Neural Network Adaptive Control for Underwater Robotic Systems

Transcription

1 Neual Netok Aaptve Contol fo Uneate Robotc Systems V.S. KODOGIANNIS Mechatoncs Goup, Dept of Compute Scence Unvesty of Westmnste Lonon, HA 3P UNIED KINGDOM Abstact: Neual netoks ae cuently fnng pactcal applcatons, angng fom soft egulatoy contol n consume poucts to accuate moellng of non-lnea systems. hs pape escbes the applcaton of neual netoks to the contol of a emotely opeate uneate vehcle, as an example of a system contanng sevee nonlneates. Neual netoks ae been use n a close-loop to appoxmate the nonlnea vehcle ynamcs. No po off-lne tanng phase an no explct knolege of the stuctue of the vehcle ae eque, an the popose scheme explots the avantages of both neual netok contol an aaptve contol. A contol la an a stable on-lne aaptve la ae eve usng the Lyapunov theoy, an the convegence of the tackng eo to zeo an the boune-ness of sgnals ae guaantee. In ths pape, a neual netok achtectue base on aal bass functons has been use to evaluate the pefomance of the aaptve contolle va compute smulaton. Keyos: neual netoks, uneate vehcles, stablty theoy, aaptve contol. Intoucton he Ocean coves ove sxty pecent of the eath s suface, yet humans have haly been able to fully exploe the epths. Unoubtely th the ecent mages captue by the Hubble Space elescope, the Russan M Space Staton an NASA s Galleo atmosphee pobe mankn knos moe about space o the oute sola system than about the Oceans. Hoeve, n the past ecae, Oceanogaphc exploaton has emege as one of the fastest gong aeas of eseach attactng huge gants. he man eason s the vast amount of mneal esouces that satelltes have mappe acoss the Oceans, not fogettng the ol an natual gas eseves. he esgn of contolles fo unmanne uneate vehcles (UUV) s challengng because of ffcultes n accuately moellng the nheently nonlnea ynamcs of UUVs n a hazaous envonment th pesstent unmoelle stubances. In geneal, eal-tme contol of non-lnea systems th unknon stuctue an paamete uncetanty emans an open aea of eseach. Dynamc moels of UUVs ae eque to esgn avance contol systems, an moels of uneate vehcles have been stue n the past []. he nonlnea ynamcs of UUVs esult n paametc an stuctual uncetantes n the ynamc moel, an ths necesstates the nee fo avance obust contol technques. Contol stateges that aess some of moellng chaactestcs have been epote n the lteatue, nclung lnea contol, obust contol, fuzzy contol, neual netoks, slng moe contol, etc. Among vaous contol technques, slng moe contol has been successfully mplemente an teste fo uneate vehcles []. Fossen et al. [3] evelope an aaptve contolle fo uneate vehcles n 6-DOF by assumng that ts ynamcs can be lnealy paametese. he emegence of neual netoks (NNs) as effectve leanng systems fo a e vaety of applcatons has esulte n the use of these netoks as leanng moels fo ynamcal systems. NN contolles have mpotant featues that ovecome the typcal ffcultes n esgnng contol systems fo uneate vehcles. Fo nstance, the ynamcs of the vehcle nee not be completely knon as a po conton fo contolle esgn. hs s vey esable fo uneate vehcle contolle to have snce the ynamc chaactestcs of uneate vehcles change th confguaton an t s mpossble to conse all the effects fom stubances. Also, the ablty of these netoks fo aaptaton an stubance ejecton an the hghly paallel natue of computaton make ths appoach sutable fo eal-tme applcatons. A NN base contol scheme fo UUV s escbe by Venugopal et al. [4] usng ect contol scheme, hee the nput to ynamcs s mplctly use fo both entfcaton an contol smultaneously. A smla ea to the above s use by Yuh [5] n hs stues on the NN-base contol scheme fo an UUV. Fuj et al. [6] popose a self-oganzng neual netok base contol system to the evelopment of the moton contol fo autonomous UUV. Kooganns et al. use seveal ffeent neual netok achtectues to evaluate a long-ange moel pectve contol scheme both fo smulaton an on-lne contol of vehcle epth [7]. An altenatve appoach to soft-computng technques s the mplementaton of appoxmate contolles base on fuzzy logc theoy. A ne fameok n slng moe fuzzy contol as pesente by eb-ollenu et al. fo selectng fee contol paametes of an nput-output lneasng contolle th slng moe contol fo the epth

2 contol of a emotely opeate uneate vehcle (ROV) [8]. In the appoach, the concept of multobjectve fuzzy genetc algothm optmsaton as aopte an a ne membeshp eghtng stategy as suggeste. In ths pape, the evelopment of a ect aaptve neual netok contolle fo uneate vehcles s popose, th a paallel nvestgaton of the pefomance an obustness ssues of the aopte contolle. Raal bass functons netoks (RBF) ae use to appoxmate the nonlnea ynamcs of uneate vehcles thout explct knolege of the plant s ynamc stuctue. he onlne eght aaptaton la of the neual netok s eve n the context of Lyapunov stablty concept. Boune-ness of all sgnals as ell as the convegence of the tackng eos to zeo ae guaantee. he contbuton of ths pape s to combne aaptve contol th neual netok achtectues to appoxmate the nonlnea an tme-vayng uneate vehcle ynamcs. ackng pefomances an obustness of the popose contolle s emonstate though compute smulaton. Vehcle Moellng he ynamc equatons of moton of uneate vehcles have been analytcally pesente n the lteatue [9]. In ths pape a nonlnea sx-egee-offeeom moel base on Fossen et al. [0] has been consee. he g boy uneate vehcle moel n the boy-fxe efeence fame can be epesente as M ν + C( ν ) ν + D( ν ) ν + g( = τ () = J ( ν () hee ν = [ u, υ, ω, p, q, ], = [ X, Y, Z, φ, θ, ψ ].In ths notaton, í enotes the lnea an angula velocty vecto th coonates n the boy-fxe efeence fame, ç enotes the poston an atttue vecto th coonates n the eath-fxe efeence fame, an ô s use to escbe the contol nput foces an moments actng on the vehcle n the boy-fxe efeence fame. he boy-fxe velocty vecto can be tansfome nto the eath-fxe efeence fame though the Eule angle tansfomaton enote by J (. M s the neta matx nclung ae mass M A, C(ν ) s the matx of Cools an centfugal tems, D (ν ) s the ampng matx, an g( ) s the vecto of gavtatonal foces an moments. he equaton of moton of uneate vehcle can be epesente n the eath-fxe efeence fame n tems of poston an atttue though the knematc tansfomatons (assumng that J ( ) s non-sngula) = J ( ν ν = J ( (3) ( ν ( ν ν ( [ = J + J = J J ( J ( ] (4) to elmnate ν an ν fom Eq.. Defnng M ( = J ( MJ ( (5) C ( ν, = J ( [ C( ν ) MJ ( J ( ] J ( (6) D ( ν, = J ( DJ g ( (7) ( = J ( g( (8) τ ( = J ( τ (9) yels the eath-fxe vecto epesentaton M ( ) + C ( ν, + D ( ν, + g ( = τ (0) he neta matx M ( ), nclung hyoynamc ae neta, s symmetc an postve efnte. he Cools an centfugal matx, also nclung ae neta effect, C ( ν,, satsfes the ske symmetc elatonshp x [ M ( C ( ν, ] x = 0. he ampng matx D ( ν, s postve efnte. A moe etale scusson on mathematcal moels of uneate vehcles can be foun n [3]. 3 Neual Netok Contolle Desgn Due to the ffculty obtanng the exact values of hyoynamc coeffcents, but also ue to the fact the coeffcents change th the confguaton of uneate vehcles, the obustness an aaptveness ae mpotant equements fo the uneate vehcle contolles. he stubances fom cuents an aves ae also vey ffcult to moel. Fossen et al. [0] eve an aaptve contol la fo uneate vehcles n 6 DOF assumng that M, C (ν ), D (ν ), an g ( ) ae lnea n the paametes an that the ynamcs can be lnealy paametese, hch s a common assumpton n the aaptve contol. he lnea paametesaton n the aaptve contol s usually val an can conse the changes an uncetantes n paametes. Hoeve, t shoul be note that ths assumpton conses the change only n paametes; the unstuctue o un-moelle ynamcs such as stubances fom cuents an aves cannot be lnealy paametese. In ths pape, an aaptve contol la s eve that oes not eque off-lne tanng. An RBF netok s use n the appoxmaton of a nonlnea functon, assumng that the nonlnea moel of the uneate vehcles s unknon. 3. Contolle Specfcatons A cetan measue of eo s efne as s = + λ () hee ë s any postve constant an =. he ese poston an atttue of the vehcle enote by, an the tme evatve, can be obtane fom a tajectoy planne. he efeence moel s

3 chosen conseng the vehcle knematcs as n [3]. he ese paametes ν an ae compute fom ν + Λν + J ( ) Ω = J ( ) Ω () = J ( ν ) (3) hee s a constant (o sloly vayng) commane nput. he esgn paametes n the efeence moel ae the matces Ë > 0 an Ù = Ù > 0 escbng the pefee ampng an stffness of the system. Ë an Ù ae usually chosen as agonal matces th postve entes on the agonal. Fo notatonal smplcty, t s convenent to ete Eq. n tems of the vtual efeence tajectoy efne as s = = λ (4) he efeence tajectoy n the boy-fxe fame can be eve = J ( ν (5) ν J ( [ J = ( J ( ] (6) No, the Eq. 0 escbng the nonlnea equaton of moton of an uneate vehcle n 6 DOF n the eath-fxe efeence coonate s consee. akng the evatve of s th espect to tme, the vehcle ynamcs can be tten n tems of s as M s = M M = ( D + C ) s + J [ τ ( Mν + C( ν ) ν + D( ν ) ν + g( )] (7) Çee, the elatonshp M + C + D + g = (8) J ( Mν + Cν + Dν + g) s use. Denotng the ynamcs of vehcle n the boyfxe efeence coonate as M ν + C( ν ) ν + D( ν ) ν + g( = f ( ν, ν, ν, (9) an takng a contol nput as τ = fˆ ( ν, ν, ν, J K s, the close-loop system becomes M s = ( D + K ) s C s + J [ fˆ( ν, ν, ν, ε] (0) + hee f ˆ( ν, ν, ν, s the estmate of the vehcle ynamcs f ( ν, ν, ν,, K s a symmetc postve egulato gan matx of appopate menson an å s use to enote the appoxmaton eo. Eq. 0 s an eo ynamcs hee the fltee tackng eo s ven by the functonal estmaton eo. Follong the appoach n [], the tackng poblem can thus be solve by fnng a aaptaton la fo ajustng f ˆ( ν, ν, ν, that ensues the boune-ness of the paamete estmates an that s ( t) 0 as t. 3. Lnea Paametesaton Conseng the Eq. (8), Eq. (9) can be paametese as M ν + C ν ) ν + D( ν ) ν + g( = Φ( ν, ν, ν, θ () ( assumng that M, C (ν ), D (ν ), an g ( ) ae lnea n the paametes. Hee, è s an unknon paamete vecto an Φ( ν, ν, ν, s a knon matx functon of measue sgnals usually efee to as egesso matx. Conse a Lyapunov functon as V = s M s + θ Γ θ () hee Γ s a postve efnte eghtng matx of appopate menson an θ = θ ˆ θ s the paamete estmaton eo. Dffeentatng V th espect to tme yels V = ( s M s + s M s + s M s) + θ Γ θ (3) Usng the symmety of neta matx s M s = s M s an the ske-symmetc popety s ( M C ( ν, ) s = 0, Eq. (3) becomes V = s ( M s + C s) + θ Γ θ (4) Conseng Eq. (7), Eq. (4) becomes V = s D s + [ J ( s] [ τ Φ( ν, ν, ν, θ ] + θ Γ (5) Let the contol nput be chosen as τ = Φ( ν, ν, ν, θˆ J ( K s (6) ) hee θˆ s the estmate paamete vecto an K s a symmetc postve egulato gan matx of appopate menson. Hence, V = s ( K + D ) s + θ [ Φ ( ν, ν, ν, J ( s + Γ θ ] (7) hen the paamete upate la th the assumpton of constant paametes ( θ = 0) θ = ΓΦ ( ν, ν, ν, J ( ) s (8) cancels out the last tem n the expesson fo V an yels V = s ( K + D ) s 0 (9) hen, the convegence of s to zeo an the bouneness of the paametes can be shon by applyng Babalat s lemma [] as follos. Snce V s negatve o zeo an V s postve efnte, V tens to a constant as t. Conseng that the neta matx M s postve efnte an s s boune, the estmaton eos θ can be shon to be boune, consequently θˆ s boune. hs makes s s boune shong that s s unfomly contnuous. heefoe, applyng Babalat s lemma [], t can be shon that V 0 an s 0 as t. he convegence of s to zeo mples the convegence of the tackng eo an to zeo snce the tackng eo s ven by s though a stable ynamcs as shon n Eq. (). θ

4 4 Neual Netok Moellng he evaton of the contol la an the aaptaton la of the eghts of the RBF netok ae consee n ths secton. hs neual netok shon n Fg. conssts of one laye of sees of neuons multple by output eghts. Such system s employe snce t s knon that a lnea supeposton of aal bass functons s the optmal soluton to a class of functon n appoxmatons gven a fnte set of ata n R [3]. Also, the elatvely smple netok stuctue enables the easy evaton of an aaptve netok upate la. he mathematcal epesentaton of the neual netok s N f ( x) = γ ) =,...,n (30) hee j= j j j γ j = exp[ x j / σ j s the nonlnea functon at noe j, taken as a Gaussan functon of the nne pouct of ts aguments. he coeffcents Fg. : Stuctue of RBF netok j epesent the cente of aal Gaussan, σ j s a measue of ts th at noe j, an j epesents the output eght fo that noe. he numbe of egee-of-feeom s enote by n, an N s the numbe of employe neuons. It s assume that thee exsts a cetan combnaton of optmal eghts of the netok that poves the satsfactoy appoxmaton of the nonlnea mappng applyng enough numbe of neuons. he Lyapunov functon canate s chosen as { V = s M s + t Γ (3) hee Γ s a postve efnte eghtng matx of appopate menson, an = ˆ s the estmaton eo of the netok output eghts. Hee, ŵ s the output eght estmate an s the optmal eght. Dffeentatng V th espect to tme yels V ( s M s s M s s M s) t{ = Γ (3) Usng the facts that s M s = s M s, the ske symmety of s ( M C ) s = 0, an the close loop system Eq. (0), Eq. (3) can be etten as V = s = s ( M s + C s) + t{ Γ D s + ( J s) [ τ f ( ν, ν, ν, ] + t{ Γ (33) he aal bass neual netok s use as an appoxmaton of the ynamcs of the plant as f ( ν, ν, ν, = ˆ γ ) + ε ( x) (34) hee ε (x) enotes the appoxmaton eo an t s assume to be boune as ε( x ) ε 0 fo x Ω hee s the oman of appoxmaton. he boun ε 0 on the appoxmaton eo can be mae smalle by selectng a lage numbe of neuons n the hen laye, an t s assume that t can be neglecte as long as enough numbe of neuons s aopte. Choosng the contol nput as τ = ˆ γ ) J ( K s (35) an assumng that the output of aal bass neual netok appoxmates the functon th suffcent pecson as long as th oman of appoxmaton s completely covee, Eq. (33) becomes ( ) ( ) [ (, )] { V = s D + K s + J s γ x + t Γ s ( D + K ) s + t{ [ γ )( J s) + Γ (36) Also, choosng the aaptve la of the netok eght as ˆ = Γγ )( J s) (37) the Eq.(33) can be shon to be negatve sem-efnte, hch mples the convegence of s to zeo an the boune-ness of applyng Babalat s lemma []. In summay, the contol la s gven by Eq. (35) an the aaptaton la s gven by Eq. (37). It shoul be note that the appoxmaton of the nonlnea functon Eq. (34) can also be expesse as the pouct of egesso matx an unknon paametes une the assumpton that the nonlnea functon s lnea n the paametes an the ynamc stuctue of vehcles s completely knon. he neual netok appoxmaton oes not eque such assumpton snce no explct knolege of the ynamc stuctue s eque. he achtectue of contolle s shon n Fg.. Fg. : Popose contolle achtectue ]

5 5 ROV Case Stuy he computatonal stuy n ths pape as base on the moel stuctue of the Noegan Expemental Remotely Opeate Vehcle (NEROV) [4]. he vehcle s contolle n all 6 DOF by sx c-moto ven thustes. he flu velocty as chosen to be zeo n all computatons. he ese veloctes an postons ee geneate by a efeence tajectoy geneato. he smulaton, usng the evelope contolle, ee pefome at 5Hz, hch mples all the measuements an the contol acton occue at a tme step of 0. secons. he tackng pefomance of the X an Y postons, the epth an the heang angle as consee n ths stuy. he cente of gavty as assume to be locate at the ogn of the vehcle coonate system. he neta matx as assume to be agonal: M ag{ m X I y q, I z, m Y, m Z = u υ ω M he mass s N, I x K p, (38) m = 85 Kg. he moments of neta aoun the x-, y- an z-axes ae I y = 50 kgm an I x = 50 kgm I x = 5 kgm,. he hyoynamc ae netas ae X u = 30 kg, Y υ = 80 kg, Z = 80 kg, ω K p = 5 kgm, M q = 30 kgm, N = 30 kgm. he C matx s assume to be as 0 m mq 0 Zω Yυ m 0 mp Zω 0 Xu u mq mp 0 Y = υ Xu u 0 C (39) 0 Zω Yυ 0 ( Iz N ) ( Iy Mq ) q Zω 0 Xu u ( Iz N ) 0 ( Ix Kp ) p Y υ Xu u 0 ( Iy Mq ) q ( Ix Kp ) p 0 he ag matx s assume to be agonal D( ν ) = ag{ u, υ, (40) ω, p, q, buoyancy. One RBF netok as use to appoxmate the nonlnea ynamcs of NEROV. In ths patcula case, the centes of Gaussan functons γ ) ee unfomly space n the state space. he th of the Gaussan functon γ ) as set to 30, an the oveall eghts of a neual netok ee ntally set to zeo. Only a sngle MIMO neual netok as employe th 4 nput an 6 output unts. he aaptve gan matx as set to 0.3I, hee I as the entty matx of appopate menson. he choce of the th of the Gaussan functon γ ) as the most ctcal facto fo oveall stablty of the system an elate to the choce of the numbe of Gaussan functons ove the state space. In othe os, the optmal th of the Gaussan functon shoul be foun conseng the th of an aea n the nput space to each neuon espons. he value of γ ) shoul be lage enough that neuons espon to enough ovelappng egons of the nput space. he tackng pefomance of the X an Y postons of the vehcle follong the specfe poston s shon n Fg. 3. It can be seen that the popose neual-netok contolle gves faly goo tackng pefomance. In ths smulaton stuy, only the neual netok contolle as use to etemne the feasblty of usng the popose neual netok achtectue. Of couse, mpove an moe obust tackng pefomance coul be acheve hen the PD an the aaptve contol nputs ae combne togethe. Bette pefomance may be obtane by futhe tunng the upate gan an nceasng the numbe of RBF centes. Hghe upate gan gave bette tackng pefomance, but hen the gan as too hgh, oscllatoy behavou as obseve. By combnng th the PD pat of the contolle, the unante oscllatoy moton coul be emove at the pce of slght ncease n the contol effot. Contol nputs ae shon n Fg. 4. Fg. 3: X an Y tajectoy (sol: ese; ashe: popose contolle) he cente of buoyancy as assume to be locate at ( x B, y B, z B ), hee x B =y B =0 an z B =0.0m s the x-, y- an z-coonates n the vehcle coonate system. he system as assume to be neutally buoyant g=(0,0,0, z B Bcèsö, z B Bsè,0), hee B=800N s the Fg. 4: Contol nputs to thustes Robustness of the popose contolle to the unmoelle stubance s shon n the Fg. 5. A snusoal stubance s ae n the ya channel. he pefomance of aaptve neual netok contolle

6 th aal bass neual netok s compae to the aaptve contolle th lnea paametesaton. A obust tackng pefomance has been acheve usng aaptve neual netok, heeas the aaptve contolle th lnea paametesaton gves egaaton n the pefomance. Fg. 5: Robustness test to the unmoelle stubances n the ya channel (sol: ese, ashe: popose, ashot: lnealy paametese) 6 Conclusons An aaptve neual netok contolle has been evelope fo an uneate vehcle n 6 DOF. One RBF netok as use to appoxmate the nonlnea ynamcs of uneate vehcle. Wthout explct po knolege of the vehcle ynamcs, the popose contol technque coul acheve mpove tackng pefomance. Results have shoe that the ynamcs of the vehcle nee not be knon explctly fo the esgn of the contolle an no lneasaton s eque to eal th nonlnea vehcle ynamcs. he popose contolle achtectue s obust an aaptve, hle t oes not nee any po tanng phase an can be apple on-lne. he next stage of ths stuy s to apply the popose aaptve contolle n the une evelopment UUV fune fom the FREESUB 5 th fameok EU poject. Refeences [] Yoege D.R. Slotne J., Robust tajectoy contol of uneate vehcles, IEEE Jounal of Oceanc Engneeng, Vol., No. 4, 985, pp [] Song F. Smth S., Desgn of slng moe fuzzy contolles fo an autonomous uneate vehcle thout system moel, Oceans 000 IEEE Conf., Vol., 000, pp [3] Fossen.I, Guance an contol of ocean vehcles, John Wley an Sons, 994. [4] Venugopal K.P., R. Suhaka R. Panya A.S., On-lne leanng contol of autonomous uneate vehcles usng feefoa neual netoks. IEEE J. of Oceanc Engneeng, Vol. 7, No. 4, 99, pp [5] J. Yuh J., Leanng contol fo uneate obotc vehcles, IEEE Contol Systems Magazne, Vol. 5, No., 994, pp [6] Fuj,., Ish K. Ua., Neual netok system fo on-lne contolle aaptaton an ts applcaton to uneate obot, Poc. 998 IEEE Conf. on obotcs an automaton, Belgum 998, pp [7] Kooganns V.S., Lsboa P.J.G., Lucas J., Neual netok moelng an contol fo uneate vehcles, Atfcal Intellgence n Engneeng, Vol., 996, pp.03-. [8] eb-ollennu A. Whte B.A, Multobjectve fuzzy genetc algothm optmsaton appoach to monto contol system esgn, IEE Poc. Contol heoy Appl., Vol. 44, No, 997, pp [9] Fossen.I., Sagatun S.I., Søensen A., Ientfcaton of ynamcally postone shps, IFAC Wokshop on Contol Applcatons n Mane Systems (CAMS 95), Noay, 995, pp [0] Fossen.I., Sagatun S.I., Aaptve contol of nonlnea systems: A case stuy of uneate obotc systems. Jounal of Robotc Systems, Vol. 8, No. 3, 99, pp [] Sanne R.M. Slotne J.J., Gaussan netoks fo ect aaptve contol, IEEE ansactons on Neual Netoks, Vol. 3, No.6, 99, pp [] Slotne J.J. L W., Apple Nonlnea Contol, Pentce Hall, 99 [3] Chen, S., Chang, E.S., Alkahm, K., Regulaze Othogonal Least Squaes Algothm fo Constuctng Raal Bass Functon Netoks, Int. J. Contol, Vol.64, No. 5, 996, pp [4] Spangelo I Egelan O, ajectoy plannng an collson avoance fo uneate vehcles usng optmal contol, IEEE Jounal of Oceanc Engneeng, Vol. 9, No. 4, 994, 50-5.