Coordinating the Motions of Multiple Robots with Specified Trajectories

Transcription

1 Coordnatng the Motons of Multple Robots wth Specfed Trajectores Srnvas Akella Seth Hutchnson Department of Computer Scence Beckman Insttute Rensselaer Polytechnc Insttute Unversty of Illnos, Urbana-Champagn Troy, NY Urbana, IL Abstract Coordnatng the motons of multple robots operatng n a shared workspace wthout collsons s an mportant capablty. We address the task of coordnatng the motons of multple robots when ther trajectores (defned by both the path and velocty along the path) are specfed. Ths problem of collson-free trajectory coordnaton arses n weldng and pantng workcells n the automotve ndustry. We dentfy suffcent and necessary condtons for collson-free coordnaton of the robots when only the robo tmes can be vared, and defne correspondng optmzaton problems. We develop mxed nteger programmng formulatons of these problems to automatcally generate mnmum tme solutons. Ths method s applcable to both moble robots and artculated arms, and places no restrctons on the number of degrees of freedom of the robots. The prmary advantage of ths method s ts ablty to coordnate the motons of several robots, wth as many as 20 robots beng consdered. We show that, even when the robot trajectores are specfed, mnmum tme coordnaton of multple robots s NP-hard. 1 Introducton Coordnatng the motons of multple robots wthout collsons as they perform a task n a shared workspace s an mportant capablty. We focus on coordnatng the motons of multple robots constraned to follow specfed trajectores. By trajectory, we mean both the geometrc specfcaton of the path and the velocty at whch the robot traverses the path. We outlne ths trajectory coordnaton problem and defne correspondng optmzaton problems where the goal s to fnd the mnmum-tme collson-free robot coordnatons when only the robo tmes can be changed. There are a number of applcatons n whch ths trajectory coordnaton task s the exact problem to be solved. Consder schedulng the motons of multple robots n a weldng, spray pantng, or assembly workcell to mnmze the cycle tme. Snce the robots have overlappng workspaces, we must coordnate ther motons to avod collsons between robots. We assume that the gven trajectory of each ndvdual robot should not be modfed snce t may take nto account collsons wth statonary obstacles, have a desred velocty profle, or have desred wat tmes at crtcal ponts. Alternatve approaches to mnmzng the completon tme, such as velocty tunng of the robots, may be napproprate; for example, a pantng robot must follow a gven trajectory to spray pant unformly. We dentfy suffcent and necessary condtons for collson-free coordnaton of multple robots and formulate the task as an optmzaton problem usng a mxed nteger programmng formulaton that can be solved usng commercal solvers. We use collson detecton software to dentfy potental collson condtons. The prmary advantage of ths method s ts ablty to handle many robots, each wth several degrees of freedom. We place no restrctons on the number of degrees of freedom of the robots. Ths approach also apples to moble robots and Automated Guded Vehcles (AGVs) movng along fxed paths wth specfed trajectores, and can also ncorporate the motons of manpulator arms on moble robots. The paper s organzed as follows. Secton 2 brefly dscusses related work. Secton 3 defnes the problem, formulates a set of optmzaton problems, and descrbes suffcent condtons for collson-free moton of multple robots. Secton 4 presents a mxed nteger programmng formulaton for coordnatng the motons of multple robots wth specfed trajectores. Secton 5 dscusses necessary condtons for collson-free moton and descrbes a follow-the-leader strategy. Secton 6 descrbes useful extensons to the basc problem. Secton 7 dscusses the computatonal complexty of the coordnaton problem. Secton 8 descrbes our prelmnary mplementaton of the planner and expermental results. Secton 9 outlnes drectons for future work. 2 Related Work Moton plannng for multple robots s a broad research area (see [11] for an overvew). In the most general case, the problem s to have each robot move from ts ntal to ts goal confguraton, whle avodng collsons wth statc obstacles or wth other robots. Ths problem s hghly underconstraned, and very few researchers have attempted to deal wth t drectly. Hopcroft, Schwartz, and Sharr [7] showed that even a smplfed two-dmensonal case of the problem s PSPACE-hard.

2 A slghtly more constraned verson of the problem s obtaned when all but one of the robots have specfed trajectores. Ths s essentally the problem of plannng a path for a sngle robot among movng obstacles, whch has been treated by Ref and Sharr [17] and Kant and Zucker [9]. One can generalze ths problem to obtan a heurstc soluton to the problem of plannng the motons of multple robots. Erdmann and Lozano-Perez [3] assgn prortes to robots and sequentally search for collson-free paths for the robots, n order of prorty, n the confguraton-tme space. At each teraton, prevous robots are treated as movng obstacles. If the problem s further constraned so that the paths of the robots are specfed, one obtans a path coordnaton problem. O Donnell and Lozano-Perez [16] developed a method for path coordnaton of two robots. LaValle and Hutchnson also addressed a smlar problem n [12], where each robot was constraned to reman on a specfed confguraton space roadmap durng ts moton. The work most closely related to ours s that of Leroy, Laumond, and Smeon [14]. They perform path coordnaton for over a hundred robots. However the sze of the largest subset of robots wth ntersectng paths s 10. In ths paper, we address an even more constraned verson of the multple robot moton plannng problem: the trajectory coordnaton problem where the trajectory of each robot, ncludng the tme dervatves along the path, s specfed. Prevous work on trajectory coordnaton has focused almost exclusvely on dual robot systems (Ben and Lee [1], Chang, Chung and Lee [2]). Shn and Zheng [19] show that for a two-robot system, generatng tme-optmal trajectores for each robot ndependently and then delayng the start tme of one of the robots leads to a mnmal fnsh tme provded the collson regon satsfes a strong connectvty assumpton. (A suffcent condton for ths assumpton s that the robots may collde only once durng ther moton.) The trajectory coordnaton problem for multple robots s closely related to jobshop schedulng problems (Garey, Johnson, and Seth [5], Lawler et al. [13]). Here space s the common resource, and there are addtonal trajectory constrants. We model coordnaton of robots wth fxed trajectores as no-wat jobshop problems (Sahn and Cho [18], Goyal and Srskandarajah [6]). 3 Problem Formulaton The general problem that we are tryng to solve can be expressed as an optmzaton problem: Gven a set of robots wth specfed paths and velocty profles on those paths, fnd a set of parameterzatons for these paths such that the total executon tme for the ensemble of robots s mnmzed, the velocty constrants on the paths are satsfed, and no collsons occur. To make ths problem more precse, we frst turn to a bref revew of paths and ther parameterzatons (Secton 3.1). Ths wll lead to a precse and straghtforward characterzaton of the set of parameterzatons under whch the robots velocty profles reman nvarant. We then develop a characterzaton for collsons that can occur between robots (Secton 3.2), and a set of suffcent condtons for collson-free coordnaton of the robots (Secton 3.4). 3.1 Trajectores and Ther Parameterzatons We denote the th robot by A, a confguraton space by C, and a confguraton by q C. By path we mean the geometrc specfcaton of a curve n confguraton space γ : ζ [0, 1] γ(ζ) = q C A dfferentable functon τ gven by τ : t [0, T ] τ(t) = ζ [0, 1] wth τ(0) = 0 and τ(t ) = 1 s a reparameterzaton of the path γ. For our problem, t s a tme varable, and T s some constant such that all robots wll have completed ther tasks pror to tme T. A path together wth a parameterzaton defnes a trajectory. By trajectory we mean a path wth the velocty of the robot specfed at every pont along the path. We wll often smplfy notaton, and denote a trajectory as γ(t) rather than explctly representng the parameterzaton. For our problem, robot veloctes are specfed a pror. One way to do ths s to specfy an orgnal parameterzaton for γ, say τ, such that the tme dervatves of τ provde the desred velocty profle. Thus, any reparameterzaton, say τ, that gves the desred velocty profle wll be such that, for any ζ value along the path, the tme dervatves of τ and τ agree. It s easy to show that all such reparameterzatons are obtaned by merely changng the start tme of task executon. Wthout loss of generalty, we wll consder only the case where the start tmes for the robots are delayed,.e., { τ (t) τ (t t = start ) : t 0 : t <, (1) n whch 0 s the tme at whch robot A begns ts moton, and τ s the orgnally specfed parameterzaton. Note that ths equaton also mples that A remans motonless untl. Ths restrcton on possble reparameterzatons leads to the followng optmzaton problem. Optmzaton Problem I: Gven a set of robots wth specfed trajectores, fnd the startng tmes for the robots such that the total executon tme for the ensemble of robots s mnmzed and no collsons occur. We now turn our attenton to a set of suffcent condtons for collson-free moton for ths optmzaton problem. As wll be seen n Secton 4, these suffcent condtons lead to an optmzaton problem that can be solved usng mxed nteger lnear programmng.

3 3.2 Collson Zones: Geometry Here we develop the representaton for the relevant nteractons between robots, usng the above termnology for an ndvdual robot movng on a path wth a specfed velocty profle. We frst develop notaton to represent the set of ponts at whch the th robot, A, could possbly collde wth the j th robot, A j. For a specfc value of ζ, the subset of the workspace that s occuped by the th robot s denoted by A (γ (ζ )). A collson between two robots corresponds to the stuaton n whch A (γ (ζ )) A j (γ j (ζ j )). For the th robot, we denote by PB j the set of values of ζ such that when robot A s at confguraton γ (ζ ) there exsts a confguraton of another robot, A j, such that the two robots collde: PB j = {ζ ζ j [0, 1] s.t. A (γ (ζ )) A j (γ j (ζ j )) } In other words, PB j s the set of all ponts on the path of robot A at whch A could collde wth A j. (Our choce of the notaton PB j derves from the usual conventon of usng the notaton CB to denote ponts n the confguraton space at whch collsons occur.) The set PB j can be represented as a set of ntervals PB j = {[ζ 1 s, ζ 1 f ],..., [ζ m s, ζ m f ]} (2) where each nterval s a collson zone, and the subscrpts s and f refer to the start and fnsh of the kth collson, ndexed by the superscrpt k, and m denotes the number of collson zones for the robot A wth A j. There s a natural correspondence between the collson zones of PB j and the collson zones of PB j. In partcular, for each collson zone n PB j there s at least one collson zone n PB j that could result n collson of the two robots. We wll refer to these correspondng pars of collson zones as collson zone pars, denoted by PI j. The set of collson zone pars can be represented by a set of pars of ntervals: PI j = {< [ζ k s, ζ k f ], [ζ k js, ζ k jf ] >}. (3) Note that the superscrpt k serves to ndex the set of collson zone pars. As we show n Secton 3.4, t s straghtforward to use PI j to establsh a set of suffcent condtons for collson free schedulng of the robots. Note that PI j and PI j contan equvalent nformaton. Conceptually, collson zone pars are generated by computng the volume swept by each robot and determnng where t ntersects the volume swept by another robot. The ntersecton regons of the swept volumes of pars of robots gve the collson zone pars. Fgure 1 s an example of two translatng robots wth specfed trajectores that overlap n two collson zones. For ths example PB 12 = {[a 1, a 2 ], [a 3, a 4 ]} and PB 21 = {[b 1, b 2 ], [b 3, b 4 ]}. Collsons can occur only when ζ 1 [a 1, a 2 ] and ζ 2 [b 1, b 2 ] or when ζ 1 [a 3, a 4 ] and ζ 2 [b 3, b 4 ]. Thus, PI 12 = {< [a 1, a 2 ], [b 1, b 2 ] >, < [a 3, a 4 ], [b 3, b 4 ] >}. Intal A 1 a a 1 4 a a 2 3 b b 1 2 b b 3 4 A 2 Intal Fgure 1: Example wth two translatng robots. Goal Goal 3.3 Collson Zones: Tmng The collson zone pars descrbe the geometry of possble collsons, but for schedulng the robots, we are nterested n the tmng of the collsons. Thus, t s useful to develop a correspondng representaton for the tmes at whch two robots mght collde. For a specfed parameterzaton, τ, the set of tmes at whch t s possble that robot A could collde wth robot A j s gven by: T B j (τ ) = {t A (γ (τ (t))) A j (γ j (ζ j )), for some ζ j [0, 1], j} = τ 1 (PB j ). As wth PB j, the set T B j (τ ) can be represented by a set of ntervals, ndexed by superscrpt k, the endponts of whch are obtaned by applyng the nverse parameterzaton (.e., τ 1 ) to the endponts of the ntervals of PB j gven n (2): T B j (τ ) = {[τ 1 (ζs), k τ 1 (ζf k )]} (4) We refer to each nterval as a collson-tme nterval. As wth collson zones, there s a natural correspondence between collson-tme ntervals n T B j and T B j, and we refer to these pars as collson-tme nterval pars. For the two robots, A and A j, we denote the set of all collsontme nterval pars by CI j. We represent CI j as a set of pars of ntervals CI j = {< I 1, I 1 j >,..., < I n, I n j >}, (5) where the frst nterval I k of each par < I k, Ik j > corresponds to robot A and the second nterval Ij k corresponds to robot A j. Durng the tme nterval I k, A s n a specfc collson zone and A j s n a correspondng collson zone durng tme nterval Ij k. Note that CI j and CI j contan equvalent nformaton. The nterval pars n CI j (τ, τ j ), ndexed by k, can be determned from the mappng specfed n (3) by applyng the approprate nverse parameterzaton to the endponts of the collson zone ntervals n each collson zone par. That s, CI j (τ, τ j ) = {< [τ 1 (ζs), k τ 1 (ζf k )], [τ 1 j (ζjs), k τ 1 j (ζjf k )] >}. (6)

4 Note that f I k and Ik j do not overlap, then the two robots cannot be n the k th collson zone par smultaneously, and therefore no collson wll occur n ths collson zone par. Ths observaton forms the bass for the suffcent condtons gven n Secton 3.4. For notatonal convenence, we ntroduce the varables Tjs k and T jf k gven by T k js = τ j 1 (ζ k js) (7) T k jf = τ j 1 (ζ k jf ) (8) where Tjs k (respectvely T jf k ) denotes the tme at whch A j enters (resp. exts) the k th collson zone f j = 0. Note that wth multple robots, the notaton Tjs k s ambguous snce t does not specfy the partcular other robot that s nvolved n the collson. When we use ths notaton, the context wll make clear whch other robot s nvolved. See Fgure 2 for a graphcal llustraton of these quanttes. A 1 A 2 T 1s T 2s T 2f T 1f tme Fgure 2: Tmelnes for robots A 1 and A 2. The bold lnes correspond to the collson-tme ntervals for the robots. Snce our parameterzatons are restrcted to those that only delay the robo tmes, we wll always have parameterzatons of the form T 1 T 2 τ (t + ) = ζ = τ(t), (9) for each value of ζ [0, 1]. Invertng the parameterzatons τ and τ we obtan τ 1 (ζ) = τ 1 (ζ) +. (10) Usng ths notaton, we can wrte CI j (τ, τ j ) as CI j (τ, τ j ) = { < [T s 1 + tstart,tf 1 + tstart ], [Tjs 1 + tstart j, Tjf 1 + tstart j ] >, < [T n s + tstart [T n js + tstart j., Tf n + tstart ],, Tjf n + tstart j ] >}. 3.4 Suffcent Condtons for Collson-free Schedulng To prevent collsons between two robots A and A j, t s suffcent to ensure that the tmes at whch A could collde wth robot A j do not concde wth the tmes at whch A j could collde wth robot A, whch can be assured f the two robots are not n any collson zone par belongng to PI j at the same tme. Ths amounts to ensurng that there s no overlap between the two ntervals of any collson-tme nterval par for the two robots. If I k Ij k = for every collson-tme nterval par < I k, Ik j > CI j(τ, τ j ), then no collson can occur. (Note that t s not necessary to also check the nterval pars n CI j, snce preventng collson of A j wth A necessarly prevents collson of of A j wth A.) Ths suffcent condton leads to an optmzaton problem: Optmzaton Problem II: Gven a set of robots wth specfed trajectores, fnd the startng tmes for the robots such that the total executon tme for the ensemble of robots s mnmzed and no two ntervals of any collson-tme nterval par overlap. In Secton 4, we wll present a Mxed Integer Lnear Program that solves ths optmzaton problem. The suffcent condton s clearly not a necessary condton. For example, n a follow-the-leader stuaton where the robots move n the same drecton along ther paths n the collson zone, the follower robot s delayed unduly snce t wats for the leader to ext the collson zone before t enters the collson zone. For now, we note that ths s a conservatve strategy that guarantees that no collson occurs between the two robots. We wll dscuss an alternatve strategy that provdes the mnmum tme collson-free schedule n Secton Assumptons We make the followng assumptons to generate a collson-free coordnaton of the robot trajectores: 1. The only movng obstacles n the workspace are the robots, and the specfed trajectory for each robot does not result n collsons wth any statc obstacles. 2. Each robot does not collde wth the other robots when they are at ther start or goal confguratons. 3. The startng velocty of each of the robots s zero. 4. Each robot path s monotonc, that s, the robot does not back up along ts path. 5. Each robot executes ts specfed trajectory, wth no changes to ts specfed veloctes, once s movng. 6. The robot motons are sampled at suffcent resoluton so that no collsons occur durng the moton between successve collson-free confguratons. 4 An Integer Programmng Formulaton We frst develop a mxed nteger lnear programmng (MILP) formulaton for Optmzaton Problem II for the two robot case, and then the general case wth multple robots. s the start tme for robot A, whch s to be computed,

5 and T s the moton tme requred for robot A to traverse ts entre trajectory when startng at tme = The Two Robot Case Frst consder trajectory coordnaton of two robots A and A j. Assume the trajectory of each robot s gven and that the robots can collde wth each other n only one regon and that the robots do not collde multple tmes n the regon. For each robot, dentfy ts collson zone and compute the tme nterval durng whch t s n ts collson zone. The collson-tme nterval [T s, T f ] of robot A, where subscrpts s and f ndcate start and fnsh tmes respectvely, ndcates when robot A j can collde wth t. The collson-tme nterval [T js, T jf ] of robot A j s smlarly computed. The maxmum completon tme for the two robots s equal to the tme when the last robot completes ts task,.e., maxmum { + T, j + T j }. Snce we wsh to mnmze the completon tme whle ensurng the robots are not n ther collson zones at the same tme, the trajectory coordnaton problem can be stated as: Mnmze max{ 0 j 0 + T f < j + T, j + T j } + T js or + T s > j + T jf Snce the objectve functon and the constrants are not lnear, we transform them to a lnear form. Let the maxmum tme for robots A and A j to complete ther motons be t complete. Clearly t complete + T and t complete j + T j. The dsjunctve or constrant can be converted to an equvalent par of constrants usng an nteger zero-one varable δ j and M, a large postve number ([15]). Here M can be chosen to be T + T j. When robot A enters the collson zone frst, δ j = 0 and the constrant + T f < j + T js s actve, and when robot A j enters the collson zone frst, δ j = 1 and the constrant j + T jf < + T s s actve. The equvalent MILP formulaton s: Mnmze t complete t complete T 0 t complete j T j 0 + T f j T js Mδ j 0 j + T jf T s M(1 δ j ) 0 0 j 0 δ j {0, 1} 4.2 The Multple Robot Case In the general case, multple robots, pars of whch may have multple collson regons, must be coordnated. Here < [T k s, T k f ], [T k js, T k jf ] > denotes the kth collson-tme nterval par for the two robots A and A j. Let N j denote the number of collson-tme nterval pars for robots A and A j,.e., N j = CI j and let N robots be the number of robots. The bnary varable δ jk s defned to be 0 f robot A enters ts k th collson zone wth robot A j before robot A j and to be 1 f robot A j enters ts correspondng k th collson zone before robot A. A vald value for M s M = N robots =1 T. The MILP formulaton to coordnate the motons of the robots s: Mnmze t complete t complete T 0, 1 N robots + Tf k tstart j Tjs k Mδ jk 0, for all < [Ts k, T f k ], [T js k, T jf k ] > CI j, for 1 < j N robots j + Tjf k tstart Ts k M(1 δ jk) 0 for all < [Ts k, T f k ], [T js k, T jf k ] > CI j, for 1 < j N robots δ jk {0, 1}, 1 < j N robots, 1 k N j 0, 1 N robots. The resultng soluton s guaranteed to be a collsonfree trajectory coordnaton strategy for all the robots. The completon tme constrants and collson-tme nterval constrants are necessary for only those robots that may collde. Note that the MILP always has a feasble soluton move the robots n sequence wth only one robot n moton at any gven nstant. Fgure 3 shows the tmelnes for two robots wth multple collson ntervals, and Fgure 4 shows the collson-free sequencng of the start tmes of the robots. A 1 A 2 tme T 1 T 2 Fgure 3: Tmelnes for robots A 1 and A 2 wth multple collson ntervals. A 1 A 2 2 tme Fgure 4: Collson-free tmelnes for robots A 1 and A 2, wth robot A 2 beng delayed at ts start.

6 5 Necessary Condtons for Optmalty We have so far computed start tmes to ensure that no two robots are smultaneously n ther shared collson zones. Ths crteron for collson avodance can be overly conservatve, for example, when two robots A and A j are movng n the same drecton n a collson zone par. We can reduce the completon tme and derve the necessary condtons for collson avodance n such cases by permttng the robots to play follow the leader. Assume robot A moves frst n ts collson zone and A j follows t. We need to compute how much earler the lead robot A should start movng n ts collson zone, before the follower robot A j can enter ts collson zone, to avod a collson. Shn and Zheng [19] proved that for two robots wth a sngle collson regon, delayng the start tme of one of the robots provdes the tme-optmal trajectory modfcaton. They compute the mnmum delay tme for the collson-free coordnaton of two robots that have a sngle collson zone par by usng a bsecton search. The delay tme of the follower robot, or equvalently, the lead tme of the lead robot, s ntalzed to a value that guarantees the lead robot wll ext ts collson zone before the follower robot enters ts collson zone. The mnmum lead tme n the collson zone for whch the lead robot can stll avod a collson wth the follower robot s then computed usng bsecton search. We extend ths dea of computng the necessary condtons for collson avodance to multple robots, where pars of robots may have multple collson zone pars. Gven two robots A and A j that have more than one collson zone par, we treat each collson zone par ndependently when computng the lead tmes usng bsecton. For the kth collson zone par, we compute the mnmum tme Tjk lead that robot A must lead robot A j by at the start of ts kth collson zone to avod a collson, and the mnmum tme Tjk lead that robot A j must lead robot A by at the start of ts kth collson zone to avod a collson. The correspondng follow-theleader constrants are +Ts k lead +Tjk < j +Tjs k when A leads through the collson zone, or j +Tjs k lead +Tjk < + Ts k when A j leads through the collson zone. The maxmum value of Tjk lead s T k, the tme taken for robot A to traverse ts kth collson zone. Snce Tjk lead T k, we defne a new varable T e k = mn{t s k + T jk lead, T f k } where Tf k = T s k + T k. T e k, the collson-free entry tme, s the tme from start n robot A s trajectory, when A enters ts kth collson zone par before A j, at whch robot A j can enter ts collson zone wthout causng a collson. Smlarly, defne Tje k = mn{t js k + T jk lead, T jf k }. The updated follow-the-leader constrants are + Te k < tstart j + Tjs k when A leads through the collson zone, or j + Tje k < + Ts k when A j leads through the collson zone. The robots A and A j do not collde when ther start tmes satsfy these follow-the-leader constrants over all ther collson zone pars. To extend ths formulaton to multple robots, we nclude these dsjunctve constrants for every par of robots that can potentally collde. The mnmum completon tme over all robots s obtaned usng the followng formulaton: Mnmze t complete t complete T 0, 1 N robots + Te k tstart j Tjs k Mδ jk 0 for all < [Ts k, T f k ], [T js k, T jf k ] > CI j 1 < j N robots j + Tje k tstart Ts k M(1 δ jk) 0 for all < [Ts k, T f k ], [T js k, T jf k ] CI j, 1 < j N robots δ jk {0, 1}, 1 < j N robots, 1 k N j 0, 1 N robots. The soluton to the above MILP solves Optmzaton Problem I and gves the mnmum tme coordnated trajectores of the robots when only ther start tmes can change. 6 Extensons Our problem formulaton so far has focused on sngle body robots wth specfed trajectores. We now dscuss useful extensons to the basc formulaton. 6.1 Artculated Robots To coordnate artculated robots wth multple lnks, we consder motons of the ndvdual lnks. An artculated robot R conssts of a set of lnks {A }. Let R[] be the robot to whch lnk A belongs. The motons of lnks of an artculated robot are separated by constant tme offsets. Let A begn movng tme T R after the frst movng lnk of R[] begns movng. That s, = R[] + T R where R[] s the start tme of robot R[]. Let N lnks be the total number of robot lnks. Note that the start tme and moton tme of a lnk may depend on the start and moton tmes of lnks that precede t n the artculated chan. Thus the formulaton for a set of artculated robots s: Mnmze t complete t complete R[] T R T 0, 1 N lnks R[] + T R + Te k tstart R[j] Tj R Tjs k Mδ jk 0 for all < [T k s, T k f ], [T k js, T k jf ] > CI j, for 1 < j N lnks and R[] R[j] R[j] + T R j + T k je tstart R[] T R T k s M(1 δ jk) 0 for all < [T k s, T k f ], [T k js, T k jf ] > CI j for 1 < j N lnks and R[] R[j] δ jk {0, 1}, 1 < j N lnks, 1 k N j R[] 0, 1 N robots.

7 The completon tme constrants are necessary for all lnks of a robot that can potentally have a collson. The collsontme nterval constrants are necessary for only those robots that have one or more lnks nvolved n a potental collson. 6.2 Specfyng Sequencng Constrants In certan tasks, t may be necessary for one robot to complete a partcular operaton or reach a certan pont before another robot performs a subsequent operaton. Ths can occur n sequenced assembly tasks, or n weldng workcells where the prmary welds must be completed before secondary welds. Consder the requrement that A has to reach q before A j reaches q j. For the unmodfed trajectores, let the tme taken for A to reach q be T q and for A j to reach q j be T qj. The sequencng constrant can then be wrtten as + T q < j + T qj. Such constrants for multple robots can be easly added to the formulaton. 7 Complexty The nteger programmng formulaton of our problem suggests t s an NP-complete problem ([4]). We frst consder the decson verson of the No-wat Jobshop Schedulng problem (Sahn and Cho [18], Goyal and Srskandarajah [6]), whch s NP-complete. Each job conssts of an ordered set of tasks, where each task s to be performed by a specfc processor. The tasks for each job must be executed n sequence wthout breaks between them. Each processor can perform no more than one task at any tme nstant, and each job can be worked on by only one processor at any tme nstant. The goal s to mnmze the makespan (.e., the maxmum tme of completon of any task). The above problem can be transformed to our Multple Robot Schedulng problem. Let each job j model the trajectory of robot A j. Let each task t k [j] model the kth trajectory segment for robot A j, where each trajectory segment s a contguous collson zone segment or collson-free segment. Let processor p model the regon r, where each regon contans one or more trajectory segments. No two trajectory segments that are n the same regon can be executed at the same tme. The length of each task s the tme taken by the robot to traverse the correspondng segment. The goal s to mnmze the completon tme of the robots. It follows that the decson verson of the Multple Robot Schedulng problem s NP-complete, and that the optmzaton problem s NP-hard. 8 Implementaton We have mplemented software n C++ to coordnate the motons of polyhedral robots wth specfed trajectores (Fgure 5) and have a prelmnary mplementaton for artculated robots. We compute the collson zones usng the PQP collson detecton package (Larsen et al. [10]). The robot confguratons are specfed at constant tme ntervals. To de- Fgure 5: Overhead vew of the paths of 20 robots, wth ther ntal confguratons ndcated by sold cubes. Num. of Num. of Collson MILP robots collson detecton tme zones tme (secs) (secs) 3 2 < Table 1: Comparson of sample run tmes for 100 frames. termne the collson zones, each robot s stepped through ts trajectory, and at each trajectory pont, all the remanng robots are moved through ther complete trajectores to detect collsons. So for N robots where each robot has T trajectory ponts, collson detecton s performed O(N 2 T 2 ) tmes. Usng the computed collson-tme nterval pars, we generate the correspondng MILP formulaton and solve t usng CPLEX [8], a commercal optmzaton package. See Table 1 for runtme data on a Sun Ultra 10 for sngle body robots. Note that the problem complexty depends prmarly on the number of collson zones, to a lesser extent on the number of robots, and s relatvely ndependent of the number of degrees of freedom of the robots. Our prelmnary experments ndcate that the MILP tme domnates the runnng tme as the number of collson zones ncreases. Example anmatons may be seen at sakella/multplerobots/.

8 9 Concluson We have developed an optmzaton formulaton that enables the mnmum tme collson-free coordnaton of multple robots wth specfed trajectores when only ther start tmes can be changed. The prncpal advantage of our MILP formulaton s that t permts the collson-free coordnaton of a large number of robots (up to 20 robots). The problem complexty depends on the number of robots and the number of potental collsons, and s relatvely ndependent of the number of degrees of freedom of the robots. Although the optmal trajectory coordnaton of multple robots s NPhard, the avalablty of effcent collson detecton software and nteger programmng solvers makes ths approach practcal. There are several ssues for future work. Developng polynomal tme approxmaton algorthms for the task of selectng start tmes and characterzng the qualty of these solutons s mportant. An alternatve approach to mnmzng the completon tme s modfyng trajectores by tunng the velocty of each of the robots. Identfyng the condtons under whch we can do ths, and developng technques to generate the optmzed trajectores s mportant. Explorng stochastc versons of the task that nvolve tmng uncertantes would be useful. Fnally, explorng applcatons of ths work n computer graphcs for choreographng anmaton characters s another nterestng drecton. Acknowledgments Srnvas Akella was supported n part by RPI and the Beckman Insttute at UIUC. Seth Hutchnson was supported by NSF under Award Nos. CCR and IIS Thanks to Prasad Akella and Charles Wampler at General Motors for suggestng the problem and helpful dscussons. Andrew Andkjar mplemented anmaton software that nterfaced wth PQP and helped generate examples. Dscussons wth Jufeng Peng helped clarfy several ponts n the paper. References [1] Z. Ben and J. Lee. A mnmum tme trajectory plannng method for two robots. IEEE Transactons on Robotcs and Automaton, 8: , June [2] C. Chang, M. J. Chung, and B. H. Lee. Collson avodance of two robot manpulators by mnmum delay tme. IEEE Transactons on Systems, Man, and Cybernetcs, 24(3): , Mar [3] M. Erdmann and T. Lozano-Perez. On multple movng objects. Algorthmca, 2(4): , [4] M. R. Garey and D. S. Johnson. Computers and Intractablty: A Gude to the Theory of NP-Completeness. W. H. Freeman and Company, New York, [5] M. R. Garey, D. S. Johnson, and R. Seth. The complexty of flowshop and jobshop schedulng. Mathematcs of Operatons Research, 1: , [6] S. K. Goyal and C. Srskandarajah. No-wat shop schedulng: Computatonal complexty and approxmate algorthms. Opsearch, 25(4): , [7] J. E. Hopcroft, J. T. Schwartz, and M. Sharr. On the complexty of moton plannng for multple ndependent objects: PSPACE-hardness of the warehouseman s problem. Internatonal Journal of Robotcs Research, 3(4):76 88, [8] ILOG, Inc., Inclne Vllage, NV. CPLEX 6.0 Documentaton Supplement, [9] K. Kant and S. W. Zucker. Toward effcent trajectory plannng: The path-velocty decomposton. Internatonal Journal of Robotcs Research, 5(3):72 89, Fall [10] E. Larsen, S. Gottschalk, M. Ln, and D. Manocha. Fast dstance queres usng rectangular swept sphere volumes. In IEEE Internatonal Conference on Robotcs and Automaton, San Francsco, CA, Apr [11] J.-C. Latombe. Robot Moton Plannng. Kluwer Academc Publshers, Norwell, MA, [12] S. M. LaValle and S. A. Hutchnson. Optmal moton plannng for multple robots havng ndependent goals. IEEE Transactons on Robotcs and Automaton, 14(6): , Dec [13] E. L. Lawler, J. K. Lenstra, A. H. G. R. Kan, and D. B. Schmoys. Sequencng and schedulng: Algorthms and complexty. In S. C. Graves, A. H. G. R. Kan, and P. H. Zpkn, edtors, Handbooks n Operatons Research and Management Scence, Vol. 4, Logstcs of Producton and Inventory, pages North-Holland, [14] S. Leroy, J.-P. Laumond, and T. Smeon. Multple path coordnaton for moble robots: a geometrc algorthm. In 16th Internatonal Jont Conference on Artfcal Intellgence (IJ- CAI 99), pages , Stockholm, Sweden, Aug [15] G. L. Nemhauser and L. A. Wolsey. Integer and Combnatoral Optmzaton. John Wley and Sons, New York, [16] P. A. O Donnell and T. Lozano-Perez. Deadlock-free and collson-free coordnaton of two robot manpulators. In IEEE Internatonal Conference on Robotcs and Automaton, pages , Scottsdale, AZ, May [17] J. Ref and M. Sharr. Moton plannng n the presence of movng obstacles. In Proceedngs of the 26th Annual Symposum on the Foundatons of Computer Scence, pages , Portland, Oregon, Oct [18] S. Sahn and Y. Cho. Complexty of schedulng shops wth no wat n process. Mathematcs of Operatons Research, 4(4): , Nov [19] K. G. Shn and Q. Zheng. Mnmum-tme collson-free trajectory plannng for dual robot systems. IEEE Transactons on Robotcs and Automaton, 8(5): , Oct