Motion synchronization in unmanned aircrafts formation control with communication delays

Motion synchronization in unmanned aircrafts formation control with communication delays

Commun Nonlinear Sci Numer Simulat 18 (2013) 744–756 Contents lists available at SciVerse ScienceDirect Commun Nonlinear Sci Numer Simulat journal h...

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Commun Nonlinear Sci Numer Simulat 18 (2013) 744–756

Contents lists available at SciVerse ScienceDirect

Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns

Motion synchronization in unmanned aircrafts formation control with communication delays Hamed Rezaee, Farzaneh Abdollahi ⇑ Department of Electrical Engineering, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran

a r t i c l e

i n f o

Article history: Received 20 March 2012 Received in revised form 7 July 2012 Accepted 15 August 2012 Available online 27 August 2012 Keywords: Multi-agent systems Formation control Synchronization Unmanned aircrafts Sliding mode control Cross coupling

a b s t r a c t This paper proposes a formation control strategy for unmanned aircrafts using a virtual structure. Cross coupled sliding mode controllers are introduced to cope with uncertainties in the attitude measurement systems of the unmanned aircrafts and unmeasurable bounded external disturbances such as wind effects, and also to provide motion synchronization in the multi-agent system. This motion synchronization strategy improves the agents convergence to their desired positions, and this is useful for a multi-agent system with faulty agents. Moreover, the proposed motion synchronization strategy is not restricted to specific communication topologies, and sufficient conditions are provided to guarantee the multi-agent system stability in the presence of communication delays. Numerical simulations are presented for a team of five unmanned aircrafts to make a pentagon formation and confirm the accepted performance of the proposed control strategy. Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction Maneuvering of unmanned vehicles maintaining a geometric formation has became one of interesting research areas in recent years with wide civilian applications in surveillance, discovering, atmospheric studies, rescue missions, fire monitoring, operations in hazardous environments, and so on. Indeed, a formation can be applied in a group of aerial robots, ground robots, and so on with a cooperative task [1–5]. Numerous researches are devoted to the leader–follower structure. In this structure, each agent follows leaders to maintain a desired relative position from them, and keep a formation. Although this structure is easy to implement and understand, the formation is sensitive to the leaders behavior, and disturbed or failed leaders affect their followers motion. Moreover, since there are not any loops in this structure, there are no feedbacks from a follower to its leaders. To cope with this issue, the virtual and behavioral structures have been developed. In the virtual structure, agents follow a virtual leader to provide a rigid body formation in the frame of the virtual leader, and a formation trajectory is defined for all the agents as a single rigid body. In the behavioral structure, no definite formation is considered, and it is useful for multi objective missions such as target seeking, obstacle avoidance, and so on [6,7]. There are two main approaches for multi-agent systems control in the literature, namely: centralized and decentralized approaches each of whom has its own advantages and disadvantages. In the centralized approach, control of each agent is based on a central controller. The ability to override the control in emergency conditions and having global information are the merits of a central controller. However, the possible malfunction of the central controller affects the performance of the whole multi-agent system. Moreover, in long maneuvers, agents may have problems to communicate with the central ⇑ Corresponding author. E-mail addresses: [email protected] (H. Rezaee), [email protected] (F. Abdollahi). 1007-5704/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cnsns.2012.08.015

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controller, cohesively. In the decentralized approach, each agent makes decisions based on its local information achieved through its own sensors and from the neighboring agents [8,9]. One of important problems in cooperative dynamic systems is motion synchronization. Synchronization means simultaneous convergence to desired positions or trajectories [10], or motion with identical speeds [11,12]. The concept of cross coupling for motion synchronization has been developed extensively in the literature with a wide range of applications for motion control of multi-axis systems [13–15], cooperative manipulator robots [16–18], etc. In recent years, there have been a number of studies on motion synchronization in multi-agent systems formation control using cross coupling. For instance, cross coupled PI controllers were applied to synchronize the errors of positions and velocities of flying wings and aircrafts, respectively in [19,20]. Synchronized position tracking PI controllers of flying wings were also applied in [6]. Two coupling topologies were employed in [21] for attitude trajectory tracking of 3-DOF helicopters based on feedforward controllers. In [22], a consensus control strategy was proposed to synchronize the acceleration of agents in a formation to reach an arbitrary velocity. By using only visual measurements, attitude tracking synchronization in a leader–follower multi-agent system was applied in [23]. In [24], attitude tracking synchronization of unmanned helicopters in a leader–follower formation was proposed by using neural network adaptive controllers, and adaptive control was applied in [25] for synchronization of position tracking errors in the presence of communication loss. For motion synchronization, coupling with more agents provides a better motion synchronization. On the other hand, it may not be feasible to couple agents in any topology. A drawback of the above mentioned approaches is that they are restricted to specific topologies that are defined for coupling agents, and the stability analysis of the multi-agents system is guaranteed for these topologies. Moreover, each agent receives the information of other agents with time-delays which may degrade the performance and stability of the multi-agent system. This issue has not been considered in the above mentioned papers. The current study is devoted to formation control of multiple-unmanned aircraft systems with 3-DOF point mass model using the virtual structure. It is intended to provide a rigid formation of unmanned aircrafts in the frame of a virtual leader and couple their behaviors to synchronize their motion. To summarize, the main contributions of this paper are stated as follows: 1. Motion synchronization in a decentralized approach without considering a specific coupling topology is applied. 2. To cope with the uncertain attitude measurement systems of the unmanned aircrafts and also wind effects, cross coupled sliding mode controllers are applied. The measured attitudes of the unmanned aircrafts are prone to uncertainties due to errors and uncertainties which intrinsic in sensors and observers. Moreover, wind effects can affect the performance of the unmanned aircrafts in formation flight. 3. The multi-agent systems stability is guaranteed in the presence of constant communication delays. In a decentralized multi-agent system, for motion synchronization it is necessary for each agent to receive the position and velocity of other agents, and this information may be received with time-delays. Communication delays can degrade the performance and stability of the multi-agent system; therefore, sufficient conditions are provided on the agents coupling to guarantee the multi-agent system stability in the presence of constant communication delays. The following notations are considered in the paper: In shows an n  n identity matrix, 0nm is an n  m matrix with zero entries,  denotes the Kronecker products, the Laplace transform of f ðtÞ is shown by f ðsÞ; detð:Þ means determinant, and diagðM1 ; M2 ; . . . ; Mm Þ shows a block diagonal matrix composed of matrices M1 ; M2 ; . . ., and Mm . The reminder of this paper is organized as follows: In Section 2, the dynamical equations of the unmanned aircrafts are given. The proposed virtual structure formulations are provided in Section 3, the sliding mode based synchronization strategy is presented in Section 4, the simulation results and numerical examples are provided in Section 5, and conclusions are given in Section 6. 2. Model definition Consider a multi-agent system with N unmanned aircrafts in R3 . The ith unmanned aircraft dynamical equations can be considered by 3-DOF point mass model as follows [8,26]:

p_ xi ðtÞ ¼ V i ðtÞ cos ðci ðtÞÞ cos ðwi ðtÞÞ; p_ yi ðtÞ ¼ V i ðtÞ cos ðci ðtÞÞ sin ðwi ðtÞÞ; p_ zi ðtÞ ¼ V i ðtÞ sin ðci ðtÞÞ; T i ðtÞ  Di ðtÞ V_ i ðtÞ ¼ g  g sin ðci ðtÞÞ; Wi ayi ðtÞ w_ i ðtÞ ¼ ; V i ðtÞ cos ðci ðtÞÞ g cos ðci ðtÞÞ api ðtÞ c_ i ðtÞ ¼ þ V i ðtÞ V i ðtÞ

ð1Þ

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where pi ðtÞ ¼ ½ pxi ðtÞ pyi ðtÞ pzi ðtÞ > denotes the Cartesian position of the unmanned aircraft, V i ðtÞ is the speed, ci ðtÞ and wi ðtÞ are the flight path and heading angles, respectively, W i is the unmanned aircraft weight, g is the gravitational acceleration, Di ðtÞ shows the drag force effect, T i ðtÞ is the engine thrust, and api ðtÞ ¼ gni ðtÞ cos ð/i ðtÞÞ and ayi ðtÞ ¼ gni ðtÞ sin ð/i ðtÞÞ are the pitch and yaw acceleration, respectively, where /i ðtÞ is the bank angle, and ni ðiÞ is the load factor which is the ratio of the lift force to the weight. Therefore, a control input ci ðtÞ ¼ ½ ati ðtÞ ayi ðtÞ api ðtÞ > for the unmanned aircraft can be considered i ðtÞ where ati ðtÞ ¼ g T i ðtÞD . Fig. 1 shows the lift force, weight, thrust force, and drag force of an unmanned aircraft, and the roll, Wi pitch, and yaw axes around which the bank, flight path, and heading angles are determined, respectively. 3. Virtual structure formulation The main objective in the virtual structure is to locate the ith unmanned aircraft in a fixed point in the frame of a virtual leader, and this can provide various rigid body formations of unmanned aircrafts in three dimensions. In Fig. 2, the frames of ^i  ^zi axes and ^ ^v  ^zv axes, respectively. To locate the the unmanned aircraft and the virtual leader are depicted by ^ xi  y xv  y ith unmanned aircraft in a fixed point in the frame of a virtual leader, the following three parameters should be adjusted: 1. The distance between the unmanned aircraft and the virtual leader which its desired value is named r i 2 Rþ . ^i plane, which its desired 2. The angle between the line linking the unmanned aircraft and the virtual leader and the ^ xi  y ^i plane and denotes the value is named 1i 2 ½ p2 ; p2 . It decomposes r i to two terms. One term is perpendicular to the ^ xi  y relative height between the unmanned aircraft and the virtual leader in the unmanned aircraft frame. Another term is on ^i plane. the ^ xi  y 3. The angle between the line linking the unmanned aircraft and the virtual leader and the heading of the unmanned aircraft ^i plane, which its desired value is named vi 2 R. It decomposes the term of r i which is on the ^ ^i plane to on the ^ xi  y xi  y ^i axes to define the relative position between the unmanned aircraft and the virtual leader on the two terms on the ^ xi and y ^ ^i plane. xi  y Moreover, to provide a formation in the frame of the virtual leader, the heading and flight path angles of the virtual leader should be considered to define the desired position of the unmanned aircraft relative to the virtual leader. Therefore, based on Fig. 2, the desired position of the ith unmanned aircraft relative to the virtual leader is defined as follows:

pdxi ðtÞ ¼ r i cos









1i þ cv ðtÞ cos vi þ wv ðtÞ ;    d pyi ðtÞ ¼ r i cos 1i þ cv ðtÞ sin vi þ wv ðtÞ ;   pdzi ðtÞ ¼ r i sin 1i þ cv ðtÞ 

ð2Þ

where pdi ðtÞ ¼ ½ pdxi ðtÞ pdyi ðtÞ pdzi ðtÞ > denotes the desired relative position of the unmanned aircraft to the virtual leader, and wv and cv are the heading and flight path angles of the virtual leader, respectively. Therefore, based on the desired relative position defined in (2), during maneuvers the formation direction changes to have an identical direction in the frame of the virtual leader. Therefore, it is necessary to determine the trajectory of the virtual leader by defining V v ðtÞ; wv ðtÞ, and cv ðtÞ, where V v ðtÞ is the speed of the virtual leader. Therefore, based on (2), the desired position of the unmanned aircraft is

Fig. 1. A diagram showing lift force, weight, thrust force, drag force, and the roll, pitch, and yaw axes around which the bank, flight path, and heading angles ^, and ^z axes, respectively, where ^ ^  ^z denotes the frame of the unmanned are determined, respectively, where roll, pitch, and yaw axes are on the ^ x; y xy aircraft.

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^v  ^zv is the frame of the virtual leader which is parallel to ^ ^i  ^zi (the frame of the unmanned Fig. 2. The virtual structure configuration where ^ xv  y xi  y aircraft).

~ di ðtÞ ¼ pv ðtÞ þ pdi ðtÞ p

ð3Þ >

where pv ðtÞ ¼ ½ pxv ðtÞ pyv ðtÞ pzv ðtÞ  is the Cartesian position of the virtual leader. Moreover, taking the time derivative of p_ xi ðtÞ; p_ yi ðtÞ, and p_ zi ðtÞ in (1) and substituting V_ i ðtÞ; w_ i ðtÞ, and c_ i ðtÞ from (1) into them yields:

€ i ðtÞ ¼ wi ðtÞci ðtÞ þ ½ 0 0 g > p

ð4Þ

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where

2

cos ðci ðtÞÞ cos ðwi ðtÞÞ  sin ðwi ðtÞÞ  sin ðci ðtÞÞ cos ðwi ðtÞÞ

6 wi ðtÞ ¼ 4 cos ðci ðtÞÞ sin ðwi ðtÞÞ

cos ðwi ðtÞÞ

sin ðci ðtÞÞ

3

7  sin ðci ðtÞÞ sin ðwi ðtÞÞ 5:

0

cos ðci ðtÞÞ

Since det ðwi ðtÞÞ – 0, by employing feedback linearization for (4) one can get:

ui ðtÞ ¼ wi ðtÞci ðtÞ where ui ðtÞ can be considered as a new control input. An important problem in flight control is unmeasurable bounded external disturbances such as wind effects. A main feature of sliding mode control is its robustness against unmeasurable bounded external disturbance, i.e., the sliding mode control can compensate the effect of unmeasurable bounded external disturbances. Another problem in flight control is uncertainties in attitude measurement systems. This uncertainty is due to malfunction of gyroscopes in the aircrafts system. Uncertainties in attitude measurement should be considered in the entries of the matrix wi ðtÞ. To model the uncertainties in attitude measurement, a coefficient of ui ðtÞ may be applied to the system. Therefore, considering all the above mentioned issues, (4) should be restated as follows:

€ i ðtÞ ¼ ai ui ðtÞ þ ½ 0 0 g > þ di ðtÞ p

ð5Þ

where ai 2 R33 is a diagonal matrix with bounded uncertain entries, and di ðtÞ 2 R3 represents the unmeasurable bounded þ external disturbance. In the next section, in general, coupled sliding mode controllers are proposed for motion synchronization of nonlinear multi-agent systems with uncertainties in agent parameters, and that will be applied for a multi-agent system when the ith agent dynamics is defined in (5). 4. Synchronized cross coupled sliding mode controllers A popular technique for robust control of nonlinear systems with uncertainty, model imperfection, and so on is sliding mode control. In this technique, the controller leads a system state to slide on a surface and this sliding guarantees the system stability [27]. In this section, sliding mode controllers are coupled to synchronize the motion of agents in a multi-agent system formation with virtual structure. Therefore, sliding mode control can guarantee the system stability when the agents are prone to unmeasurable bounded external disturbances and their dynamics are uncertain. A multi-agent system with N second order dynamic agents in R‘ is considered where the ith agent dynamical equation is as follows:

€i ðtÞ ¼ f i ðxi ; tÞ þ bi ðxi ; tÞui ðtÞ x ‘

ð6Þ ‘

where xi ðtÞ 2 R is the states vector, ui ðtÞ 2 R is the control input vector, and bi ðxi ; tÞ ¼ diagðb1i ðxi ; tÞ; . . . ; b‘i ðxi ; tÞÞ is a source min max min max of uncertainty in the system, bki ðxi ; tÞ 6 bki ðxi ; tÞ 6 bki ðxi ; tÞ where bki ðxi ; tÞ; bki ðxi ; tÞ 2 Rþ ; k 2 f1; . . . ; ‘g. Another > source of uncertainty in the agents is f i ðxi ; tÞ ¼ ½ f1i ðxi ; tÞ    f‘i ðxi ; tÞ  where fki ðxi ; tÞ 2 R; k 2 f1; . . . ; ‘g is unknown and is estimated by ^f ki ðxi ; tÞ 2 R and j^f ki ðxi ; tÞ  fki ðxi ; tÞj < F ki ðxi ; tÞ, where F ki ðxi ; tÞ is the known upper bound of the error. If we define the desired trajectory of the ith agent by xdi ðtÞ 2 R‘ , the tracking error vector can be stated by

ei ðtÞ ¼ xi ðtÞ  xdi ðtÞ:

ð7Þ

In the multi-agent system, to synchronize the convergence of the tracking error vectors defined in (7) to zero, coupled forms of the tracking error vectors are introduced such that their convergence to zero, while maintaining the system stability, synchronizes the convergence of agents toward their desired positions. Since the control strategy is decentralized, to couple tracking error vectors (for motion synchronization), each agent should send its tracking error vector to neighboring agents. Neighboring agents receive this information with a time-delay. The coupled form of the tracking error vector for the ith agent can be stated by

eci ðtÞ ¼ aii ei ðtÞ þ

X

aij ej ðt  sij Þ

ð8Þ

j2N i

where eci ðtÞ is the coupled form of the tracking error vector for the ith agent, where aij 2 R; i; j 2 f1; 2; . . . ; Ng; sij 2 Rþ represents the constant time-delay in the information that the ith agent receives from the jth agent [28–30]. This delay is the sum of the following delays [31]: 1. Transmitting delay: the time between starting and ending the transmission of an information from agent i to agent j. 2. Processing delay: the time that the information should spend at each agent to be sent or received. On the one hand, the processing delay is constant; on the other hand, the transmitting delay depends on the distance between the agents. Since the variation of the distances between the agents are not very high, the transmitting delay can be considered constant. Therefore, the total time-delay can be considered constant. This transmitted information is the

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Cartesian position and velocity errors of the ith agent with respect to the desired position and its derivative, respectively [28–30], and N i is the set of the neighbors of the ith agent. The neighbors of the ith agent can be defined as a set of agents with whom the ith agent communicates. Therefore, a coupling matrix A can be considered as follows:

2

3

a11 6a 6 21 A¼6 6 .. 4 .

a12



a1N

a22 .. .

 .. .

a2N 7 7 .. 7 7 . 5

aN1

aN2

   aNN

where aij ¼ 0 if the jth agent is not the neighbor of the ith agent, and aii can be defined arbitrary. The first step to design a sliding mode controller is to define a stable surface for the convergence of a tracking error to zero. On the other hand, in a team of N agents, the convergence of the tracking error vectors to zero can be synchronized by defining coupled sliding surfaces. These sliding surfaces should be designed such that their convergence to zero leads to the convergence of all the tracking error vectors to zero. Theorem 3 provides the coupled sliding surfaces of sliding mode controllers to provide the motion synchronization of N agents, in a decentralized approach. The following two theorems and lemma are required for Theorem 3 better understanding. Theorem 1 [32]. The eigenvalues of a matrix M with entries mij ; i; j 2 f1; 2; . . . ; mg are on disks with center mii and radius Pm j¼1;j–i jmij j which are named Gerschgorin disks. Theorem 2 [33]. Suppose a matrix MðsÞ. Therefore, the zeroes of det ðIm þ MðsÞÞ are on pthe ffi open left half plane if the eigenvalue loci of MðjxÞ does not encircle ð1 þ 0jÞ point in counter clockwise direction where j ¼ 1. kj ; j 2 f1; 2; . . . ; m2 g, respectively. Lemma 1 [34]. Consider two matrices M1 and M2 with eigenvalues ki ; i 2 f1; 2; . . . ; m1 g and ~ kj ; i 2 f1; 2; . . . ; m1 g; j 2 f1; 2; . . . ; m2 g. Therefore, the eigenvalues of M1  M2 are ki ~ The following theorem provides the mentioned sliding surfaces: Theorem 3. In the introduced multi-agent system in (6), the following equation provides the coupled sliding surfaces of N agents, and when the sliding surfaces converge to zero, it guarantees the convergence of the tracking error vectors to zero, and therefore guarantees the stability of the multi-agent system with motion synchronization, in a decentralized approach:

_ SðtÞ ¼ EðtÞ þ ðIN  GÞEc ðtÞ

ð9Þ

where

2

s1 ðtÞ

3

6 s2 ðtÞ 7 7 6 7 SðtÞ ¼ 6 6 .. 7; 4 . 5

2

ec1 ðtÞ

3

6 ec ðtÞ 7 6 2 7 7 Ec ðtÞ ¼ 6 6 .. 7; 4 . 5 ecN ðtÞ

sN ðtÞ

2

e1 ðtÞ

3

6 e2 ðtÞ 7 7 6 7 EðtÞ ¼ 6 6 .. 7; 4 . 5 eN ðtÞ

and si ðtÞ 2 R‘ shows the coupled sliding surface vector for the ith agent, G 2 R‘‘ is a matrix with real positive eigenvalues to determine the convergence rate of the tracking errors, and Ec ðtÞ is provided from a coupling matrix A with Gerschgorin disks on the open right half plane. Proof. We should show that the convergence of the sliding surface SðtÞ to zero guarantees the multi-agent system stability. _ ¼ ðIN  GÞEc ðtÞ yields: Suppose SðtÞ ¼ ½0‘N1 . In this condition, the Laplace transformation of EðtÞ

  Eð0Þ ¼ ðIN  GÞE  c ðsÞ: sEðsÞ

ð10Þ

On the other hand, the Laplace transform of (8) will be:

ci ðsÞ ¼ aii e i ðsÞ þ e

X

j ðsÞ: aij expðsij sÞe

j2N i

Therefore, the Laplace transform of Ec ðtÞ is:

 c ðsÞ ¼ ðRðsÞ  I‘ ÞEðsÞ  E where

ð11Þ

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2

a12 expðs12 sÞ

a11

6 a21 expðs21 sÞ 6 RðsÞ ¼ 6 .. 6 4 .

   a1N expðs1N sÞ

3

   a2N expðs2N sÞ 7 7 7: .. .. 7 5 . .

a22 .. .

aN1 expðsN1 sÞ aN2 expðsN2 sÞ   

aNN

By considering (10) and (11), one can get:

 ¼ ðsI‘N þ RðsÞ  GÞ1 Eð0Þ: EðsÞ

ð12Þ

The system defined in (12) is stable if the zeros of det ðsI‘N þ RðsÞ  GÞ are on the open left half plane. Two conditions are considered: (1) If s ¼ 0; RðsÞ is equal to A. By considering Theorem 1 and Lemma 1, since all the Gerschgorin disks of A are on the open right half plane and G has real positive eigenvalues, all the eigenvalues of A  G are on the open right half plane. It means that all the zeros of det ðsI‘N þ A  GÞ are on the open left half plane, and s ¼ 0 is not a zero for det ðsI‘N þ A  GÞ.   (2) If s – 0, the system in (12) is stable if the zeros of det I‘N þ RðsÞG are on the open left half plane. On the other hand, s based on Theorem 2, this condition is satisfied if the eigenvalue loci of RðjjxxÞG does not encircle the ð1 þ 0jÞ point in counter clockwise direction. Since all the Gerschgorin disks of A are on the open right half plane, based on Theorem 1, one can get

aii >

X

jaij j;

i 2 f1; 2; . . . ; Ng:

j2N i

Therefore, since j expðjsij xÞj ¼ 1, it can be said that

aii >

X

jaij expðjsij xÞj;

i 2 f1; 2; . . . ; Ng:

j2N i

Therefore, all the Gerschgorin disks of RðjxÞ are on the open right half plane. Therefore, based on Theorem 1 the eigenvalues of RðjxÞ are on the open right half plane, and since G has real positive eigenvalues, based on Lemma 1, the eigenvalues of RðjxÞ  G are on the open right half plane. Suppose aðxÞ þ jbðxÞ is an eigenvalue of RðjxÞ  G where aðxÞ 2 Rþ ; bðxÞ 2 R, i.e., an eigenvalue of RðjjxxÞG can be stated as ajðxxÞ þ bðxxÞ. Since aðxÞ > 0, it is obvious that this eigenvalue loci does not encircle the ð1 þ 0jÞ point in counter clockwise direction, and this guarantees the convergence of EðtÞ to zero and the stability of the multi-agent system. Since SðtÞ ¼ ½0‘N1  , from (10) one can conclude that if EðtÞ converges to zero, then Ec ðtÞ converges to zero, and this provides motion synchronization for the multi-agent system. On the other hand, from (9) one can obtain that the decentralized sliding surface of the ith agent is si ðtÞ ¼ e_ i ðtÞ þ Geci ðtÞ; i 2 f1; 2; . . . ; Ng. h

Remark 1. It is worth noting that Theorem 3 guarantees the stability of the multi-agent system with motion synchronization if Ec ðtÞ is provided from a coupling matrix A with Gerschgorin disks on the open left half plane if G has real negative eigenvalues. The next step to design sliding mode controllers is to make SðtÞ ¼ ½0‘N1 , i.e., to make si ðtÞ ¼ ½0‘1 ; i 2 f1; 2; . . . ; Ng. Let us define G ¼ diagðG1 ; G2 ; . . . ; G‘ Þ, and

3 ec1i ðtÞ 6 ec ðtÞ 7 7 6 eci ðtÞ ¼ 6 2i 7; 4  5 2

ec‘i ðtÞ

3 xd1i ðtÞ 6 xd ðtÞ 7 7 6 xdi ðtÞ ¼ 6 2i 7; 4  5 2

xd‘i ðtÞ

3 u1i ðtÞ 6 u ðtÞ 7 6 2i 7 ui ðtÞ ¼ 6 7: 4  5 2

u‘i ðtÞ

From [27], it can be said that to make si ðtÞ ¼ ½0‘1 , inequality s_ ki ðtÞski ðtÞ 6 gki jski ðtÞj should be satisfied for ski ðtÞ; k 2 f1; . . . ; ‘g where gki 2 Rþ . To satisfy s_ ki ðtÞski ðtÞ 6 gki jski ðtÞj and make ski ðtÞ ¼ 0, similar to [27] one can get:

  ^ ðx ; tÞ1 ^f ðx ; tÞ þ €xd ðtÞ  G e_ c ðtÞ  F signðs ðtÞÞ uki ðtÞ ¼ b i i ki ki k ki ki ki ki

^ ðxi ; tÞ ¼ where b ki

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi min max bki ðxi ; tÞbki ðxi ; tÞ and

Fki P bki ðF ki ðxi ; tÞ þ gki Þ þ ðbki  1Þj  ^f ki ðxi ; tÞ þ €xdki ðtÞ  Gk e_ cki ðtÞj

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi max min where bki ¼ bki ðxi ; tÞ=bki ðxi ; tÞ. Therefore, the proposed cross-coupled sliding mode controllers provide motion synchronization for a multi-agent system whose agents dynamical model is given in (6). Moreover, the proposed controllers guarantee the stability of the multi-agent system in the presence of communication delays. The proposed control strategy can be extended for a multi-agent system with dynamical model given in (5), if we consider ‘ ¼ 3 and

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f i ðxi ; tÞ ¼ ½ 0 0 g > þ di ðtÞ; bi ðxi ; tÞ ¼ ai : ~ di ðtÞ in (3) as the desired trajectory, ui ðtÞ can be Therefore, by applying the proposed control strategy for (5) and considering p obtained. On the other hand, the unmanned aircrafts dynamical model introduced in (5) is the linearized form of the dynamical model (1). Therefore, since wi ðtÞci ðtÞ ¼ ui ðtÞ, the unmanned aircraft control input ci ðtÞ can be obtained as follows:

ci ðtÞ ¼ wi ðtÞ1 ui ðtÞ: Remark 2. Theorem 3 guarantees the stability of the the multi-agent system for all the coupling topologies. To show the merit of a general coupling topologies, let us consider the coupling topology defined in [19]. In [19], the coupling form of tracking error vectors are considered as follows:

ec1 ¼ 2e1  e2  en ; ec2 ¼ 2e2  e3  e1 ; .. . ecn ¼ 2en  e1  en1 : It means that the ith agent should be coupled with the ði  1Þth and ði þ 1Þth agents and the multi-agent systems is stabilized based on the defined coupling topology. The mentioned algorithm cannot be applicable for a topology that the ith agent cannot communicate with the ði þ 1Þth agent due to large distance between them. In other words, [19] cannot guarantee the stability of the multi-agent system when the ith agent cannot be coupled with the ði þ 1Þth agent, although our generalized coupling topology guarantees the multi-agent system stability in this case. Remark 3. The proposed motion synchronization strategy synchronizes the convergence of all the agents to their desired trajectories. In other words, the proposed strategy synchronizes the convergence of tracking error vectors to zero. In this condition, this synchronization is independent to the agents dynamics and structures. For instance, it tries to synchronize a multi-agent system with faulty agents to compensate the faulty behavior of the agents. It should be noted that, even though the proposed approach guarantees the stability of the multi-agent system under time delayed communications, communication delays affect the synchronization performance of the multi-agent system. 5. Simulation results In this section, the accuracy of the proposed approach for motion synchronization in formation flight is shown in two examples.

Aircraft 1

Aircraft 2

Aircraft 3

Aircraft 4

Virtual Leader

Aircraft 5

Fig. 3. A pentagon formation of five unmanned aircrafts, where the virtual leader is located at the center of the pentagon.

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Table 1 The initial conditions of the unmanned aircrafts. Aircraft 1 2 3 4 5

Pi ðmÞ

V i ðm=s2 Þ

wi ðradÞ

(17, 2, 99) (10, 25, 95) (10, 2, 90) (30, 20, 120) (5, 35, 115)

46 40 45 45 45

p=4 p=4 p=4 p=4 p=4

ci ðradÞ 0 0 0 0 0

Table 2 The unmanned aircrafts formation parameters. UAV

r i ðmÞ

1i ðradÞ

vi ðradÞ

1 2 3 4 5

20 20 20 20 20

0.5p 0.1p 0.1p 0.3p 0.3p

0 0.5p 0.5p 0.5p 0.5p

Example 1. Consider a group of five unmanned aircrafts to provide a pentagon formation which is depicted in Fig. 3. The position, speed, heading angle, and flight path angle of the ith unmanned aircraft where i 2 f1; 2; . . . ; 5g are initialized with the values presented in Table 1. To achieve a pentagon formation of the unmanned aircrafts which the magnitude of its edges is 23:51 m, the formation parameters for the unmanned aircrafts are defined in Table 2. Without loss of generality, a bidirectional communication topology is supposed as shown in Fig. 4. Therefore, a coupling matrix A with Gerschgorin disks on the open right half plane is supposed as follows:

2

3 1 1 6 1 2 0 6 6 6 A ¼ 6 1 0 3 6 0 0 4 0 0

0

0

0

3

7 0 7 7 0 1 7 7 7 2 1 5 1 1 3 0

Furthermore, it is supposed that, Di ðtÞ ¼ 0; W i ¼ 5000 N; a ¼ ð1  0:2ÞI3 ; i 2 f1; 2; . . . ; 5g; g ¼ 10 m=s2 ; G ¼ I3 , communication delay is 0:5 s for all the communications, di ðtÞ entries are modeled by white noise in MATLAB SIMULINKÒ (R2008a) with power 0.1 where i 2 f1; 2; . . . ; 5g. A tracking trajectory is considered for the formation by defining the initial position of the virtual leader by (2, 8, 100) m, and its trajectory by

V v ðtÞ ¼ 45 þ t ðm=sÞ; wv ðtÞ ¼

p 4

þ

3 t ðradÞ; 10

Fig. 4. The unmanned aircrafts communication topology in Example 1. A left–right arrow ($) between the ith and jth unmanned aircrafts shows that these unmanned aircrafts exchange their tracking errors information for motion synchronization.

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300 200

z(m)

100 0 −100

Aircraft 1 Aircraft 2 Aircraft 3 Aircraft 4 Aircraft 5

−200 300 200

100 0 −100

100

−200

0

−300

−100

x(m)

y(m)

Fig. 5. The trajectories of the five unmanned aircrafts in the pentagon formation in Example 1.

40

d13(m)

d12(m)

40 20 0 0

10

20

Time(sec)

30

0 0

40

10

20

30

40

20

30

40

Time(sec)

40

d35(m)

d24(m)

40 20 0 0

20

10

20

Time(sec)

30

20 0 0

40

10

Time(sec)

45

d (m)

40 20 0 0

10

20

Time(sec)

30

40

Fig. 6. The distances between the unmanned aircrafts in the pentagon formation in Example 1 where dij represents the distance between the ith and jth unmanned aircrafts.

Fig. 7. The unmanned aircrafts communication topology in Example 2.

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cv ðtÞ ¼ 0 ðradÞ: The unmanned aircrafts trajectories in three dimensions are depicted in Fig. 5 for t ¼ 40 s, and Fig. 6 shows the distances between the unmanned aircrafts. These figures confirm reaching to a pentagon formation which the magnitude of its edges is 23:51 m. The proposed approach is independent to the dynamics and structure of the agents, and a multi-agent system behavior with unidentical agents can be synchronized. Therefore, it can be useful for a multi-agent system with faulty agents to tolerate the effect of faults. The following example shows how the motion synchronization strategy behaves robust in tolerating the faults effects on the unmanned aircrafts formation, and avoids the multi-agent system to be unstable.

300 200

z(m)

100 0 −100 Aircraft 1 Aircraft 2 Aircraft 3 Aircraft 4 Aircraft 5

−200 300 200

100 0

100 −100

0 −100

−200 −300

y(m)

x(m)

300 200

z(m)

100 0 −100 Aircraft 1 Aircraft 2 Aircraft 3 Aircraft 4 Aircraft 5

−200 300 200

100 0

100 −100

0 −100

y(m)

−200 −300

x(m)

Fig. 8. The trajectories of the five unmanned aircrafts in the pentagon formation in Example 2.

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755

Example 2. In this example, the mentioned scenario in Example 1 is repeated when 50% loss of effectiveness is applied to the yaw acceleration of the unmanned aircraft 1 and 50% loss of effectiveness is applied to the pitch acceleration of the unmanned aircraft 3. A communication topology is supposed as shown in Fig. 7. Therefore, a coupling matrix A with Gerschgorin disks on the open right half plane is considered as follows:

3 5 1 1 1 1 7 6 0 0 0 7 6 1 2 7 6 A¼6 3 0 1 7 7 6 1 0 7 6 0 3 1 5 4 1 0 1 0 1 1 4 2

Fig. 8a and b shows the trajectories of the unmanned aircrafts in Example 2 without and with motion synchronization, respectively. A comparison between Fig. 8a and b confirms that motion synchronization and cross coupling can avoid the multi-agent system to be unstable. It should be noted that the effect of the synchronization strategy in multi-agent systems performance depends on the coupling matrix A which determines which agent is coupled with which agent. 6. Conclusions A decentralized formation control strategy using the virtual structure was addressed in this paper. To cope with the problem of uncertain attitude measurements and wind effects, sliding mode controllers were proposed. By coupling the sliding mode controllers, synchronization of the convergence of agents to desired positions was achieved. The proposed approach for cross coupling provides a general communication topology such that each agent can communicate with others, in any topology. 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