1 Introduction
^{1}^{1}1Published in: XI Simpósio Brasileiro de Automação Inteligente, October, 2013. FortalezaCE, Brazil. ID: 3822.Recently, it has been possible to see a growing interest in the development of autonomous aircraft that can cooperate with police and other organizations in the solution of public security problems. The basic motivations are: the autonomous agents can deal with dangerous or unhealthy problems, like fires, violence monitoring, inspection of nuclear areas, deforestation monitoring and monitoring of areas with armed conflict, without exposing humans to the risks.
When several agents are used in the solution of these problems, some advantages arise: a) distributed systems are usually more robust than the centralized ones, and b) it is possible to make a better use of sensors, since they can be shared by the network. As an additional example, when autonomous aircraft squadrons are used in georeferencing, the visual field of the cameras increases, as showed in Figure 1, and the captured images can be mosaicked. Besides, autonomous aircraft typically fly at low heights, hence good quality images can be captured.
The latitudinal formation presented in Figure 2 is an attractive formation to deal with the georeferencing problem, since better area coverage can be achieved. However, obstacles can appear during the flight, and it may be necessary to change the squadron formation to avoid them. For example, the squadron can assume the longitudinal formation presented in Figure 3 for collision avoidance. Thus, if the first aircraft succeeds in avoiding the collision, all other aircraft in the squadron can also avoid the obstacle by using the same behavior.
The longitudinal formation is also necessary when the squadron is landing and doing lift off. Therefore, if the same squadron needs to perform georeferencing, landing, lift off and obstacle avoidance, it will eventually be necessary for the squadron to change between its latitudinal and longitudinal formations.
Therefore, the problem considered in this work is: assuming that there are aircraft in latitudinal formation, equally spaced by meters, we want to develop an algorithm that changes the squadron to the longitudinal formation, where the aircraft will also be equally spaced by meters, without collision among them.
This problem involves aircraft formation, trajectory generation and control. The nonlinear model predictive control approach of [Chao] and the leaderfollower approach of [You] and [Gu] deal with the formation problem, but changing between well defined geometric formations is not considered. Trajectory generation made by using optimization algorithms are usually found in the literature. Some examples are the works of [Cheng] and [Xu]
, but works like these do not focus on well defined geometric formations. Other techniques are also used in the trajectory definition, like the geometric moments, controlled via nonlinear gradient
[Morbidi], the modern matrix analysis used by [Coker], the combination of the hybrid navigation architecture with the local obstacle avoidance methodology and with the model predictive control [Jansen], the navigation functions [Roussos], and the variation of rapidlyexploring random trees [Neto]. But they also do not consider the changing between well defined geometric formations during the flight. Control techniques, like the reinforcement learning
[Santos] do not focus on the changing of formation.The formation reconfiguration is studied by [Venkataramanan], where the aircraft move their position inside a formation, and the same formation is considered before and after the reconfiguration. The autonomous decisionmaking architecture [Knoll] also considers this problem, but neither [Venkataramanan] nor [Knoll] consider the transition between different formations. After exhaustive literature searching, it was not found an algorithm that deals with the problem considered here. Thus, the main contributions of this work are:

An algorithm for changing from the latitudinal to the longitudinal formation of the squadron. The time complexity analysis of the proposed algorithm shows its efficiency is optimal.

A proof of correction of the proposed algorithm, that ensures its longitudinal formation features.

The simulation results by considering a case study, in which the aircraft do not collide.
2 Methodology
The proposed algorithm needs to create references to be followed by the aircraft. To do this, a set of maneuvers is specified.
2.1 The Maneuver Schemes
The proposed algorithm employs two 3D basic maneuver schemes, as shown in Figures 4 and 5. For details see [Giacomincobem]. Algorithms are used to create the references by using the specifications presented in these figures. Here they are called FW and C to implement the go forward and to turn maneuvers, respectively. The go forward interface is FW(, , , , , ) and the to turn interface is C(, , , , , , , ), where is the initial position ,
is a vector time to be filled with time intervals, and the other parameters are shown in Figures
4 and 5. These algorithms are used by the transition algorithm described at Subsection 2.2.2.2 The Transition Algorithm
The transition algorithm is shown in Algorithm 1. It is called FLATLO due to the initial characters that describe its function: From Latitudinal To Longitudinal formation. It basically performs the maneuvers presented in Figure 6 by each aircraft on the left of the squadron, and the equivalent mirrored one for each aircraft on the right of the squadron.
The interface of the transition algorithm is FLATLO(, , , , , , ), where is the aircraft index shown in Figures 2 and 3, is the aircraft airspeed, is the radius of the to turn maneuvers used by the transition algorithm, and are shown in Figure 5a, and is the aircraft initial position .
Algorithm 1 is executed by each aircraft processor, in parallel. For all aircraft, except the last one, it is calculated the forward displacement, called , at Line 4. See Algorithm 2 for details. Thereafter, each aircraft executes four maneuvers. The values that are determined between Lines 15 and 24 are used to fit each maneuver with the next one.
Each aircraft moves in direction to the longitudinal line of the squadron between Lines 26 and 30. At Line 42, the aircraft goes forward in direction to the longitudinal line. Between Lines 44 and 49, the aircraft moves to enter smoothly on the longitudinal line. Finally, the aircraft flies over the longitudinal line at Line 51.
2.3 The Aircraft Model
The simple and welltested aircraft state space model, [Anderson], is employed, and is given by
(1) 
(2) 
(3) 
(4) 
(5) 
(6) 
where the state variables are: airspeed (V), flight path angle (), flight path heading (), and the position variables (x, y, h).
A control scheme is presented in [Giacomincobem] by using the above aircraft model. Here, the references created by Algorithm 1 are submitted to this control scheme.
3 Simulation Results
The Algorithm 1 is programmed in parallel by using the C++ programming language and the GNU Message Passing Interface (MPI) Compiler. It is allocated one processor for each aircraft. The graphics are plotted by using the software Gnuplot.
The initial condition for all aircraft are: height: meters, airspeed: m/s, meters, assuming the latitudinal formation. The references are created by Algorithm 1 by using radius of meters, and different airspeeds for each aircraft, that allow all aircraft to arrive on longitudinal formation at the same time instant. The simulation is made by using the RungeKutta4 algorithm. The results are shown in Figures 7, 8 and 9.
The marks shown in Figures 7 and 8 are used to analyze the aircraft crossing. When the aircraft number zero is flying over the third mark, the aircraft number one is ahead, and when aircraft number tree is flying over fourth mark, the aircraft number two is ahead. An automatic verification did show that the minimum distance between aircraft and and between aircraft and was km and km, respectively. Therefore, there is no collision.
4 Theoretical Analysis
The theoretical analysis of Algorithm 1 is divided into two parts: its time complexity analysis and its proof of correction.
4.1 The Time Complexity Analysis
The go forward and to turn functions are executed over a fixed number of maneuvers, and all other functions used in Algorithm 1 have constant time complexity. Let be the set of maneuvers, and let be the set of references, with , and the function ref be implemented by using the functions FW or C in the Algorithm 1. Then, is a set of sets, and is the references quantity for each maneuver . The time complexity of Algorithm 1 is
where represents the total time steps for each aircraft trajectory. Since this is also the lower bound for the problem, it is concluded that the FLATLO algorithm is optimal with regard to the time complexity [Cormen].
4.2 The Proof of Correction
The proof of correction takes into account that the maneuvers executed by Algorithm 1 are basically described in Figure 6, except the first one, that is created by Algorithm 2.
Note that when an aircraft on latitudinal formation advances the distance before executing the trajectory of Figure 6, advances on its longitudinal formation. This information is used in the proof of the next theorem.
Let assume a set of aircraft initially flying in latitudinal formation, with each aircraft equally spaced by from its neighbors. Then, if the FLATLO algorithm is executed by each aircraft, the longitudinal formation, with spacing between its neighbors, is achieved.
Demonstração.
It is considered aircraft in the squadron. Clearly, if is even and is even, it can be concluded by analyzing Algorithm 2 that . Then
that is the obvious case where the two aircraft and are mirrored by the middle line. If
is odd, it follows, by analyzing Algorithm
2, that . ThenOn the other hand, if is odd and is even, it follows, by analyzing Algorithm 2, that , that is the same case that happens when is even and is odd. If is odd and is odd, then it follows, by analyzing Algorithm 2, that , which has the same result obtained when is even and is even. Therefore, for every and for every .
Similar reasoning can be applied to . Lines 32 to 40 show that , for every and every . Then, if is even
and if if odd it follows that
and therefore, for every and for every . Since this same reasoning can be made for other coordinate system bases, and since all aircraft arrive on longitudinal formation at the same time instant, then the proof follows. ∎
5 Conclusion
An algorithm for changing from latitudinal to longitudinal formation for autonomous aircraft squadrons is proposed in this paper. Despite the relevance of this problem, extensive literature review did not produce relevant results.
The proposed FLATLO algorithm time complexity is equal to the problem lower bound, hence it is optimal [Cormen].
It was proved that if the squadron is initially on latitudinal formation, with each aircraft equally spaced from its neighbors by distance , then the proposed algorithm makes the squadron to change its formation to longitudinal one, keeping the same distance from each aircraft and its neighbors.
The theoretical analysis was confirmed by the simulations, by showing that the aircraft do not collide during the formation transition, and that the references were created correctly by the proposed algorithm, such that they could be followed by a control scheme. Additionally, since the aircraft use different velocities, the number of aircraft has to be limited, and the velocities have to be verified at design time, for security reasons.
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