Formation Flight Control of Multi-UAV System with Communication Constraints

Transcription

1 do: /jatmv8208 Formaton Flght Control of Mult-UAV System wth Communcaton Constrants Rubn Xue 1, Gaohua Ca 2 Abstract: Three dmensonal formaton control problem of mult-uav system wth communcaton constrants of non-unform tme delays and jontly-connected topologes s nvestgated No explct leader exsts n the formaton team, and, therefore, a consensus-based dstrbuted formaton control protocol whch requres only the local neghbor-toneghbor nformaton between the UAVs s proposed for the system The stablty analyss of the proposed formaton control protocol s also performed The research suggests that, when the tme delay, communcaton topology, and control protocol satsfy the stablty condton, the formaton control protocol wll gude the mult-uav system to asymptotcally converge to the desred velocty and shape the expected formaton team, respectvely Numercal smulatons verfy the effectveness of the formaton control system Keywords: Three dmensonal formaton control, Jontly-connected topologes, Mult-UAV system, Non-unform tme delays, Consensus protocol Introducton Recently, wth the development of computer control, sensors, communcaton network etc, many researches on the formaton flght control have been performed Ths s because varous mssons can be successfully completed by the formaton flght, such as battlefeld reconnassance, mult-target attackng, envronment montorng and earthquake rescue and so on Mult-UAV coordnated formaton control has overwhelmng superorty n hgh effcency n performng tasks, low cost of fuel, strong robustness and more flexblty compared wth sngle UAV (Ren and Beard 2008; Cao et al 2012) Therefore, mult-uav formaton flght control has become a hot topc n UAV feld In earler years, typcal approaches for formaton control could be roughly categorzed as leader-follower, behavoral, vrtual leader/vrtual structure Most of the formaton flght researches are performed based on the leader-follower approach, where some UAVs are desgned as leaders whle others are desgned as followers (Ren 2007; Gulett et al 2000) In ths approach, the leaders track the predefned trajectory, and the followers track the nearest leaders accordng to gven schemes It s easy to analyze and mplement the leader-follower controller However, the leader s a sngle pont for the formaton, and therefore ths approach s not robust wth respect to the leader falure In recent years, the problem of mult-uav cooperatve formaton flght control based on consensus protocol has drawn substantal research effort from many studes (Kurk and Namerkawa 2013; Menon 1989; Ren 200; Seo et al 2012) Ren (2007) extended a consensus protocol, whch s ntroduced for systems modelled by second-order dynamcs, to tackle mult-uav formaton control problems by approprately choosng nformaton states on whch consensus s reached Seo (2009) 1Bejng Insttute of Technology School of Aerospace Engneerng Key Laboratory of Dynamcs and Control of Flght Vehcle Bejng Chna 2Bejng Aerospace Automatc Control Insttute Bejng Chna Author for correspondence: Rubn Xue Bejng Insttute of Technology School of Aerospace Engneerng Tyu N Rd, Hadan Bejng Chna Emal: ferenlg@13com Receved: 01/25/201 Accepted: 0/2/201

2 20 Xue R, Ca G proposed a consensus-based formaton flght control protocol and proved that the mult-uav system can form and mantan a geometrc formaton flght wth the network topology swtchng between a drected strongly-connected topology and a topology wth a spannng tree Dong et al (201) nvestgated the tmevaryng formaton control problem by applyng a consensus-based formaton control protocol, and necessary and suffcent condtons are obtaned for the stablty of the system whch contans a spannng tree n the fxed topology Then a quadrotor formaton platform was ntroduced to valdate the theoretcal results However, most of the researches about consensus-based cooperatve formaton flght control are manly focused on two systems: one s a fxed communcaton topology wthout tme delays; the other s a swtchng communcaton topology wthout tme delays as well There are few results avalable to treat the formaton control system wth jontly-connected topologes and tme delay But, n realty, the tme delay usually exsts due to transmsson rate and network congeston, and the communcaton topology of the mult-uav system wll be changed owng to communcaton jammng, complex terran, lmtaton of communcaton dstance etc Therefore, t s of great sgnfcance n both theory and applcaton to nvestgate cooperatve formaton flght control by consderng tme delay and changng topology The man contrbutons of the paper can be summarzed as follows Frst, to desgn a new formaton flght control protocol consderng two key-problems: one s the dverse and asymmetrc tme delays, and the other s the dynamcally changng topologes The topologes dscussed here may not connect all the tme but the unon of the topologes s connected n each perod of tme Second, the analyss of the complex topologes s turned to a smple research of connected component n each perod of tme accordng to the stablty analyss, and a suffcent condton for the stablty s obtaned based on Lyapunov theory The mult-uav system can shape and mantan the expected formaton wth desred velocty, when t satsfes the suffcent condton Model of the mult-uav system Ths paper consders a group system consstng of n autonomous UAVs, and the pont-mass model s used to descrbe the moton of the UAV formaton flyng The related varables are defned wth respect to the nertal coordnate system and are shown n Fg 1 (Wang and Xn 2012) ϕ h T-D 0 χ Local vertcal L Vertcal plane γ V x T-D: Thrust-drag; χ: Headng angle; L: Lft; g: Flght path angle; V: Ground speed; ϕ: Bankng angle Fgure 1 UAV model The model assumes that the arcraft thrust s drected along the velocty vector and that the arcraft always performs coordnated maneuvers It s also assumed that the Earth s flat, and the fuel expendture s neglgble, e the center of mass s tme-nvarant (Xu 2009) Under these assumptons, the moton equatons of the th UAV can be descrbed as follows: y Horzontal plane where: = 1, 2,, n s the ndex of multple UAVs under consderaton For UAV, x s the down-range; y s the cross range; h s the alttude; v s the ground speed; γ s the flght path angle; χ s the headng angle; T s the engne thrust; D s the drag; m s the mass; g s the acceleraton due to gravty; ϕ s the bankng angle; L s the vehcle lft The control varables n the UAVs are the g-load n = L /gm, controlled by the elevator, the bankng angle ϕ, controlled by the combnaton of rudder and alerons, and the engne thrust T, controlled by the throttle Throughout the formaton control process, the control varables wll be constraned to reman wthn ther respectve lmts Defne R m n as a m n real matrx set, ξ = [x,y,h ] T R 3, and u = [u x,u y,u h ] T R 3 Dfferentatng v,γ,h wth respect to tme twce and substtutng x,y,χ, one has the transformed dynamc models of the th UAV as follows: where: ξ s the poston of UAV ; u s a new control varable, (1) (2)

3 Formaton Flght Control of Mult-UAV System wth Communcaton Constrants 205 and the relatonshp between u and the actual control varable U s gven by the expressons (Xu 2009): Formaton control protocol desgn of the mult-uav system (3) () (5) We say that the control protocol u (t) solves the formaton control problem f the states of UAVs satsfy lm [ξ (t ) ξ j (t)] = r j and lm ζ (t ) = ζ (t ) = ζ * (r j = r j s the expect dstance between UAV and UAV j n formaton and ζ * R 3 s the expect velocty), e the mult-uav system can shape and mantan an expected formaton wth a desred velocty under the control protocol u (t) In ths paper, a formaton flght control protocol for the mult-uav system s desgned, and the two key-problems of non-unform tme delays and jontly-connected topologes are consdered To solve ths problem, a lnear control protocol for the th UAV s frstly presented, as follows: (7) The mult-uav system and ts behavor are descrbed n graph theory It s supposed that the mult-uav system under consderaton conssts of n UAVs and G(Γ, E, A) s an undrected graph of the mult-uav system, where Γ = {s 1, s 2,, s n } s the set of nodes, l = (1, 2, 3,, n) s the set of the number of nodes, and E = {(s,s j ) Γ Γ, j} s the set of edges At each tme, each UAV updates ts current state based upon the nformaton receved from ts neghbors Undrected graphs are used to model communcaton topologes Each UAV s regarded as a node Each edge (s, s j ) or (s j,s ) corresponds to an avalable nformaton lnk between UAV and UAV j A communcaton topology s formed when the UAVs begn to communcate to each other at any tme In realty, the communcaton topology usually swtches due to lnk falure brought by communcaton blockng, external dsturbance, hardware falure etc To descrbe the varable topologes, a pecewse constant swtchng functon σ(t): [0, p = {1, 2,, N}(σ n short) s defned, where N denotes the total number of all possble communcaton undrected graphs The communcaton graph at tme t s denoted by G σ and the correspondng Laplacan, by L σ Ths paper nvestgates the desgn of the control protocol of the mult-uav system under jontly-connected communcaton graph The state-space form of the dynamcs of the th UAV s obtaned from Eq 2, as follows: () where: a j (t) s the adjacency weght of the communcaton graph G σ ; N (t) s the neghbor set of the th UAV; k 1 > 0, k 2 > 0, and k 3 = k 1 k 2 ; τ (t) s the tme-varyng self-delay of the th UAV that may be caused by measurement or computaton, and τ j (t) s the tme-varyng delay for the th UAV to get the state nformaton of the j th UAV Here, t s not requred that τ j (t) = τ j (t) It s supposed that there are altogether M dfferent tme delays, denoted by τ m (t) {τ (t), τ j (t),, j, l), m = 1, 2,, M, satsfyng the followng assumptons 1 and 2 Assumpton 1: the tme-varyng delays τ m (t), m = 1, 2,, M (τ m n short), satsfy 0 τ m (t) h m and τ m (t) d m < 1 for specfed constants h m > 0 and d m > 0 A model transformaton s made to analyze the close-loop control performance of the mult-uav system Therefore, the concept of formaton center s ntroduced, whch s a formaton centrod of the mult-uav system A formaton of regular pentagon s consdered as an example for convenent and easy understandng of the formaton problem, as shown n Fg 2, where O s the orgn of Cartesan coordnates, O C s the formaton center, ξ (t) and ξ j (t) are postons of UAV,j n plane coordnate system, respectvely, and ξ 0 (t) s the formaton center The dstance between UAV,j and the formaton center are r and r j, respectvely Consequently, the control protocol (Eq 7) can be transformed nto: where: ξ (t) R 3 s the poston state; ζ (t) R 3 s the velocty state; u (t) R 3 s the control nput (8)

4 20 Xue R, Ca G where: r j = r j r ξ (t) o r r j ξ 0 (t) j ξ j (t) Fgure 2 Graph of regular pentagon formaton structure Accordng to the poston and velocty of the expected formaton of the mult-uav system, ξ (t) = ξ (t)ξ 0 (t) r and ζ (t) = ζ (t)ζ * are denoted, then control protocol (Eq 8) can be transformed nto: (9) It s denoted: (10) Stablty analyss of formaton flght close-loop control system Defnton of swtchng topology and related lemmas Some prelmnary defntons and results need to be presented before the stablty analyss The concept of swtchng topology s ntroduced frst It s consdered an nfnte sequence of non-empty, bounded, and contguous tme ntervals [t k, t k + 1 ), k = 0, 1,, wth t 0 = 0 and t k + 1 t k T 1 (k 0) for some constant T 1 > 0 It s supposed that, n each nterval [t k, t k + 1 ), there s a sequence of non-overlappng subntervals satsfyng t kb+1 t kb T 2, 0 b m k for some nteger m k 0 and a gven constant T 2 > 0 such that the communcaton topology G σ swtches at t kb and t does not change durng each subnterval [t kb, t kb+1 ) Assumpton 2: the collecton of graphs n each nterval [t k, t k + 1) s jontly-connected Wth the swtchng topologes defned above, t s supposed that the tme-nvarant communcaton graph G σ n the subnterval [t kb, t kb+1 ) has d σ (d σ 1) connected components wth the correspondng sets of nodes denoted by ψ kj, ψ kj,, ψ kj ; f σ denotes the number of nodes n ψ kj Then there exsts a permutaton matrx P σ R n n T 1 2 dσ such that P σ L σ P σ = dag{l σm, L σm,, L σm }, 1 2 (12) dσ (13) Under the protocol (Eq 9), the closed-loop dynamcs of the mult-uav system s: and (11) where: I n s the n-dmensonal unt matrx; denotes the Kronecker product; L sm R n n ; L σm Q s the coeffcent matrx of the varable ε(t t m ) for m = 1, 2,, M It s clear that L σ = Σ m 1 L σm and L σ = L σ T Evdently, f lm ε(t) = 0, then lm ξ (t) = 0 and lm ζ (t) = 0, e lm ξ j (t) ξ (t) = r j and lm ζ (t) = ζ *, that s, the mult-uav system can shape and mantan the expected formaton wth a desred velocty under the formaton control protocol In the followng, we prove that the mult-uav system can realze lm ε(t) = 0 under the protocol (Eq 7) M ˆ where each block matrx L σ R f σ f σ s the Laplacan of the correspondng connected component, L σm, R f σ f σ and m L σ = Σ m=1 L σm Then, n each subnterval [t kb, t kb+1 ), the system (Eq11) can be decomposed nto the followng d σ subsystems: (15) where: ε σ (t) = [ε σ1 (t),, ε (t)] R2f σ σ2fσ Lemma 1 (Ln and Ja 2010): consder the matrx C n = ni n 11 T (1 represents [1, 1,, 1] T wth compatble dmensons),

5 Formaton Flght Control of Mult-UAV System wth Communcaton Constrants 207 then there exsts an orthogonal matrx U n R n n such that T U n DU n = dag{ni n 1, 0} and the last column of U n s 1 n Gven a matrx D R n n such that 1 T D = 0 and D1 = 0, then T U n DU n = dag{u T DU n, 0}, where U n denotes the frst n 1 columns of U n Lemma 2 (Ln and Ja 2011): for any real dfferentable vector functon x(t) R n, any dfferentable scalar functon τ(t) [0, h], and any constant matrx 0 < H = H T R n n, the followng nequalty can be obtaned: Theorem 1 s proven n the followng Proof: Defne a Lyapunov-Krasovsk functon for the system (Eq 11) as follows: (17) where h > 0 s a specfed scalar value It s easy to see that V(t) s a postve defnte decrescent functon Calculatng V(t), t can be obtaned: Suffcent condtons for the mult-uav close-loop control system Theorem 1: Cconsder a mult-uav system wth non-unform tme delays and swtchng topologes, for each subnterval [t kb, t kb+1 ), f there s a common constant γ > 0 and F σ R f σ f σ, = 1, 2,, d σ such that Moreover, from (Eq 1) and Assumpton 1, V(t) can be rewrtten as: (1) then lmξ j (t) ξ (t) = r j and lm ζ (t) = ζ* that s, the mult-uavsystem can fnally shape an expected formaton wth the desred velocty Applyng Lemma 2, t can be obtaned: F σ = dag{u, I 2fσ } and U s defned as n Lemma 1, where 2Mf 2f σ σ T where: δ = [ε T T T σ (t), ε σ1 (t τ 1 ), ε σ2 (t τ 2 ),, ε σm (t τ M )] Consderng η = [ε T T T σ (t) h1, ε σ1 (t), ε σ2 (t),, ε σm (t)], where h > 0 s a constant, t s obvous that Ξ σ (δ η) = 0 Therefore: where: λ Ξσ < 0denotes the largest non-zero egenvalue of Ξ σ Therefore: (18)

6 208 Xue R, Ca G From the analyss above, system (Eq 11) s stable (Gu et al 2003), e lm V(t) = 0, thus lm ε(t) = 0; consequently, lm ξ j (t) ξ (t) = r j and lm ζ (t) ζ *, that s, the mult-uav system can shape and mantan the expected formaton wth an desred velocty under the formaton control protocol (Eq 7) Mult-UAV control system smulaton Numercal smulatons wll be gven to verfy the desgned control protocol and llustrate the theoretcal results obtaned n the prevous secton In ths paper, the drag n the UAV model (Eq 1) s calculated by (Xu 2009): G I G II G III Fgure 3 Communcaton topology of UAVs 5 (19) where: the wng area S = 371 m 2 ; the zero lft drag coeffcent C D0 = 002; the load factor effectveness k n = 1; the nduced drag coeffcent k = 01; the gravtatonal coeffcent g = 981 kg/m 2 ; the atmospherc densty r = kg/m 3 ; the weght of the UAV W = m g = 1,515 N The gust model s v w = v w, n + v w, t and vares accordng to the alttude h In the smulated gust, the normal wnd shear v w, n = 0215Ulog10(h ), where U = 227 m/s s the mean wnd speed at an alttude of 5,000 m The turbulence part of the wnd gust v w, t has a Gaussan dstrbuton wth a zero mean and a standard dervaton of 009 U The sx UAVs system wll complete the task of formaton clmbng, level flght, and gldng The communcaton topology graph of the UAVs and the expected formaton structure are shown n Fgs 3 and, respectvely The communcaton topology n Fg 3 swtches every 01 s n the sequence of (G I, G II, G III, G I ) All graphs n ths fgure are not connected, and the weght of each edge s 10, but the unon of the graphs s jontly-connected It s supposed that there are altogether three dfferent tme delays, denoted by τ 1 (t), τ 2 (t), and τ 3 (t): τ (t) = τ j (t) = τ 1 (t) for any j; τ 12 (t) = τ 23 (t) = τ 3 (t) = τ 5 (t) = τ 5 (t) = τ 1 (t) = τ 2 (t); and τ 21 (t) = τ 32 (t) = τ 3 (t) = τ 5 (t) = τ 5 (t) = τ 1 (t) = τ 3 (t) The tme delays satsfy 0 τ 1 (t) 001, 0 τ 2 (t) 002, 0 τ 3 (t) 003 and τ 1 (t), τ 2 (t), τ 3 (t) 03 It s supposed that all ntal condtons of poston, velocty, and flght path angle are randomly set The desred v 1 = ( sn (008t)) m/s and χ = 5 o It s solved that (Eq 1) s feasble for k 1 = 0, k 2 = 11, k 3 = 0 The trajectores of poston, velocty, flght path angle, headng angle, and the formed formaton are shown n Fgs 5 to 11 It s clear that the mult-uav system can complete the maneuver formaton flght task wth the expected velocty and headng angle as well as mantan the desred formaton durng the flght 10,000 0 y [m] h[m] y[m] 8,000,000 o,000 2, Fgure Expected trangle formaton dagram 5 x [m] 0 2,000,000,000 8,000 10,000 x[m] Fgure 5 3-D trajectores of UAVs formaton flyng

7 Formaton Flght Control of Mult-UAV System wth Communcaton Constrants 209 y [m] 9,000 8,000 7,000,000 5,000,000 3,000 2,000 1,000 1,000 2,000 3,000,000 5,000,000 7,000 8,000 9,000 x [m] Flght path angle [deg] Fgure Top vew of UAVs formaton flyng Fgure 11 Tme hstores of the flght path angle h [m] Dstance between vehcles [m] Dstance Dstance 23 Dstance 3 50 Dstance 5 Dstance Fgure 7 Tme hstores of the heght Fgure 9 Tme hstores of the dstance between the UAVs 0 0 Velocty [m/s] UAV Headng angle [deg] Fgure 8 Tme hstores of the velocty Fgure 10 Tme hstores of the headng angle

8 210 Xue R, Ca G Concluson Three dmensonal formaton flght control problems are nvestgated, consderng the constrants of jontlyconnected topologes and non-unform tme delays, where each UAV has a self-delay, and all delays are ndependent of each other A consensus-based formaton control protocol s desgned, and the stablty problem of the mult-uav formaton control system s turned nto the problem that looks for a feasble soluton by solvng the lnear matrx nequalty In realty, t s only necessary to study the connected components wth dfferent topology structures, makng t possble to smplfy the analyss of the whole topology structures Numercal examples are ncluded to llustrate the obtaned results n addton If the communcaton topology s jontly-connected and the non-unform tme delays satsfy the desgnng requrements, then the mult-uav system can shape the desred formaton and also mantan the expected velocty, headng angle, and expected flght path angle The problems of collson avodance constrant and the sze of the UAVs are not consdered here These challengng and meanngful problems wll be presented n future studes REFERENCES Cao Y, Yu W, Ren W, Chen G (2012) An overvew of recent progress n the study of dstrbuted mult-agent coordnaton IEEE Trans Ind Inf 9(1):27-38 do: /TII Dong X, Yu B, Sh Z (201) Tme-varyng formaton control for unmanned aeral vehcles: theores and applcatons IEEE Trans Control Syst Technol 23(1): do: / TCST Gulett F, Polln L, Innocent M (2000) Autonomous formaton flght IEEE Control Syst 20(): 3- do: / Gu K, Khartonov VL, Chen J (2003) Stablty of tme-delay systems Boston: Brkhäuser Kurk Y, Namerkawa T (2013) Consensus-based cooperatve control for geometrc confguraton of UAVs flyng n formaton Proceedngs of the SICE Annual Conference; Nagoya, Japan Ln P, Ja Y (2010) Consensus of a class of second-order mult-agent systems wth tme-delay and jontly-connected topologes IEEE Trans Autom Control 55(3): do: /TAC Ln P, Ja Y (2011) Mult-agent consensus wth dverse tme-delays and jontly-connected topologes Automatca 7():88-85 do: 10101/ jautomatca Menon PKA (1989) Short-range nonlnear feedback strateges for arcraft pursut-evason J Gud Contr Dynam 12(1): do: 10251/3203 Ren W (200) Consensus-based formaton control strateges for mult-vehcle systems Proceedngs of the Amercan Control Conference; Mnnesota, USA Ren W (2007) Consensus strateges for cooperatve control of vehcle formatons IET Control Theory Appl 1(2): do: 10109/et-cta: Ren W, Beard RW (2008) Dstrbuted consensus n mult-vehcle cooperatve control; London: Sprnger Seo J (2009) Controller desgn for UAV formaton flght usng consensus-based decentralzed approach Proceedngs of the AIAA Aerospace Conference; Seattle, USA Seo J, Km Y, Km S, Tsourdos A (2012) Consensus-based reconfgurable controller desgn for unmanned aeral vehcle formaton flght Proc IME G J Aero Eng 22(7): do: / Wang J, Xn M (2012) Integrated optmal formaton control of multple Unmanned Aeral Vehcles Proceedngs of the AIAA Gudance, Navgaton, and Control Conference; Mnnesota, USA Xu Y (2009) Nonlnear robust stochastc control for Unmanned Aeral Vehcles J Gud Contr Dynam 32(): do: 10251/10753