Experimental Validation of Cooperative Formation Control with Collision Avoidance for a Multi-UAV System

Transcription

1 Proceedings of te 6t International Conference on Automation, Robotics and Applications, Feb 17-19, 2015, Queenstown, New Zealand Experimental Validation of Cooperative Formation Control wit Collision Avoidance for a Multi-UAV System Yasuiro Kuriki and Toru Namerikawa Abstract In tis study, we consider cooperative control issues for a multi-unmanned aerial veicle (UAV) system. Specifically, we present a cooperative formation control strategy for a multi-uav system wit unidirectional network links. Our strategy is to apply a consensus-based algoritm and leaderfollower structure to te UAVs so tat tey can cooperatively fly in formation. Te leader provides eac UAV wit commands to generate a geometric configuration of te formation. Convergence is guaranteed wen te cooperative formation control algoritm is applied to te UAVs. Collisions among UAVs can occur wen tey are flying wit te cooperative control UAVs. Our strategy for collision avoidance is to apply an artificial potential approac to te UAVs. Experiments are performed on multiple commercial small UAVs to validate te proposed formation control algoritm wit collision-avoidance capability. I. INTRODUCTION In recent years, cooperative control problems for multiveicle systems ave attracted muc attention from many researcers 1. Cooperative control tecnologies ave enormous potential for application to veicles suc as unmanned aerial veicles (UAVs), artificial satellites, and autonomous mobile observation robots as well as distributed sensor networks. A multi-veicle system may be able to perform tasks more efficiently tan a single igly functional veicle. Formation control problems for a multi-veicle system ave been intensively studied in recent years, and many control algoritms ave been developed. A major strategy for formation control is to apply a consensus algoritm 2 5. We ave been studying cooperative control problems for multi-veicle systems 6 9 wit particular focus on solving formation control problems using a consensus algoritm. We expressed te UAV dynamics on te orizontal plane as a fourt-order system and te dynamics in te vertical direction as a second-order system separately, ten proposed two consensus-based control algoritms for a multi-uav system to cooperatively fly in formation on te orizontal plane and in te vertical direction 8, 9. Using tese two consensusbased cooperative control algoritms, UAVs can fly in formation in tree-dimensional space. Our proposed control algoritms guarantee convergence to te desired position for formation flying. Altoug, te network links between UAVs must be bidirectional in our proposed algoritms. In tis study, we extend our proposed control algoritm for formation flying in te vertical direction to a formation control algoritm for a multi-uav system wit unidirectional Y. Kuriki and T. Namerikawa are wit te Scool of Integrated Design Engineering, Graduate Scool of Science and Tecnology, Keio University, Kanagawa, Japan yasuiro at nl.sd.keio.ac.jp, namerikawa at nl.sd.keio.ac.jp network links. Te control algoritm we propose in tis study as an advantage in more network flexibility tan our previous control algoritms. Control problems wit collision or obstacle avoidance ave attracted muc attention in recent years, and many control algoritms ave been developed. Most strategies can be rougly grouped into rule-based and optimization-based approaces. One rule-based approac uses an artificial potential field 10, 11, and one optimization-based approac uses model predictive control (MPC) 12. An artificial potential field as been extensively applied to autonomous robot navigation. Veicles move along te negative gradient of te composite of te potential fields. We proposed an artificial potential-based collision avoidance algoritm. UAVs takes evasive action in te vertical direction if te risk of collision increases 9. We proved tat convergence is guaranteed wen our proposed control algoritm for formation flying in te vertical direction and artificial potential-based collision avoidance algoritm are simultaneously applied to multiple UAVs. In recent years, quadrotors ave attracted muc attention from academia and industry. Generally, researcers customize commercially available UAVs for teir experimental validation because te UAVs often lack enoug functionality for full experimental validation, and ten demonstrate teir fligt control algoritm using te customized UAVs 13, 14. For example, a motion capture system is built into te experimental platform to estimate te positions and attitude angles due to te lack of onboard sensors 13. However, it generally takes muc time and effort to develop experimental platform from commercially available UAVs. In tis study, we select te Parrot AR.Drone because it can save te time and effort of developing experimental platform. We sow ow our experimental platform is developed and present experiments performed using multiple AR.Drones to validate our formation control algoritm. Tis paper is outlined as follows. In Section II, we state ow to model a UAV and multi-uav system and define control objectives. Te control objectives are as follows. First, wit regard to a multi-uav system wit unidirectional network links between UAVs, UAVs fly in formation in te vertical direction. Second, wit regard to a multi-uav system wit only bidirectional network links between UAVs, UAVs fly in formation in te vertical direction wile avoiding collisions. In Section III, we propose control algoritms for te problems defined in Section II. Section IV presents experimental results to validate te proposed approac to acieve te latter objective. Finally, te concluding remarks /15/$ IEEE

2 Proceedings of te 6t International Conference on Automation, Robotics and Applications, Feb 17-19, 2015, Queenstown, New Zealand are stated in Section V. II. PROBLEM STATEMENT In tis section, we present ow to model a UAV and multi- UAV system and define te control objective. A. Modeling a UAV We assume N quadrotors wit te same motion caracteristics. Te multi-uav system consists of N quadrotors and a leader. Eac quadrotor as four propellers and a controller. Te propellers are individually given control commands by te controller. To simply model te quadrotor matematically, we assume te following. First, te quadrotor flies slowly enoug to ignore external aerodynamic forces suc as aerodynamic drag and a blade vortex interface acting on te body. Second, te propellers respond to trust commands fast enoug to ignore time delay from wen te controller gives te propellers te trust command until te propellers actually produce te commanded trust. Finally, a yawing moment is never produced. Under te condition tat overing at a constant altitude is an equilibrium point of a nonlinear quadrotor system, it can be decoupled and linearized into longitudinal, latitude, and vertical equations of motion. A linearized quadrotor model in te vertical direction is given by d dt (0) i (1) i = (1) i T totali, i {1, 2,,N}, (1) were (0) i = i, (1) i = w i and T totali = T totali /m. Also, te subscript i denotes quadrotor i. Te command T total /4 is equally assigned to eac propeller. Table I defines te above symbols. Note tat linearized quadrotor models on te orizontal plane can be expressed as a fort-order system 8. B. Modeling a Multi-UAV System We can model a multi-uav system as a group of dynamical systems were multiple UAVs and a leader excange information wit eac anoter. Tis network can be matematically described using grap teory 16. We use te grap G = (V, A) to model information interaction among N UAVs, were V = {v 1,v 2,,v N } is a set of nodes and A V V is a set of edges. Te edge (v i,v j ) in te edge set of te grap denotes a network pat from UAV i to UAV j. Tis means tat UAV j can obtain information from UAV i. A directed tree is a digrap were every node as exactly one parent node except for one node, wic is called a root. TABLE I: Definition of Symbols Symbol Definition (m) Altitude w (m/s) Rate of decent T total (N) Total trust command m (kg) Quadrotor mass Te root as no parent and as a directed pat to every oter node. A directed spanning tree of G is a tree tat contains all nodes of G. Let A R N N, D R N N,andL R N N be an adjacency matrix, degree matrix, and grap Laplacian matrix, respectively, related to te grap G. Te component of te adjacency matrix A =a ij is given by { 1, for (vj,v a ij = i ) A. (2) 0, oterwise Tis means tat a ij is set to 1 if UAV i is obtaining information from UAV j troug a network; oterwise, a ij is set to 0. Te degree matrix D is an in-degree matrix given by D = diag(deg(v 1 ), deg(v 2 ),, deg(v N )), (3) were deg(v i ) is a number of communication links arriving at node v i. Te grap Laplacian matrix L is defined as L = D A. (4) Te grap Laplacian as te following properties: if a grap as or contains a directed spanning tree, te grap Laplacian L as a single eigenvalue at zero, and all nonzero eigenvalues of te grap Laplacian ave a positive real part. C. Control Objectives We consider te following two missions: First, wit regard to a multi-uav system wit unidirectional network links between te quadrotors, N quadrotors fly in formation following teir leader in te vertical direction. Second, wit regard to a multi-uav system wit only bidirectional network links between te quadrotors, N quadrotors fly in formation following teir leader in te vertical direction wile avoiding collisions. Flying in formation means tat eac quadrotor converges to a time-variant desired position. Te time-variant desired positions are determined by te desired geometric configuration of te formation. To acieve tese control objectives, we make te following assumptions: Assumption 1: Every quadrotor must be connected to te leader, but not all quadrotors necessarily ave a direct connection wit te leader. Assumption 2: Te network connections between te quadrotors must be bidirectional. Assumption 3: Te movement of te leader must be independent of te quadrotors; tat is, te movement of te leader is not affected by any of te quadrotors. III. PROPOSED APPROACH In tis section, we present our proposed control algoritm to acieve te control objectives given in Section II. C.

3 Proceedings of te 6t International Conference on Automation, Robotics and Applications, Feb 17-19, 2015, Queenstown, New Zealand A. Formation in Vertical Direction For formation flying in te vertical direction, we consider te objectives of a group of quadrotors and of eac quadrotor separately. Te first objective is for te group of quadrotors to fly cooperatively in formation. Te second objective is for eac quadrotor to generate a geometric configuration of te formation. We apply a consensus-based cooperative control algoritm to acieve te former cooperative objective and a leader-follower structure to acieve te latter individual objective. Te leader individually provides te only directly connected quadrotors wit its own position and desired positions for formation. Te control law for quadrotor i is given by 1 T totali (t) = a ij γ k (ĥ(k) i ĥ(k) j ), j=1 k=0 i {1, 2,,N}, (5) ĥ (k) j = (k) j d (k) j,j {1, 2,,N +1}, k {0, 1},(6) were γ k R, k {0, 1} are control gains. Also, (k) j R, k {0, 1} is te state of quadrotor j in te vertical direction, and d (k) j R, k {0, 1} is te desired relative state between quadrotor j and te leader in te vertical direction. Te following teorem concerning desired convergence is derived. Teorem 1: Suppose tat a multi-uav system comprises N( 1) quadrotors expressed as (1) and a leader. In addition, assumptions 1 and 3 are satisfied. Wen control protocol (5) is applied to eac quadrotor and te control gains γ k, for k {0, 1} are selected so as to satisfy condition (7), ten all of te states of te quadrotors in te vertical direction will asymptotically converge to te desired states. γ 1 λ lr > D l cos(arg( D l )), l {1, 2,,N}, (7) 2 were λ lr is a real part of te nonzero eigenvalues of te grap Laplacian wen te eigenvalues are expressed as λ l = λ lr + iλ li C,l {1, 2,,N}. Also,D l is defined as D l =(γ 1 λ l ) 2 4γ 0 λ l. (8) Proof: By applying control protocol (5) to quadrotor i expressed as (1), we get ḣ (1) i = j=1 a ij γ 0 (ĥ(0) i ĥ(0) j )+γ 1 (ĥ(1) i ĥ(1) j ), i {1, 2,,N}, (9) were te leader is regarded as te last UAV: UAV (N + 1). We define te new state vector (k) = (k) 1 (k) 2 (k) N 0 T R () and ĥ(k) = ĥ(k) 1 ĥ (k) 2 ĥ (k) N ĥ(k) T R, for k {0, 1}. By using (k) and ĥ(k) to express (9) in matrix form, (9) can be rewritten as (1) = γ 0 Lĥ(0) γ 1 Lĥ(1), (10) were L R () () is te grap Laplacian of te multi-uav system. Here, te vector ĥ(k) consists of states of te quadrotors and commands from te leader. Hence, we separate te states of te quadrotors from te oters. Now, using d (k) =0, for k {0, 1} and te definition of te grap Laplacian, (10) can be separately expressed as ḣ (1) = γ 0 M (0) γ 1 M (1) + γ 0 M (0) + γ 1M (1) + γ 0Md (0) + γ 1Md (1), (11) were te matrix M R N N is defined as (12). Also, (k) = (k) 1 (k) 2 (k) N T R N (k), =1 N (k) R N,andd (k) =d(k) 1 d (k) 2... d (k) N T R N, for k {0, 1}, were is te Kronecker product and 1 N =11 1 T R N. M = a j=1 a1j a 1N a a j=1 a2j 2N a N1 a N2... j=1 anj (12) Equation (11) can be rewritten in matrix-vector form as d dt (0) (1) + γ 0M 0 N 0 N = γ 0M γ 1M 0 N I N (0) (1) (0) γ 1M (1) + 0 N 0 N γ 0M γ 1M d (0) d (1), (13) were I N R N N is a N-dimensional unit matrix and 0 N R N N is a N-dimensional zero matrix. Now, consider te solution to te differential equation (13). Tis differential equation means tat te states of te quadrotor are affected by not only te control gains and network structure but also te commands from te leader. To examine te stability of (13), consider te omogeneous equation of (13) expressed as (1) ρ = N ρ (14) were ρ and te matrix N are expressed as follows. (0) 0 ρ =, N = N I N. (15) γ 0 M γ 1 M Te stability of (13) is determined by te eigenvalues of te matrix N.First,letλ and s be an eigenvalue and eigenvector, respectively, of te matrix M: Ms = λs. In addition, let μ and σ be an eigenvalue and eigenvector, respectively, of te matrix N : N σ = μσ. Ten, from (14), we can get te next equality: 0 N I N γ 0M γ 1M s μs =μ s μs,σ= s μs.(16) From te lowest row block of (16) and Ms = λs, we can get μ 2 + γ 1 λμ + γ 0 λ =0. (17) Equation (17) sows te relationsip between te eigenvalues of te matrix M and tose of te matrix N. Note tat all of

4 Proceedings of te 6t International Conference on Automation, Robotics and Applications, Feb 17-19, 2015, Queenstown, New Zealand te eigenvalues λ of te matrix M ave a positive real part wen assumptions 1 and 3 are satisfied. Te eigenvalues μ, wic are te roots of (17), can be obtained by solving (17). μ l± = 1 2 ( γ 1 λ l ± D l ), l {1, 2,,N}, (18) were λ l C is te lt eigenvalue of te matrix M, μ l+ and μ l are te eigenvalues associated wit te eigenvalue λ l,andd l C is expressed as D l =(γ 1 λ l ) 2 4γ 0 λ l. (19) Tus, we can get te real part of te eigenvalues μ as ( ) Re(μ l± )= 1 2 γ 1 λ lr ± D l cos(arg( D l 2 )) l {1, 2,,N}, (20) were λ l = λ lr + iλ li, λ lr R, λ li R for l {1, 2,,N}. To acieve te control objective tat all of te states of te quadrotors in te vertical direction asymptotically converge to te desired states, all of te real parts of te eigenvalues μ must be negative. Re(μ l± ) < 0, l {1, 2,,N}. (21) From cos(arg( D l 2 )) > 0 and λ l R > 0, l {1, 2,,N}, te necessary and sufficient condition to satisfy (21) can be given by (7). Note tat condition (7) can be equivalently expressed as γ k > 0, for k {0, 1} if all of te eigenvalues λ are real numbers; tat is, if assumptions 1-3 are satisfied. Wen te control gains γ k for k {0, 1} are selected so as to satisfy condition (7), all of te eigenvalues of te matrix N ave a negative real part. Applying tese appropriate gains to te controller (5) produces te following convergence: lim ρ =0. (22) t Next, consider te particular solution to (13). One of tem can be given by (0) (1) = (0) (1) + d (0) d (1),. (23) Tis validation can be confirmed by substituting (23) for (13). Te general solution to te non-omogeneous differential equation (13) is te sum of te particular solution and te general solution to te omogeneous equation. Hence, if te control gains γ k, for k {0, 1} are appropriately selected tat is, tese gains are selected to satisfy condition (7) te general solution asymptotically converges to te commands from te leader. From te element of te first row block in (23), we prove tat all of te states of te quadrotors in te vertical direction will asymptotically converge to te desired states wen control protocol (5) wit appropriate control gains γ k is applied to quadrotor i expressed as (1). B. Formation wit Collision Avoidance In tis section, we briefly describe our proposed formation control algoritm wit collision avoidance capability 9. Te basic strategy of te collision avoidance is tat te quadrotors only take evasive action in te vertical direction if te risk of collision increases, even wen te quadrotors fly in tree-dimensional space. A safety region is set around te center of gravity of te quadrotor as sown in Fig. 1. Te eigt of te region is 2ΔH. If te safety region of a quadrotor impinges against tat of anoter quadrotor, all of te quadrotors wit overlapping safety regions begin to take evasive action. Te evasive action is continued until te safety regions no longer overlap. Te artificial potential field produced by quadrotors i and j is defined as ( ) 2 K 1 U ij = 2 ij +1 1, 2ΔH +1, for ij 2ΔH 0, oterwise i, j {1, 2,,N}, (24) were K R is a positive control parameter and ij is te altitude gap between quadrotors i and j. Teartificial potential increases as te altitude gap between quadrotors i and j closes. In addition, te artificial potential is zero wen te safety regions of quadrotors i and j do not overlap. From (24), te average artificial potential field produced by quadrotor i and te oters is given by U i = 1 N U ij, i {1, 2,,N}. (25) N 1 j=1, j i From (25), te total artificial potential field produced by eac quadrotor is given by N U c = U i. (26) i=1 We propose te following artificial force for quadrotor i to avoid collisions: f cai = i U c, i {1, 2,,N}. (27) Te artificial force acts in te opposite direction of te potential gradient; tat is, te force acts so as to decrease te potential. Terefore, te artificial force acts so as to increase te sum of te altitude gap. Our proposed formation control algoritm wit collision avoidance capability 9 is as follows: T totali = f fomi + f cai, i {1, 2,,N}, (28) Fig. 1: Safety region.

5 Proceedings of te 6t International Conference on Automation, Robotics and Applications, Feb 17-19, 2015, Queenstown, New Zealand were f fomi is control protocol (5) wit positive control gains γ k for k {0, 1}. Teorem 2: Suppose tat a multi-uav system comprises a leader and N( 2) quadrotors as expressed in (1) and tat assumptions 1 3 are satisfied. If control protocol (5) wit positive control gains γ k, k {0, 1} and control protocol (28) wit te positive control parameter K are simultaneously applied to eac quadrotor, ten eac quadrotor wit collision avoidance capability will asymptotically converge to te desired position in te vertical direction. Proof: See Section III.C in 9. IV. EXPERIMENTAL VALIDATION In tis section, we present te experimental validation of te proposed control algoritm for application to a multi- UAV platform. A. Experimental Setup We used two AR.Drones 15, a laptop, and an access point for experimental validation of control algoritm (28). Te measured data and oter information on te states of te AR.Drone were transmitted to te laptop via te access point in real time as sown in Fig. 2. Ten, te laptop calculated te control commands and sent tem back to eac AR.Drone. Note tat te laptop was regarded as te virtual leader of te multi-uav system. Te AR.Drone uses sonar sensors mounted on te back to measure te altitude and rate of climb. If one AR.Drone crosses te same vertical straigt line as anoter, it cannot measure te altitude precisely because of te geometric and electromagnetic interference. In addition, wen two AR.Drones are close togeter, te downwas of one can affect te oter, wic destabilizes te oter AR.Drone. To avoid armful effects, te experiments were conducted wit one AR.Drone keeping well away from te oter AR.Drone on te orizontal plane. Only te data concerning te vertical direction were used to validate control algoritm (28). We considered a network wit a directed spanning tree, as sown in Fig. 3. Note tat te leader was virtual. Te leader flew at a constant altitude of 1.4(m). Eac AR.Drone kept a certain relative altitude from te leader. Te relative altitudes were kept steady 0(m), and te leader provided te desired relative altitude d i =0, i {1, 2}. In oter words, te desired altitudes for two AR.Drones were kept steady at 1.4(m). Te control gains for formation in te vertical direction and control parameter for collision avoidance were selected so as to satisfy all of te conditions: γ 0 =1.25, γ 1 =3.50, and K =1.5. Eac AR.Drone ad a safety region wit a parameter of ΔH =0.5(m). Te AR.Drones were judged to be at risk of collision wit eac oter if te altitude gap was less tan 0.15(m). As sown in Table II, two experiments were carried out to validate te proposed control algoritm (28). Te first experiment (Case I) was conducted to validate only te performance of te formation control algoritm in te vertical direction. Te second experiment (Case II) was conducted to validate te performance of te formation control algoritm wit collision avoidance capability. B. Experiment Results We present te experimental results ere. Figs. 4 and 5 sow te results of Cases I and II, respectively. For Case II, te collision avoidance algoritm was switced from on to off in 23(s). In te figures, te first grap from te top sows te time istory of te altitude, te second grap sows te time istory of te absolute value of te altitude gap between two AR.Drones, and te tird grap sows te time istory of te rate of descent. Te solid lines illustrate te experimental results, and te dotted line illustrates te simulation results conducted under te same conditions as te experiments. Fig. 4 sows tat te experimental results corresponded wit te simulation results and tat eac AR.Drone was able to fly in formation at te desired altitude of 1.4(m). Tese results validate te control strategy for formation in te vertical direction (5). Fig. 5 sows tat te experimental results corresponded wit te simulation results. Tere was an altitude gap of more tan 0.2(m) between two AR.Drones wile te collision avoidance control algoritm was working. Tis indicates tat te collision avoidance control algoritm can avoid collisions between two AR.Drones. Tese results validate te formation control strategy wit collision avoidance capability (28). Fig. 3: Definition of network for experimental validation. Fig. 2: AR.Drone: control structure. TABLE II: Experimental conditions Case I Case II Formation control ON ON Collision avoidance control OFF ON OFF

6 Proceedings of te 6t International Conference on Automation, Robotics and Applications, Feb 17-19, 2015, Queenstown, New Zealand is matematically modeled using grap teory, and ten proposed a control algoritm for multi-uav system wit unidirectional network links to fly in formation in te vertical direction. Our experiments using two AR.Drones validated te proposed algoritm s effectiveness. We did not conduct experiments to validate te formation control algoritm for te orizontal plane because of a lack of a precise position sensor. Hence, developing an experimental platform wit precise positions on te orizontal plane and conducting experiments to validate te formation control algoritm on te orizontal plane remains a callenging issue. REFERENCES Fig. 4: Experimental results of Case I using formation control algoritm (5) witout collision avoidance algoritm. Fig. 5: Experimental results of Case II using formation control algoritm wit collision avoidance capability (28). V. CONCLUSIONS AND FUTURE WORK In tis study, we briefly sowed ow a linearized model of quadrotors is expressed and ow a multi-uav system 1 R. Murray, Recent Researc in Cooperative Control of Multiveicle Systems, Journal of Dynamic Systems, Measurement, and Control, Vol. 129, pp , R. Olfati-Saber, J. Fax and R. Murray, Consensus and Cooperation in Networked Multi-Agent Systems, Proceedings of te IEEE, Vol. 95, No. 1, pp , W. Ren and N. Soresen, Distributed coordination arcitecture for multi-robot formation control, Robotics and Autonomous Systems, Vol. 56, pp , W. Ren, Consensus Tracking Under Directed Interaction Topologies:Algoritms and Experiments, IEEE T. on Control Systems Tecnology, Vol. 18, No. 1, pp , Z. Meng, W. Ren et al., Leaderless and Leader-Following Consensus Wit Communication and Input Delays Under a Directed Network Topology, IEEE T. on Systems, Man, and Cybernetics, Vol. 41, No. 1, pp , C. Yosioka and T. Namerikawa, Observer-based Consensus Control Strategy for Multi-Agent System wit Communication Time Delay, Proc. of IEEE Multi-conference on Systems and Control (2008), pp , H. Kawakami and T. Namerikawa, Cooperative Target-capturing Strategy for Multi-veicle Systems wit Dynamic Network Topology, 2009 American Control Conference, pp , Y. Kuriki and T. Namerikawa, Formation Control of UAVs wit a Fourt-order Fligt Dynamics, Decision and Control (CDC), IEEE 52nd Annual Conference 2013, pp , Y. Kuriki and T. Namerikawa, Consensus-based Cooperative Formation Control wit Collision Avoidance for a Multi-UAV System, 2014 American Control Conference, 2014 (to be publised). 10 M. M. Zavlanos, and G. J. Pappas, Potential Fields for Maintaining Connectivity of Mobile Networks, IEEE T. on Robotics, Vol. 23, No. 4, pp , G. M. Atinc, et al., Collision-Free Trajectory Tracking Wile Preserving Connectivity in Unicycle Multi-Agent Systems, 2013 American Control Conference, pp , A. Ricards and J. How, Decentralized Model Predictive Control of Cooperating UAVs, Decision and Control (CDC), IEEE 43rd Annual Conference 2004, pp , N. Micael, M. Scwager, V. Kumar, and D. Rus, An Experimental Study of Time Scales and Stability in Networked Multi- Robot Systems, Experimental Robotics, pp , Springer Berlin Heidelberg, F. Augugliaro, A. P. Scoellig, and R. D Andrea, Dance of te Flying Macines: Metods for Designing and Executing an Aerial Dance Coreograpy, IEEE Robotics & Automation Magazine, Vol. 20, No. 4, pp , Parrot Co., Parrot AR.Drone 2.0, ttp://ardrone2.parrot.com/ 16 C. Godsil and G. Royle, Algebraic Grap Teory, Springer, 2001.