TuLiP: A Software Toolbox for Receding Horizon Temporal Logic Planning & Computer Lab 2

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1 TuLiP: A Softwar Toolbox for Rcding Horizon Tmporal Logic Planning & Computr Lab 2 Nok Wongpiromsarn Richard M. Murray Ufuk Topcu EECI, 21 March 2013 Outlin Ky Faturs of TuLiP Embddd control softwar synthsis Rcding horizon tmporal logic planning Computr Lab

2 Problm Dscription Problm: Givn a plant modl and an LTL spcification nsur that any xcution of th systm satisfis, dsign a controllr to Th volution of th systm is dscribd by diffrntial/diffrnc quations whr s R n,u R m,d R p must b satisfid rgardlss of th nvironmnt in which th systm oprats Assum that is of th form ' = init {z } assumptions on initial condition s(t + 1) = As(t)+ Bu(t)+ Ed(t)) u(t) 2 U d(t) 2 D ^ s ^ ^ i2i f f,i {z } assumptions on nvironmnt 2 =) s init ^ s s ^ ^ i2i g s g,i {z } dsird bhavior

3 Embddd Control Softwar Synthsis Ky lmnts to spcify th problm discrt systm stat continuous systm stat (discrt) nvironmnt stat spcification Hirarchical Approach nv Discrt plannr Continuous controllr Discrt plannr computs th nxt cll to go to in ordr to satisfy Th synthsis algorithm considrs all th possibl bhaviors of th nvironmnt Issu: stat xplosion Continuous controllr simulats th plan Constraind optimal control problm Continuous xcution prsrvs th corrctnss of th plan u sd nois u Plant Local control 3

4 Main Stps Systm modl Systm spc Continuous Stat Spac Partition Proposition prsrving partition Continuous Stat Spac Discrtization Finit transition systm Continuous controllr Digital Dsign Synthsis Discrt Plannr Gnrat a proposition prsrving partition of th continuous stat spac cont_partition = prop2part2(stat_spac, cont_props) Discrtiz th continuous stat spac basd on th volution of th continuous stat disc_dynamics = discrtizm(cont_partition, ssys, N=10) Digital dsign synthsis prob = gnratjtlvinput(nv_vars, sys_disc_vars, spc, disc_props, disc_dynamics, smv_fil, spc_fil) ralizability = chckralizability(smv_fil, spc_fil, aut_fil, hap_siz) ralizability = computstratgy(smv_fil, spc_fil, aut_fil, hap_siz) aut = Automaton(aut_fil) 4

5 Exampl: robot_simpl.py Dynamics ẋ = u x, ẏ = u y whr u x,u y 2 [ 1, 1] Dsird Proprtis Visit th blu cll infinitly oftn Evntually go to th rd cll whn a PARK signal is rcivd Assumption Infinitly oftn, PARK signal is not rcivd ' = ( park) =) ( (s 2 C 5 ) ^ (park =) (s 2 C 0 ))) This spc is not a GR[1] formula Introduc an auxiliary variabl X0rach that starts with Tru ( X0rach =((s 2 C 0 _ X0rach) ^ park)) X0rach 5

6 Manually Constructing disc_dynamics: Dsird Proprtis Visit th blu cll infinitly oftn Evntually go to th rd cll whn a PARK signal is rcivd Assumption robot_discrt_simpl.py Systm Modl: Robot can mov to th clls that shar a fac with th currnt cll Infinitly oftn, PARK signal is not rcivd ' = 3( park) ( 3(s C 5 ) This spc is not a GR[1] formula Introduc an auxiliary variabl X0rach that starts with Tru ( X0rach =(s 2 C 0 _ (X0rach ^ park))) X0rach 6 (park 3(s C 0 )))

7 Dfining a synthsis problm: SynthsisProb class Combin th thr stps of digital dsign synthsis: gnratjtlvinput computstratgy Automaton synthsiz SynthsisProb Filds of SynthsisProb: nv_vars sys_vars spc disc_cont_var disc_dynamics Usful mthods: chckralizability(hap_siz='xmx128m', pick_sys_init=tru, vrbos=0): chcks whthr th problm is ralizabl gtcountrexampls(rcomput=fals, hap_siz='xmx128m', pick_sys_init=tru, vrbos=0) rturns th st of initial stats from which th systm cannot satisfy th spc synthsizplannraut(hap_siz='xmx128m', priority_kind=3, init_option=1, vrbos=0) synthsizs th plannr that nsurs systm corrctnss 7

8 Exampl: robot_simpl2.py Dynamics ẋ = u x, ẏ = u y whr u x,u y 2 [ 1, 1] Dsird Proprtis Visit th blu cll infinitly oftn Evntually go to th rd cll whn a PARK signal is rcivd Assumption Infinitly oftn, PARK signal is not rcivd ' = ( park) =) ( (s 2 C 5 ) ^ (park =) (s 2 C 0 ))) This spc is not a GR[1] formula Introduc an auxiliary variabl X0rach that starts with Tru ( X0rach =((s 2 C 0 _ X0rach) ^ park)) X0rach 8

9 Computr xrcis 1 Synthsiz a ractiv plannr with th following spcifications Systm variabls: X0,...,X8 Xi = 1 if robot in Ci, Xi = 0 othrwis. Environmnt variabls: obs {1,4,7}, park {0,1} C6 C7 C8 C3 C4 C5 C0 C1 C2 Dsird Proprtis Visit th blu cll (C8) infinitly oftn Evntually go to th grn cll (C0) aftr a PARK signal is rcivd Avoid an obstacl (rd cll) which can b on of th C1, C4, C7 clls and can mov arbitrarily Assumption Infinitly oftn, PARK signal is not rcivd Th obstacl always movs to an adjacnt cll Constraints (or discrt dynamics) Th robot can only mov to an adjacnt cll, i.., a cll that shars an dg with th currnt cll 9

10 Computr xrcis 2 Synthsiz intrsction logic for th car with th following spcification C3 C6 Dsird Proprtis Evntually go to C6 If thr is a car at on of th C3, C4, C7 clls at initial stat, nd to wait until it disappars bfor going through th intrsction Go through th intrsction only whn C2 and C5 ar clar No collision with othr cars Assumption?? (find a st of nontrivial assumptions that rndr th problm ralizabl) Constraint Th robot can only mov forward to an adjacnt cll, i.., a cll that shars an dg with th currnt cll C2 C1 C5 C4 C7 10

11 Rcding Horizon Framwork for LTL Spcifications Ida: Rduc th synthsis problm to a st of smallr problms of short horizon Considr a spcification of th form = init i I f f,i = i I s s,i g Organiz clls into a partially ordrd st P =({W j }, ) g whr is th st of goal stats, i.., all clls in W 0 satisfy g Assum that for ach j, thr xist a proposition and a mapping F such that th following shorthorizon spcification is ralizabl j = ( W j ) always k I f Infinitly o+n f,k = always Infinitly o+n dscribs rcding horizon invariants F(W j ) W j, j = 0dfins intrmdiat goal for starting in W j 2 1 Partial ordr condition guarants that w mov closr to goal W 0 W 1... W 4 F(W 4 )=W 2, F(W 3 )=W 1, F(W 2 )=W 0, F(W 1 )=W 0, F(W 0 )=W 0 k I s s,k ( F(W j )) W W 0 W 1 W 2 W 3 W 4

12 Ky Elmnts Original spcification: Short horizon spcification: i j = = init W i j k i I f I f f,i f,k = = i I s s,i i I g g,i s,k F i (W i j) k I s Partially ordrd st W i 0 W i 1... W i p = V P i =({W i 0,...,W i p}, g,i ) W i 1 p W i 2 p W i 3 p W i n p W0 i is th st of goal stats, i.., all clls in satisfy W i 0 W i 0 g,i Wi j, j = 0 g,i Rcding horizon invariant W i 1 j 1 W i 1 1 W i 2 j 2 W i 2 1 W i 3 j 3 W i 3 1 W i n jn W i n 1 init = is a tautology Mapping F i : {W0,...,W i p} i {W0,...,W i p} i F i (W i j) g,i Wi j, j = 0 W i 1 0 W i 2 0 W i 3 0 W i n 0 12

13 Rcding Horizon Tmporal Logic Planning SynthsisProb systm modl systm spc ShortHorizonProb W j F j RHTLPProb shprobs Problm ShortHorizonProb A class for dfining a short horizon problm Usful mthods computlocalphi(): automatically comput that maks this short horizon problm ralizabl. RHTLPProb A class for dfining a rcding horizon tmporal logic planning problm Contains a collction of shorthorizon problms Usful mthods computphi(): automatically comput for this rcding horizon tmporal logic planning problm if on xists. validat(): chck whthr all th sufficint conditions for doing rcding horizon tmporal logic planning ar satisfid 13

14 Exampl: autonomous_car_road.py Autonomous vhicl navigating an urban nvironmnt Traffic ruls No collision Stay in th travl lan unlss thr is an obstacl blocking th lan Progrss rquirmnt Rach th nd of th road Assumptions Obstacl may not block a road Obstacl is dtctd bfor th vhicl gts too clos to it Limitd snsing rang Obstacl dos not disappar whn th vhicl is in its vicinity 14

Lecture 8 Receding Horizon Temporal Logic Planning & Finite-State Abstraction

Lecture 8 Receding Horizon Temporal Logic Planning & Finite-State Abstraction Ufuk Topcu Nok Wongpiromsarn Richard M. Murray AFRL, 26 April 2012 Contents of the lecture: Intro: Incorporating continuous