Reactive execution for solving plan failures in planning control applications

In eg a ed Compu e -Aided Enginee ing 22 (2015) 343–360 343 DOI 10.3233/ICA-150493 IOS P ess Reac i e execu ion o sol ing plan ailu es in planning con ol applica ions Cesa Guzmana, Pablo Cas ejona, E a Onaindiaa,∗and Je emy F ankb aUni e si a Poli ècnica de València, Valencia, Spain bNASA Ames Resea ch Cen e , Mo e Field, CA, USA Abs ac . We p esen a no el eac i e execu ion model o planning con ol applica ions which epai s plan ailu es a un ime. Ou p oposal is a domain-independen eg ession planning model which p o ides good-quali y esponses in a imely ashion. The use o a eg essed model allows us o wo k exclusi ely wi h he su icien and necessa y in o ma ion o deal wi h he plan ailu e. The model pe o ms a ime-bounded p ocess ha con inuously ope a e on he plan o eco e om incoming ailu es. This p ocess gua an ees he e always exis s a plan epai o a plan ailu e a any ime. The model is es ed on a simula ion o a eal-wo ld plane a y space mission and on a well-known ehicle ou ing p oblem. Keywo ds: Reac i e planning, dynamic execu ion, moni o ing plan execu ion, eac i e execu ion agen , unp edic able en i on- men 1. In oduc ion The applica ion o A i icial In elligence AI plan- ning echniques is helping indus ies o imp o e hei e iciency and pe o mance in a g ea a ie y o ap- plica ions: manu ac u ing and elecommunica ion ne - wo ks [31,37], educa ion [20], ou e planning [10,41], mili a y and ci ilian coali ion ope a ions [36], space explo a ion [9], e c. In gene al, e en hough much o he esea ch on AI planning is aimed a gene a ing domain-independen planning echnology, he applica ion o planning o in- dus y gi es ise o special-pu pose sys ems, which a e expensi e o ex end o o he cases. On he o he hand, he e exis ew sys ems ha in eg a e au oma ed plan- ning and plan execu ion and his is, pe haps, one o he main causes o he ela i ely low deploymen o au o- ma ed planning applica ions [22]. While he p ima y ocus o planning is on delibe a i e ools o calcula e plans ha achie e ope a ion goals, he ocus o execu- ∗Co esponding au ho : E a Onaindia, Depa men de Sis emas In o má icos y Compu ación, Uni e si a Poli ècnica de València, Spain. Tel.: +34 963 877 755; Fax: +34 963 877 359; E-mail: [email p o ec ed].es. ion is on de eloping con ol me hods o e ela i ely sho ime spans o ensu e he plan ac ions a e execu ed s ably [3]. Mos planning and execu ion (P&E) sys ems ollow an in eg a ed app oach in which he execu ion moni o - ing sys em is in eg a ed wi h he planne o ice e sa. Sys ems like IXTET-EXEC [29] o TPOPEXEC [46] wo k unde a con inual planning app oach [6], in e - lea ing planning and execu ion in a wo ld unde con- inual change. Ano he example can be ound in [9], whe e he planne o a spacec a con inuously ope - a es on he plan execu ion o epai ailu es. IDEA (In- elligen Dis ibu ed Execu ion A chi ec u e) [1] is a eal- ime a chi ec u e ha p oposes a uni ied iew o delibe a ion and execu ion whe e he planne is em- bedded wi hin he execu o ; and T-REX [32](Teleo- Reac i e EXecu i e) is a delibe a i e P&E sys em o AUV con ol inspi ed om IDEA. This ype o uni ied app oaches allows only o a s ic and con olled in e - lea ing o P&E, making i di icul o ha e a gene al- pu pose planne o di e en ypes o execu o sys- ems. Some o he a o emen ioned P&E sys ems [29,46], deal wi h empo al plans and ocus on a uni ied ap- p oach ha accommoda es lexible plan execu ions, bu ISSN 1069-2509/15/$35.00 c 2015 – IOS P ess and he au ho (s). All igh s ese ed This a icle is published online wi h Open Access and dis ibu ed unde he e ms o he C ea i e Commons A ibu ion Non-Comme cial License. 344 C. Guzman e al. / Reac i e execu ion o sol ing plan ailu es in planning con ol applica ions hey a e no conce ned wi h p o iding esponses in a imely ashion. In con as , he wo ks in [1,32], besides gene a ing plans o e ela i ely long ime pe iods, hey also in oduce a eac i e planne o allow obus pe - o mance in dynamic en i onmen s. The e m eac i e planning has been app oached om di e en pe spec- i es [8]: –Responding e y quickly o changes in he en i- onmen h ough a eac i e plan lib a y ha s o es he bes cou se o ac ion o each possible con in- gency. –Choosing he immedia e nex ac ion on he basis o he cu en con ex ; in his case, a delibe a i e p ocedu e can be used o speci y he nex ac . –Using mo e complex cons uc s in o de o handle execu ion ailu es o en i onmen al changes. The i s app oach, used by ea ly P&E sys ems, im- plies s o ing a plan o each possible s a e o he wo ld, an op ion which is no a o dable in highly dynamic en i onmen s. Hie a chical con ol s uc u es p o ide a mechanism o choose he immedia e nex ac ion when a quick esponse is equi ed in unp edic able en i- onmen s [7]. This is he app oach ollowed by he models ha emphasize eac i eness, compu ing jus one nex ac ion in e e y ins an , based on he cu - en con ex [32]. The use o mo e complex s uc- u es allow o conside mo e delibe a i e (long ho i- zon) esponses a he han sho - e m eac i eness bu , in p ac ice, none o hese eac i e amewo ks ha e e e exploi ed he idea o p o iding quick delibe a i e esponses. They eac o a change o ailu e bu hey do no gua an ee a esponse wi hin a limi ed ime. A key aspec o eac i e planning is ha i is no only abou p o iding quick esponses o changes bu also p ese ing he plan s abili y [18] and gua an ee ha he ope a ion goals achie ed by he plan a e s ill eachable a e he plan epai . In con as o con ol applica ions ha use condi ion moni o ing o an icipa ing and e- ac ing o aul s [4,40], eac i e planning is abou e- pai ing a aul when is de ec ed while ying o main- ain he execu abili y o he es o he plan. The ocus o his wo k is on he de elopmen o a eac i e P&E sys em capable o p o ide as delibe - a i e esponses o epai a ailed ac ion wi hou ex- plici ly ep esen ing con ingency b anches and eligible plans o each possible s a e o he wo ld. Ou sys em does no simply e u n he immedia e nex execu able ac ion bu i ope a es o e a planning ho izon ha is longe han he minimum la ency in e al s a ing a he cu en execu ion ime. This idea was exposed, hough ne e exploi ed,in IDEA [13], which in p ac ice wo ks wi h he minimal planning ho izon in o de o educe he eac i e planne ’s sea ch space; ha is, he sho e he planning ho izon, he mo e eac i i y, bu also he less con ex ual in o ma ion o epai he ailu e. The idea o planning ho izon has been also exploi ed in non- eac i e dynamic planning applica ions [47]. In his wo k, we p opose a no el eac i e P&E model ha keeps ack o he execu ion o a plan and epai s he incoming ailu es. The execu o - epai ing sys em inco po a es a eac i e planning p ocedu e which is exclusi ely used o plan epai and i is inde- penden o he delibe a i e planne ha compu es he solu ion plan o he p oblem. Unlike he in eg a ed P&E app oaches men ioned be o e, ou s is a highly modula and econ igu able P&E a chi ec u e. The e- ac i e planne is specialized in small plan epai s ha mus be accomplished p omp ly. Addi ionally, i p e- compu es an ini ial sea ch space which encodes solu- ion plans o po en ial ailu es in a agmen o he plan named plan window (we use his e m equi a- len ly o he concep o planning ho izon in IDEA [1]). The cons uc ion o he sea ch space is a ime-bounded p ocess subjec o he agen ’s execu ion la ency and he numbe o execu ion cycles in he plan window, whose objec i e is o ha e a solu ion plan a ailable when a ailu e a ises. Once he sea ch space is buil , he agen p oceeds wi h he plan execu ion, and simul aneously he eac i e planne compu es he sea ch space o he nex plan window. The eac i e model is hus capable o deduce s ic limi s o he leng h o he plan win- dow and compu e he la ges sea ch space wi hin he ime limi . Then, i a ailu e occu s, he co esponding sea ch space is used o ind a eco e y plan. This gi es ou eac i e model an any ime-like beha iou . Ou model con ibu es wi h se e al no el ies: (a) i is a domain-independen P&E model ha can be ex- ploi ed in any applica ion con ex ; (b) i is indepen- den o he delibe a i e planne ; (c) i ades o de- libe a i e and eac i e mechanisms o p o ide good- quali y esponses in a imely ashion; (d) i a oids deal- ing wi h unnecessa y in o ma ion om he wo ld, han- dling speci ically he in o ma ion ele an o he ail- u e; and (e) i pe o ms a ime-bounded delibe a i e p ocess ha pe mi s o con inuously ope a e on he plan o epai p oblems du ing execu ion. This pape is o ganized as ollows. Sec ion 2 p e- sen s some ela ed wo k and Sec ion 3 ou lines he gene al P&E a chi ec u e whe e he eac i e model is in eg a ed. The wo ollowing sec ions p esen some o mal concep s and in oduce he eac i e planning model o eco e om plan ailu es. Sec ion 6 p esen s C. Guzman e al. / Reac i e execu ion o sol ing plan ailu es in planning con ol applica ions 345 a mo i a ion example on he Ma s space mission. Sec- ion 7 p esen s he model e alua ion, Sec ion 8 dis- cusses some limi a ion o he model and, inally, he las sec ion concludes and ou lines u u e esea ch lines. 2. Rela ed wo k Classical planning e e s gene ically o planning o s a e- ansi ion sys ems ha adop a se ies o assump- ions like ha he sys em is de e minis ic and s a ic, ha ac ions a e ins an aneous ansi ions and ha he planne is no conce ned wi h any change ha may oc- cu in he sys em while i is planning [21]. In con as , eac i e planning ope a es in a imely ashion wi h highly dynamic, non-de e minis ic and unp edic able en i onmen s, assuming unce ain y in he wo ld and he exis ence o mul iple ou comes due o ac ion ail- u es o exogenous e en s [34]. Ou eac i e planne is no a empo al planne bu i is designed o e u n imely esponses in highly dynamic en i onmen s. Rega ding non-de e minis ic planning, he s udy o inding he sequence o ac ions o e en s ha explain he cu en obse ed s a e o he wo ld is called diagno- sis. Mos o he esea ch on diagnosis pu he emphasis in mal unc ioning componen s and ob aining a plan o eco e om he componen ailu e a he han mon- i o ing a plan execu ion [5,43]. Pa icula ly, he wo k in [43] p o ides a o mal cha ac e iza ion o diagno- sis and i s ela ion o planning and, in [5], au ho s p o- pose an e olu iona y s a egy ha success ully diag- noses se e al ypes o componen ailu es, such as he sepa a ion o a body pa o he comple e ailu e o a senso o mo o . Planning in non-de e minis ic en i onmen s has also been add essed om a p obabilis ic pe spec i e, ep- esen ing p obabili ies o e he expec ed ac ion ou - comes o belie s a e space. Planning based on Ma ko Decision P ocesses is a well-known app oach o deal wi h non-de e minism when an accu a e p obabili y dis ibu ion on each s a e ansi ion is a ailable [27,33]. In highly dynamic en i onmen s whe e exogenous e en s equen ly occu , i is no possible o ha e a model o he unce ain y in he wo ld. Likewise, a Fi- ni e S a e Machine (FSM) can also be used o imple- men a eac i e beha iou [39] bu FSM equi es an ex- plici modeling o a ini e se o plans (s a es and an- si ions) and i is pa icula ly aimed a choosing only he immedia e nex ac ion. When a plan is o be exe- cu ed in an unp edic able en i onmen , i is imp ac i- cal o ha e all po en ial epai plans explici ly ep e- sen ed and he emphasis is no only on sho - e m eac- i i y bu also on p ese ing he achie abili y o he op- e a ion goals. Fo his eason, a ime-bounded eac i e planne ha calcula es p omp ly eco e y plans o e a planning ho izon is he mo e sui able solu ion. 3. An a chi ec u e o planning and execu ion Ou wo k akes place in he con ex o PELEA [24], a single-agen a chi ec u e in which an agen is endowed wi h capabili ies o gene a ing a plan, execu ing, plan moni o ing and, op ionally, lea ning. Ou ul ima e goal o ex ending his model o a mul i-agen con ex is dis- cussed in Sec ion 8. Be o e add essing his issue, ou pu pose is o ha e an agen equipped wi h a eac i e execu ion mechanism ha enables he agen o epai a plan a un ime, hus a oiding he need o eso o he delibe a i e planne each ime a ailu e occu s. In he ollowing, we will e e o he concep o agen , speci ically o execu ion agen , as an au onomous en- i y capable o pe o ming easoning and communica - ing wi h o he en i ies o he sys em like he delibe a- i e planne , which o e s a planning se ice. In ou app oach, a planning se ice p o ides execu- ion agen s wi h independen plans o be execu ed. An agen , which is an ex ension o a PELEA1agen [24], execu es and moni o s one ac ion a a ime and calls i s epai ing mechanism o a eco e y plan whene e a ailu e a ises. In case he agen is unable o sol e he ailu e by i s own, i will eques he planning se ice anewplan. The ocus o his pape is on he epai ing mecha- nism o he execu ion agen . The planning module em- bedded in o he execu ion agen is a eac i e planne , which is used o eco e om ailu es a un ime. The componen s o an execu ion agen (see Fig. 1) a e: –Execu ion module (EX). The EX is ini ialized wi h a planning ask, which cu en s a e is ead om he en i onmen h ough he senso s. The EX is esponsible o eading and communica ing he cu en s a e o he es o modules as well as execu ing he ac ions o he plan in he en i on- men . –Moni o ing module (MO). The main ask o he MO is o e i y ha he ac ions a e execu able in 1A mo e de ailed desc ip ion may be ound a h p://se e g ps. dsic.up .es/pelea/. 346 C. Guzman e al. / Reac i e execu ion o sol ing plan ailu es in planning con ol applica ions Fig. 1. Flow o he eac i e execu ion model. he cu en s a e be o e sending hem o he EX. When he EX epo s he MO he s a e esul ing om he execu ion o some ac ion o he plan, he MO checks whe he he nex ac ion o he plan is execu able in he esul ing s a e. This p ocess is called plan moni o ing, e i ying whe he he al- ues o he a iables o he ecei ed s a e ma ch he expec ed alues o no . O he wise, he MO will de e mine he exis ence o a plan ailu e. –Reac i e Planne module (RP). The RP is used when a plan ailu e is de ec ed by he MO and a eco e y is equi ed. The RP uses a p e- compu ed sea ch space, called epai ing s uc- u e, o p omp ly ind a plan ha b ings he cu - en s a e o one om which he plan execu ion can be esumed (see Sec ion 5). In case he RP is no able o ind a plan wi h i s epai ing s uc u e, he MO eques s he planning se ice a new plan. The EX, MO, and RP modules o an execu ion agen ope a e he Reac i e Execu ion Model. The con- ol low o he eac i e execu ion model is shown in Fig. 1. An ac ion plan Π o sol ing a planning ask is calcula ed by he planning se ice. Πconsis s o a se- ies o ac ions o be execu ed a gi en ime s eps, each o which makes a de e minis ic change o he cu en wo ld s a e. The elapsed ime om one ime s ep o he nex one de ines an execu ion cycle; i.e., he mon- i o /ac ing/sensing cycle o an ac ion execu ion. The model ollows se e al execu ion cycles, pe o ming he scheduled ac ion in each cycle un il he plan execu ion is comple ed. Ini ially, he MO ecei es he plan Π om he plan- ning se ice and, be o e sending Π o execu ion, i pe - o ms wo ope a ions: 1. I sends Π o he RP, which c ea es a epai ing s uc u e o a agmen o Πo leng h lcalled plan window,whe elis he numbe o ac ions o execu ion cycles in he plan window. The e- pai ing s uc u e associa ed o he plan window con ains in o ma ion, in he o m o al e na i e plans, o epai a ailu e ha a ec s any o he l ac ions included in he plan window. 2. When he ime o he RP expi es, and a epai - ing s uc u e has been calcula ed o a pa icula plan window, he MO moni o s he a iables o he i s ac ion o he window. I he sensed al- ues o he ac ion a iables ma ch he equi ed al- ues o he ac ion o be execu ed, he MO sends he scheduled ac ion o he EX o i s execu ion (see Fig. 1). O he wise, a ailu e is de ec ed and he MO calls he RP, which will make use o he epai ing s uc u e o ix he ailing ac ion. No- ice ha a ailu e ha occu s in he i s ac ion o a window is due o an exogenous e en (e.g. o he agen s change he wo ld s a e) and no due o an e oneous execu ion o he p eceding ac ion in he plan. The MO ecei es he esul o he sensing ask om he EX a e execu ing he ac ion, i upda es he plan window acco dingly by elimina ing he al eady exe- cu ed ac ion and p oceeds wi h he nex ac ion o he plan window. Fo ins ance, assuming a plan o i e ac- ions and a epai ing s uc u e o a plan window o l=3([a1,a 2,a 3]), he plan window will be upda ed o [a2,a 3]a e success ully execu ing a1,and heRP will use he same epai ing s uc u e o ix a po en ial ailu e in he emaining ac ions o he plan window, ha is, a2o a3. Subsequen ly, he RP will c ea e a new epai ing s uc u e, o example, o he plan window [a4,a 5]. The low goes on as long as no plan ailu es a e encoun e ed. In case ha a non-execu able ac ion is ound, he MO epo s he ailu e o he RP. Then, he RP uses he epai ing s uc u e associa ed o he plan window o he non-execu ableac ion and ob ains a new plan Π ha sol es he ailu e and eplaces he old plan Π, a aining likewise he goals o he planning ask. The RP is cons an ly wo king while he EX is ex- ecu ing he ac ions o he plan. Hence, besides ha - ing a epai ing s uc u e eady o a end a ailu e in an ac ion o he cu en plan window, he RP is also gene a ing he subsequen s uc u e o he nex plan window. Typically, he ime o he RP o compu e he epai ing s uc u e o he subsequen plan window is he ime ha he EX will ake o execu e he ac- ions included in he cu en window. The e o e, he C. Guzman e al. / Reac i e execu ion o sol ing plan ailu es in planning con ol applica ions 347 mo e ac ions in he cu en plan window, he mo e ime he RP will ha e o c ea e he nex epai ing s uc u e and, consequen ly, he longe he window associa ed o his epai ing s uc u e. This wo king scheme gi es ou model an any ime beha iou , hus gua an eeing he RP can be in e up ed a any ime and will always ha e a epai ing s uc u e a ailable o a end an immedia e plan ailu e. On he o he hand, some simila i ies be ween ou model and he li e cycle o a scien i ic wo k low can be ound. Following [23], we can do his analogy: he modeling phase is equi alen o he planning ask mod- eling; he deploymen phase amoun s o he plan Πcal- cula ed by he planning se ice; and he execu ion and moni o ing phase would be he same in ou model. Un- like scien i ic wo k low, ou model does no include an analysis phase; howe e , successi ely epe i ions o he deploymen ( epai plan) and execu ion phases happen when a plan ailu e a ises. 4. Fo mal model In his sec ion, we o malize he concep o planning ask, pa ial s a e and a solu ion plan o a ask as a se- quence o pa ial s a es [21]. Ou planning o malism is based on a mul i- alued s a e- a iable ep esen a ion whe e each a iable is assigned a alue om a mul- iple alue domain ( ini e domain o a a iable). Fo modeling planning p oblems, we used PDDL3.1,2 he mos ecen e sion o he Planning Domain De ini ion Language [17] (PDDL). De ini ion 1. Planning ask A planning ask is gi en by he 4- uple P=V,I,G,A: –Vis a ini e se o s a e a iables, each associa ed o a ini e domain, D , o mu ually exclusi e al- ues ha e e o objec s in he wo ld. ∈Vmaps a uple o objec s o an objec po he planning ask, which ep esen s he alue o .Fo exam- ple, in a plane a y Ma s o e s domain,3a o e (B) can be placed a any o he waypoin s w1,w2o w3. Hence, he a iable loc-B ep esen s he loca- ion o o e B,andDloc-B={w1,w2,w3}. A a iable assignmen o luen is a unc ion on a a iable such ha ( )∈D ,whe e e ( ) 2PDDL syn ax de ini ion in oduced in 2008 by M. Helme (h p: //ipc.in o ma ik.uni- eibu g.de/PddlEx ension/). 3Ou PDDL speci ica ion o his domain can be ound a h p:// se e g ps.dsic.up .es/planin e ac ion/ esou ces/. Fig. 2. Plan as a sequence o pa ial s a es. Va iables a e loc-B:lo- ca ion o o e B;link-w1-w2: map o a el om w1 o w2;com- : communica ion o he esul s o analyzing he ock . Unde lined a iables a e he p econdi ions o he ac ion Na iga e. is de ined. A luen is ep esen ed as a uple  ,p, meaning ha he a iable akes he alue p. A o al a iable assignmen o s a e applies he unc ion o all a iables in V. A s a e is always in e p e ed as a wo ld s a e. A pa ial a iable as- signmen o pa ial s a e o e Vapplies he unc- ion o some subse o V. –Iis a s a e ha ep esen s he ini ial s a e o he planning ask. –Gis a pa ial s a e o e Vcalled he p oblem goal s a e. –Ais a ini e se o ac ions o e V. An ac ion a is de ined as a pa ial a iable assignmen pai a=p e,e o e Vcalled p econdi ions and e - ec s, espec i ely. I an ac ion is execu able in a s a e, i.e. i s p econdi ions hold in such a s a e, he alues o he s a e a iables ( luen s) change as speci ied in he e ec s. An ac ion plan, ΠA, ha sol es a planning ask Pis a sequence o ac ions ΠA=a1,...,a n ha applied in he ini ial s a e Isa is ies he goal s a e G. An ac- ion ai∈ΠAis execu able in a wo ld s a e Si he lu- en s con ained in Ssa is y he p econdi ions o ai;i.e. p e(ai)⊆S. The esul o execu ing aiin a s a e Sis anews a eS ha con ains he luen s o Swhich a e no modi ied by e (ai)plus he luen s as speci ied in e (ai). Then, execu ing ΠAin he ini ial s a e I esul s in a sequence o s a es S1,...,S nsuch ha S1is he esul o applying a1in I,S2is he esul o applying a2in S1,..., and Snis he esul o applying anin Sn−1.AplanΠAis a solu ion plan i G⊆Sn[21]. A plan can also be iewed om he poin o iew o he wo ld condi ions ( luen s) ha a e necessa y o he plan o be execu ed. Tha is, ins ead o iewing a plan as he esul o i s execu ion, we can iew a plan as he necessa y condi ions o i s execu ion. Thus, a plan can also be de ined as a sequence o pa ial s a es, a he han wo ld s a es, con aining he minimal se o 348 C. Guzman e al. / Reac i e execu ion o sol ing plan ailu es in planning con ol applica ions Fig. 3. Plan as a sequence o pa ial s a es o he plan ΠA. Va iables a e loc-B: loca ion o o e B;loc-L: loca ion o lande L;loc- -w3: loca ion o he ock ;ha e-B: indica es i Bhas he ock ;link-wi-wj: map o a el om wi o wj; is-w3-w2: loca ion w2is isible om w3. com- : esul s o analyzing he ock a e communica ed. luen s ha mus holdin hewo ld o heplan obe execu ableinsuchawo lds a e. In he example depic ed in Fig. 2, he pa ial s a e Gis he goal s a e Go a planning ask P,andi con ains wo luen s. Le ’s conside he las ac ion o aplanis(Na iga e Bw 1w2), which achie es he e - ec loc-B,w2. Then, he necessa y luen s o be able o execu e he ac ion and achie e he luen s in Ga e ep esen ed in s a e G. We can obse e ha Gdoes no only con ain he luen s ha ma ch he p econ- di ions o he ac ion Na iga e (i.e., loc-B,w1and link-w1-w2, ue, which ep esen ha he loca ion o o e Bmus be he waypoin w1and a link be ween w1and w2mus exis , espec i ely) bu also he luen com- , ue. This is because his luen (communi- ca ing he esul s o analyzing he ocks ) is a goal o G ha is no achie ed by he e ec s o he ac ion Na iga e. The eby, com- , ueis achie ed ea lie in he plan and i mus hold in Gin o de o gua an ee ha i is sa is ied in G. The s a e Gin Fig. 2 is called a eg essed pa ial s a e because i is calcula ed by eg essing he goals in G h ough he ac ion Na iga e. Likewise, he same e- g ession can be applied o he es o ac ions o a gi en plan ΠA.Le aibe an ac ion and Ga goal s a e such ha P=p e(ai),E=e (ai)and E⊆G.Thepa - ial s a e Gin which aican be applied is calcula ed by he eg essed ansi ion unc ion Γ(G, ai),de inedas: G:= Γ(G, ai):=G E∪P(1) Gis a pa ial s a e ha ep esen s he minimal se o luen s ha mus hold in he wo ld s a e in o de o achie e Gby means o he execu ion o ai. No ice ha Gincludes P, he p econdi ions o ai, plus he luen s which a e in Gbu a e no p oduced by E(G E); i.e., he luen s ha a e achie ed be o e Gin he plan and mus keep hei alues un il G. The eg essed pa ial s a e app oach was i s used by PLANEX [11] o supe ise he execu ion o a se- Fig. 4. Plan ΠA o a plane a y Ma s o e domain. quence o ac ions. Plans a e ep esen ed by means o a iangle able ( his s uc u e p o ides suppo o plan moni o ing) and he p oblem goals a e eg essed om he las column o he able, including ac ion p econdi ions, h ough he emaining ac ions o he plan. Roughly, he eg ession o a luen o e an ac ion ( h ough he eg essed ansi ion unc ion Γ)isasu i- cien and necessa y condi ion o he sa is ac ion o he luen ollowing he execu ion o he ac ion. The wo k in [19] o malizes his concep in he si ua ion calculus language whe eas we apply he same o maliza ion in PDDL. De ini ion 2. Solu ion plan as a sequence o pa ial s a es Gi en a solu ion plan ΠA=a1,...,a n o a planning ask P, he eg essed plan o ΠAis de ined as a ch onologically o de ed sequence o pa ial s a es G0,G 1,...G n,whe e: Gn:= G G0⊆I Gi−1:= Γ(Gi,a i) A eg essed plan is deno ed by ΠG0−Gn,whe eG0 is he ini ial pa ial s a e and Gnis he inal pa ial s a e o he plan ΠA. De ini ion 2 speci ies he ele- an luen s a each ime s ep o he success ul execu- ion o ΠA, whe e each ai∈ΠAis he ele an ac ion o achie ing Gi om Gi−1. Hence, a eg essed plan ΠG0−Gnde i ed om ΠAdeno es he luen s ha mus hold in he en i onmen a each ime s ep o success- ully execu e he ac ions in ΠA. In o he wo ds, his de ini ion allows us o disce n be ween he luen s ha a e ele an o he execu ion o a plan and hose ones ha a e no . I exploi s he idea o anno a ing plans C. Guzman e al. / Reac i e execu ion o sol ing plan ailu es in planning con ol applica ions 349 Fig. 5. Repai ing s uc u es o a o e Bin a Plane a y Ma s Domain. a: eg essed plan ΠG0−G5 o a1,a 2,a 3,a 4,a 5,b1: he epai ing s uc u e o he plan window [a1,a2,a3], and b2: he epai ing s uc u e o he plan window [a4,a5]. wi h condi ions ha can be checked a execu ion ime o con i m he alidi y o a plan [11]. Tha is, i he luen s o a pa ial s a e Gihold in a wo ld s a e S (Gi⊆S) hen he ac ions comp ised in he plan ag- men ΠGi−Gna e execu able in S, hus gua an eeing he goals o he planning ask a e achie ed. Figu e 3 shows he eg essed plan ΠG0−G5de i ed om he solu ion plan shown in Fig. 4, and calcula ed h ough he successi e applica ion o he eg essed ansi ion unc ion Γ. The plan in Fig. 4 shows he ac- ions o a o e o ga he a ock sample, communi- ca e he esul s o analyzing he ock and na iga e back o he ini ial posi ion. Ac ion a2, o ins ance, c ea es he luen ha e-B, in G2; and he pa ial s a e G1 is he esul om applying Γ(G2,a 2), which includes p e(a2)( he luen s which a e unde lined in node G1 o Fig. 3) plus he luen s ha a e in G2bu a e no p oduced by e (a2). The e o e, i he senso eading e u ns a wo ld s a e in which all o he luen s in G1 hold, hen ac ion a2is execu able in such a wo ld s a e; i he luen s in G2occu in he subsequen wo ld s a e hen a3is execu able and so on. The las pa ial s a e, G5, comp ises G, he goals o he planning ask. In e ms o plan moni o ing, G5 ep esen s he luen s ha sa is y he p econdi ions o a ic i ious inal ac ion, a , whe e p e(a )=Gand e (a )=∅,i.e.Gis moni- o ed by checking he p econdi ions o a . As a inal ema k, we no e ha he eac i e execu ion model is de ined a he same g anula i y le el han he planning model, and bo h use PDDL as he speci ica- ion language. This eases he communica ion be ween he planning se ice and execu ion agen s and a oids he o e head o ansla ing a high-le el planning spec- i ica ion in o a low-le el desc ip ion as i happens in o he models [14]. 5. Reac i e execu ion model The key concep o ou eac i e execu ion model is he epai ing s uc u e. Gene ally speaking, gi en a so- lu ion plan ΠA, a epai ing s uc u e Tis a pa ial-s a e sea ch ee ha encodes eco e y plans o a plan win- dow o ΠA. Since nodes in Ta e pa ial s a es, he e- ac i e execu ion model only handles he minimal da a se ha is necessa y o ca y ou a epai ing ask. Figu e 5 shows wo epai ing s uc u es, T1and T2, o he eg essed plan in Fig. 3 ( o simplici y, we do 350 C. Guzman e al. / Reac i e execu ion o sol ing plan ailu es in planning con ol applica ions no show all he pa ial s a es ha would be gene a ed). T1is he sea ch ee associa ed o he plan window [a1,a 2,a 3], o , equi alen ly, o he eg essed subplan ΠG0−G3. The leng h o he window is h ee (l=3), because i comp ises h ee ac ions, and he pa ial s a e G3is called he oo node (G )o T1. A pa h in T1 ep esen s a ( eg essed) plan o each he oo node G3. Speci ically, a pa h in a epai ing s uc u e is in e - p e ed as a eco e y plan ha leads he cu en wo ld s a e o ano he s a e om which he execu ion o he plan ΠAcan be esumed. All eco e y plans in T1 ha e one hing in common: hey e en ually guide he execu ion o he plan owa ds G3, he oo node o T1. Fo ins ance, suppose ha Sis he se o luen s ha ep esen s he s a e o he wo ld s a e such ha G 16 ⊆S(see Fig. 5 b1). The applica ion o he plan Π=a1,a 4,a 8,a 6 o Swill each he pa ial s a e G3, om which he es o he plan ΠA,a4,a 5, can be execu ed. The ee T2(see Fig. 5 b2) is associa ed o he plan window [a4,a 5], o o he eg essed plan ΠG3−G5.The numbe o epai ing s uc u es necessa y o keep ack o he execu ion o a plan ΠAdepends on he ime limi o c ea e he sea ch ees, which, in u n, delimi s he size o he ee. Two pa ame e s de e mine he size o a sea ch space T, he leng h o he plan window associ- a ed o T(l), and he dep h o he ee (d). In gene al, he la ge he alue o l, he mo e al e na i es o ind a eco e y plan; and he deepe he ee, he longe he eco e y plans comp ised in T. The minimum alue o dmus be l+1in o de o ensu e ha he ee com- p ises a leas one ac ion ha epai s he i s ac ion o he plan window associa ed o T. On he o he hand, he maximum alue o dis de e mined by he a ailable ime o build T. Pa icula ly, in T1,d=6,which e- sul s om l=3and he ime limi o build T1(Sec ion 5.3 explains in de ail how o es ima e he maximum size o a epai ing s uc u e). In he ollowing, we explain (1) he p ocess o build a epai ing s uc u e, (2) how o ind a plan in a sea ch ee o epai a ailu e and (3) he analysis o es ima e he size o he sea ch ee. 5.1. Building a epai ing s uc u e T The cons uc ion o he epai ing s uc u e Ts a s a e es ima ing he size o T; i.e., when he alues o l and da e known. The gene a ion p ocess, shown in Al- go i hm 1, consis s in expanding T om he oo node G ia he applica ion o he eg essed ansi ion unc- ion Γ(G, a) ollowing Eq. (1) (line 6 o Algo i hm 1). The algo i hm is a classical backwa d cons uc ion o a planning sea ch space [21], whe e a node Gis ex- panded un il dep h(G)=d;i.e.,G eaches he max- imum dep h ee (line 4), o Gis supe seded by an- o he node ha exis s in he ee (lines 7 o 13 de ine a mechanism o he con ol o epea ed s a es which is de ailed below). Inpu : G ,d Ou pu : T 1: Q←{G },T←{G } 2: while Q =∅do 3: G← emo e i s node om Q 4: i dep h(G)<d hen 5: o all {a|a∈Ais a ele an ac ion o G}do 6: G←Γ(G, a) 7: i G/∈T hen 8: i ∃G∈T |G⊂G hen 9: ma k Gas supe se o G 10: else 11: Q←Q∪G 12: se ansi ion (labeled a) om G o G 13: T←T∪G 14: else 15: Q←∅ 16: e u n T Algo i hm 1: Gene a ing he epai ing s uc u e T. The pu pose o Algo i hm 1 is o gene a e mul i- ple eg essed plans om G . Unlike he applica ion o Γ(G, a)in De ini ion 2, which depa s om a gi en so- lu ion plan ΠA, such a plan does no exis when build- ing a ee T. Ac ually, he aim o Algo i hm 1 is p e- cisely o ind he ele an ac ions o a node G(line 5), and e en ually c ea e a plan ΠA ha links wo pa icu- la pa ial s a es. The ope a ion o eg essing a luen in a node G o e an ac ion achecks whe he ais a ele an ac ion o achie e o no . An ac ion ais ele an o ,and o igina es an a c (G,G)in T, i i does no cause any con lic wi h he luen s in Gand G. The cons uc ion o Thas hen o check wo consis ency es ic ions: (1) ha e (a)does no con lic wi h he luen s in G,and (2) ha p e(a)does no con lic wi h he luen s in G. We de ine Φ(G,G)as he unc ion ha e u ns whe he o no a con lic be ween wo se s o luen s Gand Gexis s. Φ(G,G)holds i ∃ ,p∈Gand ∃ ,p∈Gand p=p. De ini ion 3. Rele an ac ion Gi en a luen ∈G, ais a ele an ac ion o i he ollowing condi ions hold: 1) ∈e (a)and C. Guzman e al. / Reac i e execu ion o sol ing plan ailu es in planning con ol applica ions 351 2) ¬Φ(e (a),G)and 3) G=Γ(G, a)∧¬Φ(p e(a),G ) The cons uc ion o T ollows he applica ion o De ini ion 3 o each luen o a pa ial s a e Gwhich has no eached dep h(G)=d(lines4and5o Algo- i hm 1), and he expansion con inues un il no new pa - ial s a es a e added o he ee. The sea ch space Tis ac ually a g aph due o he exis ence o mul iple pa hs ha each he same pa ial s a e om he oo node du - ing he cons uc ion o T. Mul iple pa hs a e o igina ed because o ac ions like (Communica e ock B L w3w2) and (Communica e soil B L w3w2), which can be ex- ecu ed in ei he o de , o he exis ence o e e sible ac- ions like (Na iga e Bw 1w2)and(Na iga e Bw 2w1). Consequen ly, Tmay con ain many edundan pa hs. A se o s a e a iables induce a s a e space ha has a size ha is exponen ial in he se , and, o his ea- son, planning, as well as many sea ch p oblems, su - e om a combina o ial explosion. E en hough nodes in Ta e pa ial s a es ha con ain a less luen s han wo ld s a es, he la ge size o he epai ing s uc u es a e some imes una o dable o a eac i e sys em. Wi h he aim o educing he size o T, we only conside o expansion he luen s o G ha a e ela ed o he ele an a iables, ha is, he a iables in ol ed in he p econdi ions o he ac ions o he plan window. Thus, gi en a plan window [a1,a 2,a 3], we app oxima e Tby expanding only he luen s ela ed o he ele an a i- ables in ol ed in he se p e(a1)∪p e(a2)∪p e(a3), which is ac ually he se o luen s ha migh need o be epai ed. The ime complexi y o Algo i hm 1 e- sponds o he classical complexi y o he gene a ion o a ee, ha is O(ˆ bd),whe eˆ bis he es ima ed b anching ac o o T ha is de ailed in Sec ion 5.3. The gene a ion p ocess makes wo nodes in Tbe connec ed by a unique simple pa h. Since we a e in e - es ed in keeping only he sho es (op imal) pa hs, he cons uc ion o Tp unes epea ed s a es (line 7 in Al- go i hm 1) and a oids he expansion o supe se nodes (lines 8 and 9). Le ’s assume ha Tcon ains a pa h om a node G o he oo node G o T. A node G such ha G⊂Gis said o be a supe se o node G.In his case: –Gs ands o he minimal se o luen s ha mus hold in Sin o de o execu e he ac ions o he pa h ha eaches G . –The bes eco e y plan om Gis also he bes pa h om Gbecause he RP e u ns he sho es plan o G . All in all, a epai ing s uc u e encodes he op imal pa h be ween each pai o nodes o which a eco e y plan can be ound. Once he RP has c ea ed T,i com- munica es he MO all he a iables in ol ed in T. 5.2. Repai ing a ailu e When an ac ion o he plan window associa ed o a epai ing s uc u e T ails, he RP inds a way o keep he plan going, ei he by eaching a pa ial s a e in T om which o execu e he aul y ac ion again o a he ano he s a e om which o execu e a la e ac ion o he plan window. Le Tbe a epai ing s uc u e o a eg essed plan ΠG0−G associa ed o he plan window [a1,...,a ]o aplanΠA.Whenanac ionin[a1,...,a ] ails, a e- pai ing ask de ined as R=S, G is ac i a ed, whe e S4is he se o luen s o he cu en wo ld s a e and G is he a ge s a e we wan o each in T. The node G a ies depending on he ailed ac ion and he pa icula epai ing ask o such ac ion. Since se e al eco e y plans can be ound o ix a aul y ac ion, he RP will successi ely execu e a epai ing ask un il one o hem is success ul o ixing he ac ion. This way, i he e - oneous ac ion is a1, he RP will i s y he epai ing ask R=S, G0; o he wise, i will y R=S, G1 and so on un il G =G ; i he ailu e occu s in a2, he i s a emp will be R=S, G1and he las a - emp will be o G =G ; in he case ha he ail- u e a ec s a , only wo epai ing asks can be ealized, R=S, G −1and R=S, G . Mo e o mally, gi en R=S, G  o a aul y ac- ion a, he RP applies a modi ied b ead h- i s sea ch om G un il a node Gs ha sa is ies Gs⊆Sis ound in T.Gsis a consis en s a e wi h S, a s a e ha com- p ises all he necessa y luen s o execu e in he cu - en wo ld s a e he plan o med wi h he ac ions om Gs o G .I Gs⊆Sis ound, he eco e y plan om Gs o G is conca ena ed wi h he plan om G o G (unless G =G ), and wi h he plan om G o Gn,whe eGnis he las s a e o he o iginal plan ΠA which con ains G, he p oblem goal s a e. I Gs⊆S is no ound, he RP will execu e he subsequen e- pai ing ask R=S, G +1un il one o hem success- ully e ie es a eco e y plan o Ris in oked wi h G =G and a plan is no ound. In his la e case, T does no comp ise he necessa y in o ma ion o ind a 4Technically speaking, he MO does no communica e he RP all o he luen s in Sbu only he alues o he a iables ha appea in T; hese a iables we e sen by he RP o he MO a e building T. 358 C. Guzman e al. / Reac i e execu ion o sol ing plan ailu es in planning con ol applica ions Table 4 Summa y o s a is ics o RP, LPG-ADAPT and LAMA pe o - mance S abili y (%) ΔΠ ATime (ms) μσ μσ μ σ RP 92 19 0.97 0.85 1.85 4.33 LPG-ADAPT 85 19 2.40 1.94 49.47 4.72 LAMA 51 27 1.33 1.52 62.83 36.99 alyze i , o calib a ing he o e ’s came a again (e.g., ailu e 2 o p oblem 4). Failu es o ype C we e ound in wo cases, which could be epai ed because hei e- spec i e T1included pa hs in ol ing he second o e (e.g., ailu e 3 o p oblem 6). He e, a ha dwa e ailu e p e en he o e om analyzing he soil in a speci ic loca ion, ou model epai s he ailu e using he second o e ha explo es he a ea seeking o soils, analyzes he soil and communica es he esul s o he lande . In he ailu es o ype D, he RP akes ad an age o he posi i e ailu e, which achie es he e ec s o he nex ac ion o execu e and, consequen ly, he RP p oceeds wi h he ollowing ac ion in ΠA. The pe o mance esul s in Table 3 and he sum- ma y o s a is ics in Table 4 show ha ou RP pe - o ms admi ably well in all he measu ed dimensions. Rega ding s abili y, RP ou pe om s LPG-ADAPT and LAMA. LAMA is he app oach wi h he wo s a e o s abili y (51%), which is easonable since he plan- ne does no epai a plan bu i compu es a new plan. In Table 3 we can see ha he plan quali y o numbe o ac ions o Π Ais sligh ly highe wi h he RP han wi h LAMA in some cases (e.g., ailu e 1 o p ob- lem 12 o ailu e 3 o p oblem 7), and lowe in some o he cases (e.g. ailu e 2 o p oblem 12 o ailu es 1 and 2 o p oblem 10). LAMA is able o ind sho e plans in a ew cases because i compu es a plan o he new si ua ion wi hou being subjec o keep he ac ions in ΠA. Ne e heless, all in all, he RP e u ns plans o be e quali y han LAMA as Table 4 shows ( he mean alue in he inc ease o he numbe o ac ions is 0.97 in RP agains 1.33 in LAMA). The compa i- son be ween RP and LPG-ADAPT clea ly bene i s RP in bo h s abili y and quali y o he eco e y plan, pa - icula ly in he mos complex p oblems (10 o 12). As o he compu a ion ime, inding a eco e y pa h o he oo node is mo e cos ly since he epai ing mech- anism explo es he en i e sea ch space. Howe e , RP shows ou s anding esul s compa ed o LPG-ADAPT and LAMA, which p o es he bene i o using he RP o epai plan ailu es in eac i e en i onmen s besides a oiding he o e head o communica ing wi h a de- libe a i e planne . In conclusion, we can a i m ha ou model is a obus eco e y mechanism o eac i e planning ha also p o ides good-quali y solu ions. 8. Limi a ions and ex ensions o he model The esul s in Sec ion 7 show ha ou eac i e ex- ecu ion model mee s he pe o mance needs o a e- ac i e plan epai and ha i also ou pe o ms o he epai ing mechanisms. Howe e , he model p esen s some limi a ions ha we in end o o e come in he u- u e. One limi a ion is he machine dependency o he es ima ion model explained in Sec ion 5.3. In o de o ep oduce he expe imen s, o o expo hem o o he sys ems, he aining o he es ima ion model mus be epea ed o adjus he alue o ¯ Γ o a pa icula p o- cesso . Assuming se e al agen s execu ing hei plans in a common en i onmen , a epai ing ask o an agen migh cause con lic s in he plan o he o he s. Addi- ionally, he occu ence o ailu es could es ic he ca- pabili ies o he agen s, p e en ing hem om achie - ing some goal. The e o e, a mul i-agen app oach whe e execu ion agen s ac , coo dina e and join ly e- pai a ailu e is desi able [2,16,35]. A communica- ion p o ocol ha helps agen s eques , p o ide and ag ee on a pa icula eco e y plan is a desi able ap- p oach o a mul i-agen epai sys em [25]. Pa icu- la ly, he p esen wo k ep esen s a i s s ep owa ds a mul i-agen P&E sys em capable o coo dina ing agen s plans while minimizing c owd-e ec s [44]. Ou model can be easily ex ended o pa allel and empo al planning. The pa allel execu ion o se e al ac ions o an agen is achie able by g ouping oge he he p econdi ions and e ec s o he pa allel ac ions in o a new ac ion. The exis ence o g ouped ac ions would a oid he duplica ed s a es ha a ise om he mul iple se ializa ion o he ac ions in he g oup. On he o he hand, handling du a i e ac ions in empo al planning would in ol e c ea ing a eg essed pa ial s a e a each ele an ime poin o an ac ion and adding luen s o ep esen he ongoing execu ing ac ions a each execu- ion cycle. 9. Conclusions and u u e wo ks This pape p esen s a eac i e execu ion model, which comp ises a RP, o eco e om ailu es in plan- ning con ol applica ions. The model is embedded in o a P&E sys em whe e an execu ion agen ecei es a plan om a delibe a i e planne and i s mission is o moni o , execu e and epai he gi en plan. P o iding ime-bounded esponses in eac i e en i- onmen s is a di icul and some imes un easible ask C. Guzman e al. / Reac i e execu ion o sol ing plan ailu es in planning con ol applica ions 359 due o he unp edic abili y o he en i onmen and he impossibili y o gua an ee a esponse wi hin a gi en ime. An al e na i e solu ion o o e come his di icul y is wo king wi h ime-bounded da a s uc u es a he han designing ime-bounded easoning p ocesses. By ollowing his app oach, ou model ensu es he a ail- abili y o a epai ing s uc u e, o sea ch ee, wi hin a gi en ime, which is la e used o ix he ac ion ailu es du ing he plan execu ion. Se e al ea u es ha e been conside ed in o de o ha e a sea ch ee gene a ed in due ime: (1) he ee is o med o pa ial s a es which con ain a less luen s han wo ld s a es; (2) he ee is limi ed o a pa icula agmen o he plan and ee dep h ha a e calcula ed by an es ima ion model; and (3) he expansion o he ee only conside s he ele an a iables ha migh po- en ially ail du ing he plan execu ion.Unde hese c i- e ia, we show he esul s ob ained o wo di e en do- mains, a simula ion o a eal NASA space p oblem, and a ehicle ou ing domain. The esul s co obo a e ha he e is a 95% likelihood o ob ain a epai ing s uc- u e in ime. Addi ionally, he exhaus i e expe imen- a ion on he epai ing asks con i m ha he epai ing s uc u e oge he wi h he sea ch eco e y p ocess is a e y sui able mechanism o ix ailu es ha ep esen sligh de ia ions om he main cou se o ac ion in a planning con ol applica ion. The esul s suppo se - e al conclusions: he accu acy o he model o gene - a e epai ing s uc u es in ime, he use ulness o a sin- gle epai ing s uc u e o epai mo e han one ac ion in a plan agmen while eusing he o iginal plan as much as possible, and he eliabili y and pe o mance o ou eco e y sea ch p ocedu e in compa ison wi h o he well-known classical planning mechanisms. The cu en RP can be ex ended in se e al di e en di ec ions as, o ins ance, by including he necessa y machine y o deal wi h empo al plans. Ou nex u u e wo k is o exploi his model o a mul i-agen eco e y mechanism in which agen s dynamically o m a eam- wo k a execu ion ime and wo k oge he in he epai o a plan ailu e. 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