e collson avodance control algortm of te UA formaton flgt Jalong Zang Janguo Yan Xaoun Xng Dongl Yuan Xaole Hou Xaoqao Q 2 Pu Zang 3 Scool of Automaton Control, ortwest Polytecncal Unversty, X an {zl07@mal.nwpu.edu.cn; yg03@nwpu.edu.cn; xxaoun@nwpu.edu.cn; yuongl@nwpu.edu.cn; ou.xaole@nwpu.edu.cn } 2 Scool of Mecano-Electronc Engneerng, Xdan Unversty, x an { qxaoqao35}@63.com 3 Scool of Economcs Management, Saanx Fason Engneerng Unversty, X an {878946452}@63.com Abstract A collson avodance control algortm for a mult-ua system based on bdrectonal network connecton structure s proposed n ts paper, wc can effectvely avod te collson of between UAs, between te UA te obstacle, aceve te UA cooperatve formaton flgt complete te msson. In order to avod te collson well, te proposed consensus-based algortm te leader-follower control strategy are smultaneously appled to te formaton control so tat we can ensure te convergence of te formaton. e tree UAs composed of te trangular formaton as te control obect, te leader flgt pat as expected pat, te followers track te leader to mantan te trangle formaton flgt. And te safety dstance of between te UAs, between te nsde of te UA formaton te obstacle keep te safety dstance. Eac of te UAs as te same forward velocty eadng angle rate n te orzontal plane as well as tey keep te relatve dstance constant n te vertcal drecton. s paper proposes a multple UAs consensus-based collson avodance algortm based on te artfcal potental feld metod. e smulaton experments are performed on multple UAs to valdate te proposed control algortm ave some reference sgnfcance for engneerng applcaton. Introducton A mult-ua formaton system as more sgnfcant advantages tan a sngle UA, suc as te long te tme of fgt, te bg combat radus, te wde range of nvestgaton, te g combat effcency strong searc ablty so on n te mltary feld. And sprayng pestcdes, aeral potograpy, terran survey ar refuelng n te cvlan feld. erefore, t as attracted muc more attenton from many scolars becomes a ot topc of researc[], especally focusng on te pat plannng[2-4], cooperatve formaton control, nformaton sarng fuson[5-7], obstacle avodance as well as oter aspects of researc, wc as been made some obvous acevements. Also, a mult-ua formaton system s only a small part of applcaton n engneerng practce. However, most of tem are based on teoretcal level of researc. e securty problem of te UA formaton fgt s one of te key factors, especally te obstacle avodance researc s partcularly mportant. e obstacle avodance problem for a mult-ua formaton system as been studed well n recent years, te proposed many control algortms ave been developed for te problems. e obstacle avodance control algortm s rougly grouped nto two categores tat rule based approaces optmzaton based approaces. One of te rule based approaces s an artfcal potental feld based approac[8-9]. An artfcal potental feld s not only appled to te UA obstacle avodance problems, but as been extensvely appled to autonomous robot navgaton, ten ts basc teory s tat te UAs move along te negatve gradent of te composte of te potental felds. However, te optmzaton-approaces s a model predctve control(mpc)[0]. e man control algortm n process of te formaton control s appled n te consensus-based control teory to desgn te controller well. e consensus-based algortm for te UA cooperatve formaton control s a knd of dstrbuted control metod, wc as te advantage of avng network structure flexblty[-4], aceves mult-cannel compound control obstacle avodance. e key problem of te mult-agent system obstacle avodance control s ow to apply a consensus-based algortm to cope wt t well [5-7], t wll greatly smplfy te complexty of te problem we studed. erefore, a consensus-based algortm te artfcal potental-based metod are appled to te formaton so tat tey can effectvely avod obstacle n te tree dmensonal space. s paper s organzed as follows. In Secton 2, we state te buldng model for te tree UAs formaton system. Frstly, wt regard to a tree UAs formaton system wt bdrectonal network lnks. Secondly, wt regard to a sngle UA, t s decoupled te lnear system of te orzontal lateral drecton te vertcal drecton
to desgn te collson avodance algortm well. In Secton 3, we proposed te control metods for te problems defned n Secton 2, ncludng te leader-follower control sceme, te artfcal potental-based metod, te collson avodance scemes of desgn. Secton 4 presents expermental results to valdate te proposed control algortm to aceve te collson avodance effectvely, ncludng te setup ntal condton te analyss of smulaton expermental results. Fnally, te concludng remarks are stated n Secton 5. 2 Modelng a Mult-UA system Suppose tat tere are (=3) UAs wt te same moton caracterstcs, ncludng - followers a leader, wc s composed of a UA formaton system. Wt regard to a mult-ua system wt bdrectonal network lnks between te UAs, tat s, te between te any two UAs of te formaton system excange nformaton wt eac oter. s network can be matematcally descrbed usng grap teory. So te UA formaton system of te network topology, as sown n fgure : eader Follower Follower2 Fg. e UA formaton system of te network topology A grap G conssts of a set of te nonempty fnte sets G a set of te dsordered pars. Were s a set of nodes, s te order of grap G, E s a set of edges te dsordered number n E s usually expressed n te form of te lne to connect te two nodes, called edge. Also, s te number of sdes of E grap G [8]. In ts paper, we use a grap G, A to establs te nformaton nteracton relatonsp of between n UAs, were v, v2, v3,, v s a set of nodes, A s a set of edges. e edge v, v n E represents tat tere s a network pat from te UA to te UA, tat s, te UA can obtan use nformaton from te UA. A drected tree s a grap G, ncludng all of nodes. Every node acts as one parent node except for tself, called a root node. In a drected grap, te root node cannot be drectly connected to te parent node, can be drectly connected wt oter nodes. However, n te undrected grap, te edge v, v notes tat UA te UA nformaton wt eac oter, n oter words, v, v R are te same. et A R, can excange D v, v R be an adacency matrx, a degree matrx, a grap aplacan matrx. Were represents te relatonsp of between nodes n grap teory, based on te grap teory, te negbor matrx gven by a A A a, v, v A () 0, v, v A From te above equaton, f te UA s drectly obtanng nformaton from te UA, so,oterwse. e degree matrx D a a 0 represents an n-degree matrx gven by D dag(deg( v),deg( v2),,deg( v )) (2) were s te communcaton sum number of v deg( v ) node wt oter nodes. e grap aplacan matrx s defned as[3]: n n [ l ] R, l a From te equaton above, te matrx two propertes: () l 0,,, l a n s (3) as te followng l 0,,2, n Defnton n denotes a -dmensonal column vector of all elements tat are, column vector of all elements tat are 0. Accordng to te defnton of te aplace matrx, te followng equaton olds: n=0n[20]. (2) = D A, f a grap G as a drected spannng tree, te matrx as a sngle egenvalue at zero, all nonzero egenvalues of te grap aplacan ave postve real part. 0 n denotes an -dmensonal 3 Collson avodance algortm In ts secton, we furter study ow to avod obstacle for te formaton system based on te artfcal potental feld metod two knds of obstacle scemes to valdate te proposed control algortm, wc can aceve te obstacle avodance between te UAs. Amng at te obstacle avodance of te UA formaton system n te tree dmensonal space, te control algortm s proposed to effectvely avod obstacle n te vertcal drecton n te orzontal plane. In te paper, a consensus-based control algortm s appled to avod te collson n te vertcal drecton only to take evasve acton. However, te trangular formaton system can fly as te control comm, avod te collson between tem effectvely. s secton manly study ow to avod te collson about te obstacle of movng usng
te consensus-based algortm n te vertcal drecton. For formaton flgt n te vertcal drecton, te controlled obectves are rougly grouped nto two categores. e one category s tat te tree UAs consst of te formaton system as a control obectve, te oter s tat eac of te UAs act as a control obect. e control obectve cooperatve formaton flgt s realzed usng te consensus-based algortm wle keepng te geometrc confguraton of te formaton. However, eac of te UAs can aceve te formaton flgt usng te leader-follower control strategy. e leader provdes te drectly connected two followers wt ts own posture velocty to ensure tat te follower can track te leader keepng te trangle formaton. In te paper, te desred control algortm as an advantage n bdrectonal connecton between te UAs n topology, te nformaton transmsson between tem s smoot, wc prevent nformaton from cloggng duo to te overload of nformaton. e control law for UA s gven by ˆ k t a ˆ total k k0 ˆk k k,2,, k,,2,,, 0, (4) k (5) were te meanng of symbol denotes a leader, k R k 0, are control gans,, k R, k 0, are te state of te UA, k R, k 0, are te desred relatve state of between te UA te leader n te vertcal drecton. As sown n Eq. (), ndcates weter tere s nformaton acquston between te UA te UA, tat s, wen tere s nformaton acquston between te UAs, a a s set to one, oterwse a s set to zero. In te paper, te proposed control algortm for te UA formaton system wt collson avodance capablty s as follows: f f,2,, (6) were, from ca f s te control algortm for formaton n te from vertcal drecton, f ca s te obstacle avodance of te artfcal potental feld. eorem : Suppose tat te lnear model of te UA 2 UAs formaton system comprses a leader as expressed n Eq. (6) tat assumptons ()-(3) are satsfed. Also, te Eq. (6) wt te postve control gans k 0, s satsfed wt eac of te UAs,ten, k all of te states of te UAs n te vertcal drecton wll asymptotcally converge to te desred fgt states. Proof: By applyng control algortm (6), we can get ˆk ˆk a f k ca k0,2,, were we defne te new state vector k k k k R 2 0 ˆ ˆ ˆ ˆ 2 R k k k k k 0,, (7), ten te new force vector 0 fca f ca R. Usng new defned symbols, we can rewrte (3) as: ˆ0 ˆ f (8) o were te matrx s te grap laplacan of te multple UA system, te state vector conssts of te states of follower comms from te leader. In te paper, te followng denttes concernng te rows of te grap lapacan old, as follows n Eq. (9): ˆ k a a a a a 2 a 0,2,, k Accordng to te 0 k 0,, te Eq. (9), so te Eq. (8) s expressed as: 0 0 0 0 0M 0M 0M M (0) 0 0 0Md 0Md fca M R were te matrx vector k k k k k 2 ca d, te (9) s defned as (), te state are respectvely k k k R. Also s te kronecker product, R. Here, te matrx M s smlar to te grap laplacan, but not equal to te matrx. a a2 a a2 a2 a2 M () a a a Accordng to te matrx M, te Eq. (2) s smplfed n a matrx vector as: 0 0 d 0 I 0 + dt 0M M f ca (2) 0 0 0 I + 0 I d + 0M M + 0M M d were I s a -dmensonal unt matrx, R 0 R s a -dmensonal zero vector. o valdate te stablty of te Eq. (2), te omogeneous Eq. (8) s expressed as:
0 0 d 0 I 0 + dt 0M M (3) f ca Here, we construct te followng yapunov cdate functon, t conssts of te total energy of te mult-ua system. e expresson s as sown: 0 M U c (4) 2 2 e tme dervatve of te functon s gven by: M U M f U (5) 0 c ca c o avod obstacle well between te leader follower, artfcal force s proposed, as follow n Eq. (6): f U,2,, (6) ca c, From (22), we can get te Eq. (23): Uc Uc fca (7) Hence, from (2) (23), we can get M (8) e yapunov cdate functon represents te energy functon of te artfcal feld between te UA formaton system. roug te study of te functon, we can get te UA formaton system wt collson-avodance capablty to aceve te control of te ground staton effectvely te purpose of avodng obstacle. e grap G as a drected spannng tree f te assumptons ()-(3). Hence, from te property of te grap aplacan, as a sngle egenvalue at zero, te oterwse ave postve real part. Accordng to te property te relatonsp between te te M, we can obtan te matrx s postve defnte. In addton, te artfcal potental feld parameter s postve value, we can get Uc 0. erefore, wen te control gan are postve smultaneously, we get 0 0 k M, ten wen te control gan 0, 0 0 0 0 K are zero smultaneously, we get. And also, wen only Uc, we can also get. ote tat wen tere s no overlap of te safety regons between te UAs, we can get Uc 0. For te dervatve of te yapunov cdate functon, we can get 0 wen te control gan s postve M 0. We can get 0 f 0, 0 Uc 0, oterwse, 0. e asymptotc stablty can be solved by yapunov teorem nvolvng asalle s prncple[2]. Wen tere s no overlap n te safety regon, te partcular soluton of te Eq. (29) can be gven by k 0 0 0 0 + d + (9) + d s valdty can be confrmed by substtutng te Eq. (9) for te Eq. (2), so t s no need to explan t. Because te leader provde te desred comms of eac of followers, not te desred nput comms, we can use 2 + 0 d 2 0. ote tat 2 + 0 d 2 0 are te nput of te leader te nput error of between te leader te follower. In concluson, te general soluton of te non-omogeneous dfferental te Eq. (3) s te sum of between te general soluton of te omogeneous equaton te partcular soluton. Hence, te general soluton of Eq. (3) asymptotcally converges to te (20) wen control gans k 0, te artfcal, k potental feld parameter are selected to be small postve. 0 0 0 + d +, (20) + d From te (20), we can get te result of convergence to te comm from te leader. Accordng to te element of te frst row block n (9), t s proved tat every UA as te capablty of te collson avodance can asymptotcally converge to te desred traectory for te trangle formaton flgt. k t 4 Smulaton experment analyss In ts secton, we valdate te proposed te consensus-based control algortm usng te smulaton experment for applcaton to a mult-ua platform. k 5.5 4. e setup ntal condton e relatve te ground velocty of te UA te movng velocty of te obstacle are 56 m/s, te mass of eac of te AUs s 90kg, te ptc rate lmt for UA s 0deg/s, te yaw rate lmt for UA s 2 deg/s. o satsfy te buldng model smulaton of all of te condton n vertcal drecton, we set up te control gans 0.5, 3 te control parameter for collson k 5.5. And te alttude of safety regon for every UA s 3m, te radus of safety of regon s 3.5m, te alttude gap of between tem s no more tan.2m, te relatve dstance of between tem s no more tan 0m. In order to valdate te proposed control algortm, te smulaton experments can be grouped nto two categores, te one s carred out wtout addng te collson avodance algortm, te oter s carred out wt addng te collson avodance algortm. e UA cooperatve formaton flgt control algortm as been publsed n te lterature, as a good effect on te collson avodance [8]. By te comparson analyss of two group smulaton experments, te proposed control algortm can not only mprove te control metod of publsed lterature, but aceve te effect of te collson avodance well. 4.2 Smulaton results Accordng to te ntal condtons assumptons of te smulaton, we sow te smulaton results, as sown n fg.2 to 4. e fg. (a) (b) of te followng fgures represent respectvely te smulaton result of te wtout
te control algortm te smulaton result of addng te control algortm. e UA rate of clmb(m/s) (a)e tme response wtout collson avodance algortm e UA rate of clmb(m/s) Artfcal potental force() Artfcal potental force() 5 5 (b) e tme response wt collson avodance algortm Fg. 2 e rate of clmb of te UA (b) e tme response wt collson avodance algortm Fg. 3 e rate of clmb of te UA otal rust comm for collson avodance() (a)e tme response wtout collson avodance Algortm 5 (a)e tme response wtout collson avodance Algortm otalrust comm for collson avodance() (b) e tme response wt collson avodance algortm Fg. 4 e rate of clmb of te UA e fg. 2(a) sows tat te rate of clmb as te same of between te two followers a leader from s to 2s, te trend of curves decreases frstly ten ncreases slowly, eventually remans constant value, n oter words, te tree UAs keep te trangle
formaton flgt. As we can be seen from fg. 2(b), te two curves bot present a contnuous oscllaton trend from 3s to 30s, eventually converge to te constant value after 30s. e UA formaton system takes evasve maneuver f te UA formaton system encounter te obstacle, ten recover te trangle formaton flgt mmedately. e fg. 3 (a) sows te wole process of te UA formaton system, ncludng from takng off to te assembly formaton, troug te encountered te obstacle to te loose formaton, ten te assembly trangular formaton. e curve of te leader presents a trend of te decrease frstly ten ncrease slowly, owever, te trend of te two followers s opposte from s to 30s. Because te relatve dstance of between te followers a leader s ncrease f te UA formaton system encountered te obstacle. And also, te relatve dstance decrease frstly ten keep constant, wle te artfcal potental feld presents a trend of te decrease frstly ten ncrease slowly. From te fg. 3 (b), we can obtan tat te desred constant value of artfcal potental feld of between te two followers a leader s 45. By addng te collson avodance algortm, te relatve dstance of any two UAs n te formaton remans constant, te artfcal potental force of between te two followers te a leader are equal to 5, wc can aceve te purpose of te collson avodance. o ensure te UA formaton as te geometrc confguraton of te trangle te desred traectory flgt, te total trust of every UA provded s equal ter drecton s consstent, so te curve presents a orzontal trend. From te fg. 4(b), te tree UAs cooperatvely flgt from s to 30s, te UA encounter te obstacle after 30s, te curve presents an oscllaton dvergence trend. Wen te UA formaton encounter te obstacle avod te collson successfully, te relatve dstance of te nsde of follower n te formaton te movng obstacle, presents a trend of te ncrease frstly ten decrease slowly. At te same tme, te artfcal potental force presents a trend of te decrease frstly ten ncrease slowly. Based on te above analyss, te proposed control strateges can effectvely avod te collson between te UAs, between te UA formaton system te obstacle troug te comparatve analyss of smulaton experment, can aceve te purpose of te collson avodance n teory. e man deas of collson avodance can be dvded nto two categores. e one s tat te collson avodance algortm combned wt artfcal potental feld s appled to avod te collson n te vertcal drecton, te oter s tat te artfcal potental feld s appled to avod te collson n te orzontal plane. 5 Concluson future researc Amng at te tree UAs formaton system te obstacle as te controlled obectves, a lnear matematcal model for te UA formaton system s bult by usng te grap teory, te UA formaton cooperatve control te collson avodance algortm s proposed n te tree dmensonal space. yapunov teory was appled to analyze te stablty of te proposed for te collson avodance. e smulaton experments are carred out to valdate te convergence valdty of te desred control algortm under te condton of buldng model smulaton ypotess. e mproved control algortm of te artfcal potental feld leader-follower control strategy s appled to te complcated collson avodance n te tree dmensonal space, t can be smplfed as te collson avodance control n te orzontal plane te control n vertcal drecton. e relatve dstance of between any two UAs n te formaton s less tan n te safety regon n te vertcal drecton, t must take evasve maneuver to avod te collson. 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