Managing non-determinism in symbolic robot motion planning and control

Transcription

1 2007 IEEE Interntionl Conference on Robotics nd Automtion Rom, Itly, April 2007 ThD11.4 Mnging non-determinism in symbolic robot motion plnning nd control Mrius Kloetzer nd Clin Belt Center for Informtion nd Systems Engineering Boston University 15 Sint Mry s Street, Boston, MA {kmrius,cbelt}@bu.edu Abstrct We study the problem of designing control strtegies for non-deterministic trnsitions systems enforcing the stisfction of Liner Temporl Logic (LTL) formuls over their set of sttes. We focus on nite trnsition systems with inputs, which re often encountered when solving motion plnning problems by using discrete quotients induced by given prtition of the stte spce. Our pproch solves the problem conservtively using LTL gmes, nd consists of the following three steps: (1) the originl trnsition system is trnsformed into trnsition system on which n LTL gme cn be plyed, (2) solution of the LTL gme on the new trnsition system is obtined, nd (3) n interfce between this solution nd the initil trnsition system is constructed. The correctness of the method is ensured by design. The dvntges nd conservtiveness of our pproch re discussed nd illustrted by simple exmples. I. INTRODUCTION Motion plnning nd control of robots with nontrivil dynmics or kinemtics is usully two step process. In the rst step, (cellulr) decomposition of the C-spce is constructed, nd discrete pth is generted by serch in the quotient grph [1], [2]. In the second step, reference trjectory comptible with the robot dynmics or kinemtic is generted, nd robot control lws re designed to follow the trjectory. A very ttrctive lterntive to this is simultneous plnning nd control, in which the genertion of the discrete solution in the prtition quotient is performed t the sme time with the ssignment of vector elds in the regions of the prtition (robot control lws), while tking into ccount the restrictions imposed by robot under-ctution nd speed constrints [3], [4], [5], [6]. In ddition, enrichment of the speci ction lnguge from the clssicl plnning tsk go from A to B to temporl logic speci ctions (e.g., visit either A or B, rech A nd then B in nitely often ) leds to symbolic pproches to simultneous plnning nd control, where discrete bstrctions re used to provide forml link between the continuous robot control system nd the discrete representtion of the environment, nd lgorithms resembling model checking re used to provide solution to the discrete problem [7], [8]. All the pproches enumerted bove for simultneous plnning nd control fce common problem: the restrictions imposed by the robot dynmics or kinemtics cn, in generl, led to non-deterministic representtions of the discrete problem. For exmple, the prepres reltionship between collection of policies in [4] results in nondeterminism of the discrete bstrction. Recent results in control of f ne systems in simplices [9] nd of multi-f ne systems in rectngles [10] show tht even though controllers determining the trnsition of robot to speci c neighbor might fil to exist, controllers gurnteeing the trnsition to set of neighbors cn be found. In the trnsition system corresponding to the discrete prt of the problem, this mens non-determinism. Motivted by the bove, in this pper we focus on the discrete prt of simultneous motion plnning nd control, nd consider the following problem: given non-deterministic trnsition system with inputs, nd Liner Temporl Logic (LTL) formul over its set of sttes, determine set of initil sttes nd control strtegy so tht the produced trjectory stis es the formul. This problem is quite generl, nd, to the best of our knowledge, there is no vilble computtionl frmework providing solution. However, it is relted to LTL gmes [11]. Also, it is possible tht solutions cn be found by using results from the control of discrete event systems from speci ctions given s ω- regulr expressions (lnguges) over inputs [12] or by using tree utomt on in nite objects [13]. In this work, we dvocte the use of LTL gmes [11] for the construction of (conservtive) solution to the problem. Our fully utomtic computtionl frmework consists of three min steps. First, we convert the originl trnsition system into form in which n LTL gme cn be plyed. Second, we nd solution to the LTL gme in the form of feedbck utomton. Third, the solution of the gme is used to generte control strtegy for the initil trnsition system. We illustrte our pproch for the cse of tringulted plnr environment, where the ssignment of f ne feedbck controllers in the tringles using the method developed in [9] leds to non-determinism. The reminder of the pper is orgnized s follows. Section II provides some de nitions necessry throughout the pper. The problem is formulted in Section III nd our solution is presented in Section IV. Simultion results re given in Section V nd we conclude with nl remrks in Section VI /07/$ IEEE. 3110

2 II. PRELIMINARIES For nite set A, we use A, A ω, nd 2 A to denote its crdinlity, the set of ll in nite words over A, nd its power set (the set of ll its subsets), respectively. A. Trnsition Systems nd Liner Temporl Logic De nition 1: [Non-deterministic trnsition system] A - nite non-deterministic trnsition system is tuple T = (Q, Σ, δ, h, O), where: Q is nite set of sttes, Σ is nite input lphbet, δ : Q Σ 2 Q is trnsition function, h : Q O is the observtion mp, O is the set of observbles. For given stte q Q, the set of vilble (fesible) inputs is denoted by Σ q (i.e., Σ q is the set of σ i Σ for which δ(q, σ i ) 1). An input word σ Σ ω is denoted by σ = σ 1 σ 2 σ 3... A trjectory or run of T produced by n input word σ strting from q is n in nite sequence r Q ω, r = r 1 r 2 r 3... with the property tht r 1 = q nd i 1, r i+1 δ(r i,σ i ). A trjectory r of T produces word w O ω de ned s w = w 1 w 2 w 3..., w i = h(q i ), for ll i 1. A trnsition system T = (Q, Σ, δ, h, O) for which the observtion mp is identity (i.e., the sttes re of interest Q = O, nd cn be observed) is denoted for simplicity by T =(Q, Σ,δ). De nition 2: [Syntx of LTL formuls] An LTL formul over O is recursively de ned s follows: Every observble o O is formul, nd If φ 1 nd φ 2 re formuls, then φ 1, φ 1 φ 2, φ 1, φ 1 Uφ 2 re lso formuls. The semntics of LTL formuls re given over words of trnsition system T. De nition 3: [Semntics of LTL formuls] The stisfction of formul φ t position i N of word w, denoted by w i φ, is de ned recursively s follows: w i o if o = w i, w i φ if w i φ, w i φ 1 φ 2 if w i φ 1 or w i φ 2, w i φ if w i+1 φ, w i φ 1 Uφ 2 if there exist j i such tht w j φ 2 nd for ll i k < j we hve w k φ 1 A word w stis es n LTL formul φ, written s w φ, if w 1 φ. The symbols nd stnd for negtion nd disjunction. The Boolen constnts nd re de ned s = π π nd =. The other Boolen connectors (conjunction), (impliction), nd (equivlence) re de ned from nd in the usul wy. The unry temporl opertor is clled next, nd the binry temporl opertor U is clled the until opertor. Two useful dditionl temporl opertors, eventully nd lwys cn be de ned s φ = Uφ nd φ = φu, respectively. Formul φ mens tht φ becomes eventully true, wheres φ indictes tht φ is true t ll positions of trjectory. Remrk 1: In generl, the semntics of LTL formuls re given over in nite words in the power set of set of tomic propositions. We use the simpli ed semntics of De nition 3 motivted by the problem formulted in Section III. ABüchi utomton is nite stte utomton ccepting in nite strings. The de nition of Büchi utomton nd its cceptnce condition is beyond the scope of this pper, nd we refer the interested reder to [13]. For ny LTL formul, there exists non-deterministic Büchi utomton ccepting ll nd only the runs stisfying it [14] (this utomton is lso clled genertor of the LTL formul). B. LTL Gmes An LTL gme is de ned on grph G =(V, E, h, O), where V is the set of vertices, E V V is the set of edges, O is set of observbles, nd h : V O is n observtion mp. The gme speci ction is n LTL formul over the observble set O [15] (see lso [11] for the cse of grphs with no observbles). There re two plyers in the gme: P (the plyer) nd A (the dversry). The set of vertices V is prtitioned in two sets: V p, which is the set of sttes from which the plyer cn choose move (n edge leding to the next vertex), nd V, which is the set of sttes from which the dversry hs choice of n edge. We ssume tht the current vertex from set V is lwys known, nd not just its observble. An (in nite) ply consists of n in nite sequence of sttes resulted from n in nite sequence of trnsitions (edges) chosen by the two plyers. The plyer P wins ply if the produced word w O ω stis es the LTL formul. Strting from given initil stte, the plyer hs winning strtegy if, whenever the current stte is in V p, she mnges to choose edges such tht she wins the current ply, no mtter wht edges the dversry chooses when the current stte is in V. The gol of n LTL gme is to nd set of initil sttes W s V from where the plyer hs winning strtegies nd winning strtegy for plys strting in those initil sttes. The stndrd lgorithm for solving n LTL gme involves the trnsformtion of the non-deterministic Büchi utomton corresponding to the LTL formul into deterministic genertor [16]. Motivted by the simplicity of exposition nd by some complexity issues, we ssume tht the LTL formuls we re deling with ccept deterministic Büchi genertors (there re cses when this is not true [17], nd lgorithms for solving LTL gmes become more complicted). For more detils of the steps involved in solving LTL gmes the interested reder is referred to [16], [15]. The result of solving n LTL gme on G =(V, E, h, O) (s outlined bove) is set of initil sttes W s V (if non-empty) nd winning strtegy W, which is strtegy (s in De nition 4) gurnteeing the stisfction of the LTL formul whenever the initil stte is in the set W s. De nition 4: A strtegy is n utomton W = (S, V, O, s 0, τ, π), where: S is the nite set of sttes (the memory), V is the input lphbet, O is the set over which the LTL formul φ ws de ned, 3111

3 s 0 S is the initil stte (initil memory), τ : S O S is the memory updte function, π : S V p V is the function giving the chosen move when the current stte of G is in V p. The sttes S result from the sttes of the deterministic utomton (e.g., Büchi) corresponding to the LTL formul. The current vertex of G gives the input of W nd it is used for generting the plyer s moves, while the corresponding observble is used for updting the internl memory of W. Thus, given the current stte (memory) of W, the strtegy π depends on the current stte of G, while the memory updte function τ depends only on the observble of this stte. The reltion between grphs nd trnsition systems is immedite. When plying n LTL gme on trnsition system, insted of choosing the next stte, we choose n input producing the desired trnsition. This is possible if plyer s sttes hve only deterministic outgoing trnsitions, nd then the function π from the winning strtegy will tke vlues in the input set of the trnsition system insted of V. III. PROBLEM FORMULATION AND APPROACH In this pper we consider the following problem: Problem 1: Given non-deterministic trnsition system T =(Q, Σ,δ) nd n LTL formul φ over Q, nd set of fesible initil sttes Q 0 Q nd control strtegy such tht formul φ is stis ed by ll resulting in nite words. Remrk 2: If the trnsition system T ws deterministic (i.e., δ : Q Σ Q), then solution to Problem 1 could be found by using n ide similr to model checking [18], [7], [19]: the LTL formul φ is trnsformed into Büchi utomton B φ nd the product utomton T B φ is computed. In this product utomton n ccepting run with speci c structure is found nd projected to run of T. Since T is deterministic, control strtegy implementing the desired run cn be constructed. We propose to use the frmework of LTL gmes to provide solution to Problem 1. For this, we need to reformulte the problem from grph to trnsition system, nd then de ne prtition of the set of sttes into plyer s sttes nd dversry s sttes. Remrk 3: One quick wy of solving this would be to lbel s dversry s stte ech stte from where there exists t lest one input yielding non-determinist trnsition. However, such n pproch would be pretty conservtive, becuse if such n dversry s stte hd more thn one fesible input, we wouldn t use our freedom of choosing one from these inputs nd thus restricting the set of possible next sttes. Our pproch involves three min steps. First, we crete new trnsition system T g with observbles, where we choose the plyer s sttes by looking for sttes nd inputs with deterministic behvior. T g will, in generl, hve more sttes nd inputs thn T, nd its observble set will be Q. Second, we serch for winning strtegy for the LTL gme on T g. Finlly, when such strtegy exists, we dpt it to control strtegy providing t ech step the input to be pplied to our initil trnsition system T. IV. CONTROL STRATEGY USING LTL GAMES We strt by trnsforming T into nother trnsition system T g =(Q g, Σ g,δ g,h g,q), where: Q g is the nite set of sttes, prtitioned into sets Q p (plyer s sttes) nd Q (dversry s sttes), Σ g is the new lphbet, Σ Σ g, δ g : Q g Σ g 2 Qg is the trnsition function, h g : Q g Q is the observtion mp. The construction of T g is described in Section IV-A. Here we give some de nitions nd outline how the gme is plyed. The trnsition function δ g is de ned s: its restriction to Q p Σ g is deterministic, i.e. q Q p nd σ Σ g, δ g (q, σ) 1, its restriction to Q Σ g is non-deterministic, nd there is only one fesible (enbled) input in every stte from Q, i.e. q Q, there exists n unique input σ Σ g such tht δ g (q, σ) 2 nd ll other inputs σ Σ g \ {σ} re not fesible t q (or δ g (q, σ ) =0). These properties of δ g show tht when the current stte is in Q p, the plyer P cn lwys choose the next stte from fesible set (by pplying fesible input), nd when the current stte is in Q, P hs no control on the next stte (we only know tht the next stte will be in subset of Q g, given by the unique fesible input). Thus, the structure of T g is pproprite for n LTL gme s described in Section II-B. By solving the LTL gme over T g with winning condition φ, one cn obtin set of winning initil sttes, W s Q g, nd, if W s, winning strtegy in the form of nite utomton, similr to the one from De nition 4. When W s is nonempty, the winning strtegy is W =(S, Q g, Q, s 0, τ, π), where: S is nite set of sttes, Q g is the input lphbet, Q is the set over which the LTL formul φ ws de ned, s 0 S is the initil stte, τ : S Q S is the memory updte function, π : S Q p Σ g is the function giving the choice of the next pplied input (move) when the current stte of T g is in the set Q p. The difference between this winning strtegy nd the one from De nition 4 is tht here π tkes vlues in Σ g insted of Q g. Becuse of the mentioned properties of δ g, every move the plyer should mke when the ply is in Q p corresponds to n unique input from Σ g. In order to obtin n input to be pplied to T g in every stte, the mp π is extended to S Q g, by letting π(s, q) =σ, where σ Σ g is the only fesible input in stte q, q Q. The winning strtegy given by utomton W is implemented in feedbck form. At ech step, the following ctions re performed ( step is the intervl between two successive trnsitions of T g ): (1) it reds the current stte of T g, (2) ccording to function π, it pplies n input from the set Σ g to T g, nd (3) it updtes its stte ccording to function τ. If the initil stte of T g is in the set W s, this feedbck control strtegy gurntees tht the gme will be won. Note 3112

4 q 2 q 3 q 4 b () 1 1 q 2 b b 2 q 4 3 q 3 b q 4 1 q 4 Fig. 1. () Trnsition system T : nd q 3 re det sttes, q 2 is ndet, nd q 4 is mix. (b) Obtined T g: q 4 from T ws split in two ndet sttes in T g nd two new inputs ppered in Σ g. In T g, the observbles re plced ner the sttes, nd the dversry s sttes re doubly encircled. All deterministic trnsitions re represented with continuous lines, non-deterministic ones with dshed lines, nd ech input is plced ner the corresponding trnsition(s). tht this feedbck controller ssumes tht we cn lwys red the current stte of T g nd not only its observble. A. Construction of T g We present here the min ides used for trnsforming the initil trnsition system T into the new trnsition system T g. Due to spce constrints, we do not include ny lgorithm nd we refer to [20] for full pseudo-codes. For creting the sttes nd the observtion mp of T g, we lbel the sttes from Q by function l : Q {det, ndet, mix}: stte with only deterministic outgoing trnsitions or without outgoing trnsitions is lbelled with det, stte with only one fesible input, for which the outgoing trnsitions re non-deterministic, is lbelled with ndet, nd stte with more fesible inputs, from which t lest one yields non-deterministic trnsitions, is lbelled with mix. Ech mix stte is split in one or more ndet sttes nd t most one det stte in the new trnsition system T g (for simplicity of writing, we ssume tht the ttributes ndet nd det for sttes of T g hve exctly the sme mening s in T ). The observble of ll these new sttes is the old mix stte. Ech det (respectively ndet) stte from T corresponds to det (respectively ndet) stte in T g. Thus, T g will hve only det (in the subset Q p ) nd ndet sttes (in the subset Q ). For better understnding we present in Fig. 1 simple exmple of trnsforming trnsition system T into T g. The input lphbet Σ g nd the trnsition function δ g re creted fter the set of sttes Q g = Q p Q is obtined. T g hs lrger input set thn T becuse the det sttes from T must remin det in T g. This cn be explined by following the exmple in Fig. 1: in T, the det stte q 3 hs trnsition with input b to the mix stte q 4. q 3 corresponds to one stte of T g ( 3), but q 4 is split in two sttes ( 4 nd q 2 4). Since we need deterministic trnsitions for going from 3 to either 4 or q 2 4, we introduce the new inputs b 1 nd b 2. Thus, by pplying b 1 when T g is in 3, the next stte is 4 nd there is only one next fesible input (). In T this hs the effect of pplying input b while in q 3 nd enforcing input when q 4 is reched. Similrly, when b 2 is pplied to T g while in 3, this induces the sequence of inputs b, b strting from q 3 in T. Although the dversry hs full control on the trnsitions from 4 nd q 2 4 in T g, the plyer hs control in going to either b 2 b 1 (b) q 4 q 2 4 or q 2 4, thus restricting the dversry s power. Following the sme ide, ech det stte of T g (including those obtined by splitting mix sttes of T ) might dd some new inputs to the lphbet Σ g. In order to dpt these new inputs to inputs of T, mp α :Σ g Σ is creted, which will be used by the controller for T. For ll inputs in set Σ, α is just the identity mp. The ndet sttes of T g (from dversry s set Q ) will not dd new inputs. Ech of these sttes will hve only one fesible input (from set Σ), nd thus the dversry power in these sttes comes only from the non-determinism induced by tht fesible input. B. Solution to Problem 1 We now hve ll the necessry informtion to present the solution to Problem 1. The set of initil sttes of T from where the controller gurntees the stisfction of the LTL formul φ is: Q 0 = {q Q q W s s.t.h g (q )=q} (1) The set Q 0 from (1) is in ccordnce with the solution of the LTL gme for the trnsition system T g : if the initil stte of T g is in the set W s, then the strtegy generted by utomton W gurntees the stisfction of formul φ by ny produced run of T g. Of course, Q 0 = if nd only if W s =, cse when we conclude tht we cnnot solve Problem 1. If Q 0, the control strtegy for the trnsition system T will be feedbck controller, which will lso include the interfce between T nd the ply on T g, supervised by W. Agin, we brie y describe the opertions performed by this controller nd we refer to [20] for n lgorithm ccompnied by detiled explntions. At ech discrete step, correct stte of T g is rst picked: this stte must be in concordnce with the previous stte of T g nd the previously pplied input, nd its observble must equl the current stte of T. Then, bsed on the winning strtegy W, n input for T g is found, which is then dpted to n input for T by using mp α. Whenever the resulted input for T g is in the set Σ g \Σ, the dversry s power in the next stte of T will be reduced (for exmple, see the cse depicted in Fig. 1, when input b 1 or b 2 ppers). For better understnding, we complete the exmple from Fig. 1 by considering the formul φ = (q 4 ), mening visit q 4 nd then. By using the presented pproch, we obtin Q 0 = {q 3,q 4 }. When T strts from q 3, the rst two inputs pplied by the controller re b, (corresponding to input b 1 of T g ). If the resulted stte is q 3, the input will be pplied nd is reched. After is visited, the controller will keep pplying ny of the fesible inputs in the visited sttes (in our implementtion, will be lwys pplied). Note tht by trying to solve the sme problem by using the pproch from either Remrk 2 or Remrk 3, we would get no solution for the problem (in the rst cse there re no deterministic trnsitions out of q 4, nd in the second cse, once q 4 is reched, the dversry cn keep choosing the self-loop in this stte). 3113

5 C. Conservtiveness nd Complexity Our pproch is conservtive in the following sense: when creting the trnsitions of T g from ndet (dversry s) sttes, we don t restrict the set of inputs to be pplied to T t the next discrete step. For illustrting this, consider the exmple in Fig. 1 nd the formul φ =. Our pproch gives Q 0 = {,q 3,q 4 }, while one cn observe tht φ cn be lso stis ed by strting from q 2 nd keep pplying input, which eventully leds to being visited. However, pproches from Remrks 2 nd 3 give only strtegies from Q 0 = {,q 3 }. This problem does not pper in the cse of det sttes of T g, becuse of the inputs in {Σ g \ Σ}, which restricts the set of choices for the next input. If there re no trnsitions between mix nd ndet sttes of T, then this source of conservtiveness does not pper. Another source of conservtiveness in our pproch is tht we ssumed n LTL formul over the sttes of T.A more generl setting would involve trnsition system T with n observtion mp nd n LTL formul over the set of observbles. If sttes of T hving the sme observble were indistinguishble (not like in the cse of T g ), the proposed method wouldn t be suitble, becuse we couldn t nd stte of T g (with the correct observble) nd be sure tht the fesible inputs in tht stte re lso fesible in the rel stte of T. However, this formultion is motivted by our prticulr motion plnning ppliction, where we ssume tht the environment is prtitioned nd the resulting regions re lbelled (see exmple in Section V). Finlly, when our pproch fils in providing solution, we declre Problem 1 to be unfesible. Other pproches could use n itertive procedure obtining t ech step ner discrete representtion (tilored to speci c problem) nd trying to solve Problem 1 for tht re ned trnsition system. The upper bound complexity of the presented pproch is in generl high, becuse n LTL gme on grph T g is solvble in polynomil time in the size of the grph nd in double-exponentil time in the size of the LTL formul [21], [22]. However, there hve been isolted some LTL frgments ccepting deterministic Büchi utomt nd gurnteeing much lower complexity [22]. V. CASE STUDY To illustrte the pproch proposed in the pper, we consider simple plnning nd control problem for plnr continuous system evolving in rectngulr tringulted environment (see Fig. 2). The dynmics, rectngulr bounds, nd control constrints re given in eqution (2). We ssume tht the environment is prtitioned in 8 tringles (lbelled by q i,i=1,...,8 in Fig. 2), nd the tsk is speci ed by the LTL formul φ = q 4 q 7. In other words, the tsk requires tht regions q 4 nd q 7 be visited (in ny order) by the trjectories of the closed loop system. [ ] [ ] [ ẋ = x + u x [ 3, 4] [ 3, 3], u [ 1, 1] [ 1, 1] ], (2) !1!2 q 4 q 2 q 3!3!3!2! Fig. 2. Tringulr prtition (thin blck lines) nd two continuous trjectories strting from region q 3 nd stisfying formul φ. 4 q q 3 q 2 17 q Fig. 3. Trnsition system T corresponding to the prtition in Fig. 2. Deterministic trnsitions re represented with continuous lines, nondeterministic ones with dshed lines, nd ech input is plced ner the corresponding trnsition(s). A forml de nition of the semntics of the formul over continuous trjectories of the closed loop system is beyond the scope of this pper, nd cn be found in [19]. However, intuitively, trjectory stis es the formul if the word produced by the sequence of regions visited by the system s time evolves stis es the formul. As in [7], [19], we strt by constructing trnsition system T with sttes Q = {,...,q 8 } corresponding to the prtition elements nd with trnsitions cpturing the bility of designing f ne feedbck controllers such tht tringle is either left in nite time to neighbor, or becomes n invrint for the closed loop system. The construction of such controllers is computtionlly ef cient, since it cn be reduced to opertions on polyhedrl sets [19]. However, unlike in [7], [19], we don t restrict our ttention to controllers mking tringle n invrint or driving ll initil sttes in tringle to one neighbor (this utomticlly induces deterministic quotient, with trnsitions lbelled by the corresponding controllers). Enbled by recent results on control of f ne systems in simplices [9], we exhustively check for existence of controllers mking tringle n invrint, driving ll initil sttes in tringle to neighbor, two neighbors, nd three neighbors. While drmticlly 7 q q 6 q 7 q 7 67 q q 5 q

6 reducing the conservtiveness of the pproch, it induces non-deterministic trnsition system. The obtined trnsition system T is shown in Fig. 3, where the lbels (inputs) stnd for f ne feedbck controllers with n obvious mening. Note tht, motivted by the prticulr form of the formul (which is rechbility type formul) we do not consider the controllers corresponding to invrince, nd, therefore, T does not hve ny self trnsitions. By using the pproch from Section IV, we conclude tht the formul φ cn be stis ed by trjectories strting from ny tringle q i,i = 1,...,8. The LTL formul φ ccepts deterministic Büchi genertor B with 4 sttes, nd the constructed trnsition system T g hs 11 sttes. The winning strtegy W hs 4 sttes nd it is utomticlly generted, together with the set Q 0, fter providing the inputs T nd B. When implementing the (discrete) controller to the continuous system [19], discrete trnsition in T tkes plce when the current tringle is left, nd ech input to be pplied to T is mpped to n f ne feedbck controller s described in [9], pplied s long s the continuous trjectory evolves in the current tringle. The most interesting cse is tht of trjectories inititing in tringle q 3, becuse its corresponding discrete stte hs only non-deterministic outgoing trnsitions. We show two continuous trjectories strting in region q 3 nd corresponding to the winning strtegy we found s in Section IV. Even though the continuous trjectories rech different sequences of tringles, they both stisfy the formul. Only the rst prt of ech trjectory is shown (becuse the trjectories re in nite); fter region q 7 is hit, the trjectories will keep oscillting between tringles q 7 nd q 2. Solving the sme problem by ignoring the nondeterministic trnsitions of T nd using the pproch from Remrk 2, one would obtin tht the formul could be stis ed by strting from ny tringle except q 3 (note tht stte q 3 doesn t hve deterministic outgoing trnsitions in T ). By using the method from Remrk 3, the dversry would hve control on ll the outgoing trnsitions from sttes q 2,q 3,q 4,q 5, nd winning strtegies exist only when strting from sttes q 6,q 7,q 8. In the lter cse, q 7 will be rst visited, nd then q 4 (if q 4 ws visited rst, smrt dversry would keep the ply between sttes q 4 nd q 3, nd thus the gme would be lost). VI. CONCLUSIONS In this pper, we developed control strtegy for nondeterministic trnsition from speci ction given s n LTL formul over its set of sttes. The method is bsed on LTL gmes, nd consists of three steps: the construction of new trnsition system, the solution of n LTL gme, nd the genertion of the control strtegy for the initil system. The problem is motivted by nd pplied to plnning nd control of continuous systems from symbolic speci ctions given s temporl logic formuls. VII. ACKNOWLEDGMENTS This work ws prtilly supported by NSF CAREER nd NSF t Boston University. The uthors wish to thnk Pulo Tbud for useful discussions on this topic. REFERENCES [1] J. C. Ltombe, Robot Motion Plnning. Kluger Acdemic Pub., [2] H. Choset, K. M. Lynch, S. Hutchinson, G. Kntor, W. Burgrd, L. E. Kvrki, nd S. Thrun, Principles of Robot Motion: Theory, Algorithms, nd Implementtions. MIT Press, Boston, [3] S. Lindemnn nd S. LVlle, Computing smooth feedbck plns over cylindricl lgebric decompositions, in Proceedings of Robotics: Science nd Systems, Cmbridge, USA, June [4] D. Conner, H. Choset, nd A. Rizzi, Integrted plnning nd control for convex-bodied nonholonomic systems using locl feedbck control policies, in Proceedings of Robotics: Science nd Systems, Cmbridge, USA, June [5] C. Belt, V. Isler, nd G. J. 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