⊂R3 ∆V

(13)

which also solves the dynamic restrictions issues, if a dynamic solution exists. IV. P ROPOSED M ETHOD A. Problem Solution

Therefore, aircraft have velocity limitations which must be take in account when the guidance is chosen. The equation (11) is now reformulated and is added the velocity restriction, minimize

1) Dynamic Collision Avoidance Analysis: When the matter are aircraft flying, implicitly the theme is about dynamics and the above formulation shall take that into account. The dynamic behaviour is extremely important because, for instance, if in presence of an evasion guidance vector which should be attained in some period of time, the aircraft must be able to do that, i.e, the dynamic constraints of aircraft must be in accordance with the guidance vector solution. So, the thought that arise is – How introduces the dynamic behaviour on the formulation (12)? – The answer, indeed, it is simple, because the formulation (12) also solve the dynamic problem. The theorem (3) show how does it.

(12)

Formulated the collision avoidance problem, the next logical step will be solve it. Despite the simple formulation achieved above, its resolution can be more complicated. The trivial procedure would be submits the problem (12) to a solver or use the concepts of mathematical programming for try to solve it. But in current work it is propose a quasi

on the other hand, the collision cone equation can be represented as,

⃗1 V

.

P1

⃗2 V

Vˆ proj · rˆ = cos δCAD

⃗rCAD1

.P2

⃗r1

⃗rCAD2

eˆ ⃗r2

⃗0 V ⃗1 ⃗2 eˆ ∆V1 ∆V1

⃗02 V

⃗01 V

eˆ

 ˆ proj · n  ˆ=0 V Vˆ proj · rˆ = cos δCAD   ˆ proj 2 kV k =1

eˆ ⃗3 V ⃗r3

⃗rCAD3

~. where n ˆ is the normal vector produced by ~r and V Joining the equations (14), (15) and adding the restriction of unit vector, the system to be solve take the next form,

⃗2 ∆V

P0 . ⃗03 V

.P3

~ through direction ~e Fig. 3. Variation of Velocities Vector ∆V

analytical solution, or at least an simple algorithm. The next explanation will try to demonstrate it. The theorem (2) is the key of an attempt in to achieve a solution for the problem (12). If analyzing the theorem (2) it ~ is a variation that produces an alteration can be seen that ∆V in all relative velocity vectors of object/aircraft at the origin of the referential system. That change in the relative velocity vectors must be sufficient for those vectors leave the interior of the respective collision cone. ~ shifts all relative velocity In figure (3) can be seen that ∆V vectors in same direction. This fact is extremely important because if was chose a direction eˆ, it is possible to calculate ~ needed for the relative velocity vector get out of the the ∆V collision cone. So, the first task must be choose a random direction eˆ and ~0i through that direction, project the relative velocity vector V into the collision cone. The difference between the relative ~ vector, but how velocity vector and its projection is the ∆V the projection can be found? – The procedure is to construct a ~0i and the direction eˆ, after plane formed by relative vector V that it will be searched the interception direction of the plane with the collision cone and projecting the relative velocity vec~0i through eˆ into the corresponded interception direction. tor V ~0i and the The plane formed by relative velocity vector V direction eˆ that contain the velocity projections Vˆ proj is given by, Vˆ proj · n ˆ=0

(15)

(14)

(16)

where Vˆ proj is the projection or projections if a relative ~0i exists. velocity vector V Notice, that the system of equations may not have solutions, and that happens when the direction eˆ produces an angle of ~0i , or the zero or π radians with the relative velocity vector V ~ V0i is out, or in the revolution surface. Constructed and solved the system (16), it is necessary analyze the solutions because, the system is quadratic and is only necessary one solution. First of all, it is need to know ~0i remain inside or outside of if the relative velocity vector V collision cone. If inside (figure (4)) it has two solution. The ~0i , right solution can be identified by angles analysis between V eˆ and the solutions vectors. As it can be seen in figure (4) ~0i and eˆ it is known the supplementary angle θ1 formed by V vectors, θ2 and θ3 are the angles formed by eˆ and solution one and two respectively, lastly, θ4 and θ5 are the angles formed ~0i and solution one and two respectively. The solution is by V found when the sum of θ1 + θ2 + θ4 or θ1 + θ3 + θ5 is equal to π radians. ~0i The second case is when the relative velocity vector V is outside of the collision cone (figure (5)). In that case, both solutions are valid because the direction eˆ produce a projection ~0i into the two straight lines of cone surface but, it will of V ~ k which be choose the projection that produce the higher k∆V is always the projection in the opposite side of the collision cone. The algorithm (1), that try to apply the described concepts and assumes that there is a system which gathered the aircraft data and compute the evasion velocity vector for each them. The priority scheme it is not covered on current work, on algorithm (1) the priority is defined by index j, being 0 the must higher priority. The algorithm (1) try to show as shall be done the procedure ~ through one direction eˆ. Notice that the choice for to find ∆V ~ is the vector with the maximum norm because its norm for ∆V ~0i get out of respective collision cone. allows all vectors V ~ is based on one direction eˆ but it is The return vector ∆V ~ , then, the procedure should be need to find the minimum ∆V ~ k on a finite number o directions eˆ search the maximum k∆V and inside this set of possible variations select the minimum ~ k. k∆V

⃗rCADi

.

⃗1 V

Pi

⃗ri

eˆ

θ5

⃗ proj2 V 0i

⃗0 V

θ1

eˆ θ4

⃗ proj1 V 0i

⃗0i V θ3 θ2

P0 .

Fig. 4. Inside of Collision cone

⃗rCADi Pi

.

⃗i V

⃗ri

eˆ ⃗0i V

⃗ proj1 V 0i

⃗ proj2 V 0i

eˆ ⃗0 V

P0 .

Fig. 5. Outside of Collision Cone

V. P REDICTIVE C OLLISION AVOIDANCE Previously was addressed the problem of compute the solution for the generalized collision avoidance problem, if that exist. Therefore, the solution was found for an instantaneous time t which is the time when data were gathered. However, the computation of the solution spends some time until be available, during that time aircraft kept their trajectories (notice, the solution that was found is a velocity vector). So, in that moment the solution does not work anymore, because aircraft are not in the same place when data were gathered. For solve that issue, it will be necessary some kind of trajectory predictivity based on computed solution, and aircraft will not take a velocity vector but a four-dimensional (4D) waypoint, i.e, a three-dimensional position plus time. In that sense, it was developed a trajectory calculation method for solve the differential geodetic navigation equations based on pseudospectral technique [19] (due to the restricted

Algorithm 1 Centralized Collision Avoidance Algorithm ~i , i = 1 . . . N ; ~ej , j = 1 . . . M Input: ~ri , V ~k , k = 2 . . . N Output: ∆V 1: N : Number of Aircraft 2: M : Number of directions e ˆ 3: for j = N − 2 to 0 do 4: for k = 1 to M do 5: eˆk ← ~ek 6: for i = j to N − 1 do ~ji = V ~j − V ~i 7: Vˆji ← V 8: rˆji ← ~rji = ~rj − ~ri 9: n ˆ ← eˆk × Vˆji 10: [s, Vˆjiproj ] ← Solve the system of equations (16) 11: s : Number of Solutions 12: if s = 0 then ~ji ← ~0 13: ∆V 14: else 15: if s=1 then 16: return Vˆjiproj1 ∨ Vˆjiproj2 17: else if s = 2 then 18: θ1 ← arccos(−Vˆji · eˆi ) 19: θ2 ← arccos(Vˆjiproj1 · eˆi ) 20: θ3 ← arccos(Vˆjiproj2 · eˆi ) 21: θ4 ← arccos(Vˆjiproj1 · Vˆji ) 22: θ5 ← arccos(Vˆjiproj2 · Vˆji ) 23: if k(θ1 + θ2 + θ4 ) − πk <  then 24: Vˆjiproj ← Vˆjiproj1 25: else if k(θ1 + θ3 + θ5 ) − πk <  then 26: Vˆjiproj ← Vˆjiproj2 27: end if 28: end if 29: end if 30: θ = π2 : Angle of Rotation 31: eˆi ← eˆi cos θ + (ˆ n × eˆi ) sin θ + n ˆ (ˆ n · eˆi )(1 − cos θ) ~ proj ← Vˆ proj (ˆ ~ji 32: V eTi Vˆjiproj )−1 eˆTi V ji ji ~ji ← V ~ proj − V ~ji 33: ∆V ji 34: end for ~k ← max k∆V ~ji k2 35: ∆V 36: end for ~j ← min k∆V ~k k2 37: ∆V ~j ← V ~ j + ∆V ~j 38: V 39: end for

space, we do not demonstrate here the all discretization). After disctretization it is still possible find the destination waypoint on a computational efficiency way. Both two procedures (the above escape velocity vector and four-dimensional waypoint computation), can be understood as control method in some sense, because the solution of collision avoidance problem is calculated based on previous velocity vectors, which can be assumed as control variables of navigation dynamic system. Furthermore simple philosophy of proposed procedures can be understood, in same time, as control method which has predictive features.

The goal is give to ACAS an automated feature, and the ability to predict the future behavior of aircraft. So, the ability into predict the future aircraft trajectories aside with the solution of collision problem is itself a predictive control method. The set of techniques whose compose the MPC world have a lot of different approaches, but a common principles must be required: 1) The prediction of the state in a time horizon H, 2) The optimization of some criterium, 3) Application of the control to the system. In its core the MPC is a control technique which the future system behavior in time horizon is predict and the controls are calculated taking into account the past and the future of the system. A couple of specifications are translated to a mathematical formalism where a optimization process is applied and the controls on the discretized time horizon are computed. After the optimization process, as say it, the controls are given in each time division of the horizon, but one of must interesting characteristic of MPC is to specify on optimization step the future behavior dependent, for instance, only of the firsts controls, remaining constant until the end of horizon. So, now it is intended to make a relation of MPC concepts and the formulation of collision avoidance problem presented above. Part of the work, implicitly, was done. The two first items are almost solved, the prediction of state is the solution itself (the evasion vector) and the optimization of some criterium, indeed can be any choice. However, above we have proposed a criterium suitable for the current problem, which is the minimization process for find the evasion vector that better satisfies the dynamic constraints. Nevertheless the controls are not yet computed, it will be again necessary a prediction process, so, with a discretization of navigation equations a 4D waypoint on time horizon H shall be found (on our case, we have used a pseudosepectral technique based on Chebyshev polynomials and Gauss-Lobatto collocation points applied to the geodetic navigation equations) and assigned to each aircraft. Hence, a possible procedure, for an generalised ACAS, can be developed based on following sequence, 1) After collect all aircraft data in a time t, solve the problem (12) where the evasion solutions are computed; 2) For a time horizon H, predict all aircraft trajectories from time t to time t + H, applying the discretization technique to navigation dynamic model and using the solutions found in the previous step. 3) Extract the four dimensional waypoints from predict trajectories, whose will be sent to each aircraft. VI. A PPLICATION The example presented here intends to show the effectiveness of proposed method. Before that, it is necessary to clarify somethings, the performance of algorithm depends of various parameters. So the present results depend of machine, the operating system, the language compiler and the way as algorithm was implemented.

Fig. 6. Ground Project of Flight Plans of Ten UAVs

TM

R The used processor was an Intel Core i7 CPU 920 @ 2.67GHz×8, the Operating Systems was Fedora 20-64 and it was built a software package implemented in C Language compiled by gcc 4.8.2. The developed software package implement also a real time aircraft navigation simulator that interact with ACAS. For the present simulation was used 10 aircraft with same basic dynamic parameters. Velocity range [50, 80] (m/s), path angle range [−0.26, 0.26] (rad), heading range [−π, π] (rad), velocity variation range [−1.0, 1.0] (m/s2 ), path angle variation range [−0.02, 0.02] (rad/s), heading variation range [−0.002, 0.002] (rad/s), and as show in Figure 6 the flight plan of each aircraft has the two waypoints in same altitude h, which put it in collision with all others aircraft. The simulation which has approximately 500 seconds of duration where the ACAS was engaged after 3 seconds. After simulation begins and every 3 seconds the ACAS collects the aircraft data and compute the respective evasion maneuvers if they are in collision, if not the aircraft take the guidance vector for respective destination waypoint. The implemented algorithm, also was tested for 10, 14 and 20 aircraft which respective maximum computation time was 1, 2 and 4 seconds. The final result can be observed in the Figures 7 and 8 whose represent the ground projection and altitude trajectories. The ACAS was calibrated to ensure a minimum separation distance of 1000 m, but in some cases that minimum distance was not fulfilled (just the aircraft pair A9 and A3 , where the minimum distance was 996.3 meters). Due to the prediction of the system, it can happen, when aircraft have a distance nearly to the minimum distance allowed the ACAS will spend some time (in configuration of current system the ACAS send the evasion guidance vector after 3 seconds of reception of aircraft data) and during that time the aircraft continue the movement. If they are in collision then during the 3 seconds probably the safety distance will be compromised.

A1 A6

A2 A7

A3 A8

A4 A9

A5 A10

ϕ [rad]

0.6780

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

0.6760

0.6740 −0.1620 −0.1600 −0.1580 −0.1560 λ [rad] Fig. 7. Longitude vs Latitude

A1 A6

A2 A7

A3 A8

A4 A9

A5 A10

2000

h [m]

ACKNOWLEDGMENT

1500

1000

0

200 400 Time [sec] Fig. 8. Altitude vs Time

VII. C ONCLUSIONS It is proposed a new formulation of collision avoidance problem for multiple aircraft based on geometric approach. It is also proposed a resolution method suitable for real implementations as well as a way for solve the aircraft dynamic issues based on a predictive control philosophy. The tests made until now indicate that algorithms have real time features, with the possibility of a reduction of computational time when was parallelized. On current work only was presented the formulation and the resolution, but, in future, must be study the management of priorities and the integration on a real ACAS. Also, should be relevant to make an study that validate the hole system for an exhaustive number of scenarios.

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