Reciprocal Collision Avoidance for Multiple Car-like Robots

Transcription

1 Recprocal Collson Avodance for Multple Car-lke Robots Javer Alonso-Mora, Andreas Bretenmoser, Paul Beardsley and Roland Segwart Abstract In ths paper a method for dstrbuted recprocal collson avodance among multple non-holonomc robots wth bke knematcs s presented. The proposed algorthm, bcycle recprocal collson avodance (B-ORCA), bulds on the concept of optmal recprocal collson avodance (ORCA) for holonomc robots but furthermore guarantees collson-free motons under the knematc constrants of car-lke vehcles. The underlyng prncple of the B-ORCA algorthm apples more generally to other knematc models, as t combnes velocty obstacles wth generc trackng control. The theoretcal results on collson avodance are valdated by several smulaton experments between multple car-lke robots. I. INTRODUCTION AND RELATED WORK In ths paper, a novel collson avodance strategy for a group of car-lke robots s presented. Varous applcaton areas throughout research and ndustry have seen an evergrowng nterest n moble robots. Industral and servce robots are mostly non-holonomc, and often desgned as car-lke vehcles. A partcular example of car-lke vehcles deployed n an ndustral settng are the MagneBkes [], compact robots wth bcycle knematcs desgned for the collaboratve nspecton n power plants. Ths and all other applcatons, where multple car-lke robots nteract n ther workspaces, requre recprocal collson avodance methods. Movng a vehcle on a collson-free path s a wellstuded problem n robot navgaton. The work n [], [] and [] presents representatve examples of collson avodance methods for sngle moble robots. Bascally, smlar approaches as n the sngle robot cases can be appled n the context of collson avodance for multple robots. However, the ncrease n robot densty and collaboratve nteracton needs methods that scale well wth the number of robots and avod collsons as well as oscllatons. The collson avodance approaches are extended n [] among others for multple robots by decouplng path plannng and coordnaton. In ths lne, [6] presented a method based on velocty profles and schedulng to navgate several cars n a common envronment. Collsons are then avoded but some of the cars need to pause and stop completely to let others move ahead freely. Other work nvestgated potental felds [7] and cooperatve control laws [8] to drect a group of robots to ther objectves whle avodng collsons. Decentralzed control helps lowerng computatonal cost and ntroduces addtonal robustness and flexblty to the multrobot system. The problem of navgatng car-lke robots n Ths work was partally supported by ALSTOM. J. Alonso-Mora, A. Bretenmoser and R. Segwart are wth the Autonomous Systems Lab, ETH Zurch, 89 Zurch, Swtzerland {jalonso,andrbre,rsegwart}@ethz.ch J. Alonso-Mora, P. Beardsley are wth Dsney Research Zurch, 89 Zurch, Swtzerland {jalonso,pab}@dsneyresearch.com dynamc scenaros has also been studed, wth a great nterest n navgaton among humans [9]. A successful approach for ths knd of scenaros based on a dynamc wndow was proposed n []. Our approach bulds on Optmal Recprocal Collson Avodance (ORCA) [] for holonomc robots and extends t to robots wth car-lke knematcs by usng a trajectory trackng control [], whch s specfc for ths type of knematcs. However, the concepts here proposed apply to other knematc models n general snce the trajectory trackng controller s seen as a module that can be replaced to adapt the collson avodance method to the partcular knematcs of other systems. ORCA s a collaboratve collson avodance method based on velocty obstacles, where each holonomc robot makes a smlar collson avodance reasonng and collson-free moton s guaranteed wthout oscllatons. Furthermore, n our approach, ORCA could be substtuted by other samplng-based collson avodance methods, such as Recprocal Velocty Obstacles [] or Hybrd Recprocal Velocty Obstacles []. A formal extenson of ORCA to dfferentally-drven robots was presented by the authors n []. That work shares wth ths paper the dea of extendng ORCA to robots wth non-holonomc knematcs by trackng a holonomc trajectory. ORCA was also extended to navgatng smple arplanes wth car-lke knematcs n D space [6], where a set of trajectores s precomputed. Nevertheless, safety s not fully guaranteed as collsons may arse n the transent before reachng the desred velocty. In ths paper we ntroduce a formal approach where ths s taken nto account by enlargng the radus of the robots. As an alternatve, [7] presented the acceleraton velocty obstacles for agents wth holonomc acceleraton capabltes, whch explctly takes nto account acceleraton lmts and results n trajectores wth contnuous velocty (ths was not the case for RVO and ORCA). Nevertheless, t does not generalze to general knematcs and cannot be drectly appled to carlke vehcles. In contrast, n our approach the contnuty n velocty and actuators s acheved thanks to the trajectory trackng strategy. In contrast to purely determnstc methods, n [8] a method for recursve probablstc velocty obstacles s studed, and n [9] collson-free trajectores are found by usng Gaussan processes. The remander of the paper s structured as follows. Secton II gves an overvew of our collson avodance algorthm. Secton III descrbes the knematcs of the robot, whereas Secton IV presents the trajectory trackng controller and Secton V gves an overvew of optmal recprocal

2 collson avodance for holonomc robots. In Secton VI the B-ORCA algorthm s descrbed n detal. In Secton VII the smulaton experments are dscussed. Fnally, Secton VIII concludes and gves an outlook on our future work. II. OVERVIEW OF THE B-ORCA ALGORITHM Bcycle recprocal collson avodance (B-ORCA) presents an effcent method for avodng collsons n a scenaro wth multple car-lke robots. The method s fully dstrbuted and the nformaton requred by each robot n order to avod collsons ncludes the poston, velocty and radus of ts neghbors. The B-ORCA algorthm does not only offer oscllaton-free recprocal collson avodance among multple possbly heterogeneous robot unts (.e. the robot knematcs may not be of the same type), but also avods collsons wth dynamc and statc obstacles. Lkewse to [], the man dea s that a robot wth gven knematc constrants s able to track a holonomc trajectory wthn a certan maxmum error bound. Therefore, by enlargng the radus of the robot by ths bound, t can be treated as holonomc. In ths case, a collson-free trajectory s effcently computed followng []. By usng a standard trajectory trackng controller [] and precomputng the maxmum trackng errors, a set of holonomc trajectores s obtaned that can be tracked wthn the gven maxmum error bound. Ths set s ntroduced as a further constrant n the selecton of collson-free nputs for the robot. Furthermore, the controller of [] guarantees contnuty n the drvng velocty and acceleraton of the robot, as well as n the steerng angle, and respects the knematc lmts (maxmum drvng velocty, drvng acceleraton, steerng angle and steerng velocty) of the vehcle. Nevertheless, lkewse to ORCA, a crcular robot-shape s requred. III. ROBOT KINEMATIC MODEL In ths work the robots are consdered to be non-holonomc car-lke vehcles. A smplfed car model wth a fxed rear wheel and a steerable front wheel, as shown n Fgure, s used. The generalzed coordnates are q = (x,y,θ,φ), where x, y represent the poston of the rear wheel, θ the orentaton of the car and φ the steerng angle. If the car of length L has rear-wheel drvng, the knematc model s gven (n accordance wth []) by ẋ cosθ ẏ θ = snθ tan φ/l v + v, () φ where v and v are the drvng and steerng velocty nputs, respectvely. The model sngularty at φ = ±π/ s avoded by restrctng the range of the steerng angle to φ < φ max < π/. Furthermore, both nputs are lmted to v v max and v v max, as well as the drvng acceleraton v a max. The parameters of the bcycle robots (see Secton IV-D) used n the smulaton experments of ths work are those of the nspecton robot MagneBke as descrbed n [] and those of a faster car-lke vehcle. Fg.. Schema of a car-lke robot, wth extended radus ǫ and desred velocty v d. Its mddle pont s denoted by p. IV. TRAJECTORY TRACKING One of the underlyng concepts of the B-ORCA algorthm s that a car-lke robot tracks a constant-speed straght-lne trajectory whle stayng wthn a known trackng error. A. Trajectory trackng controller The trajectory trackng controller [] s obtaned by applyng full-state lnearzaton va dynamc feedback to the non-lnear system of Equaton (). The two system outputs and ther dervatves are gven by [ [ ] x ξ cosθ z =, ż =, y] z = ξ snθ [ ξ tanφsnθ/l+ξ cos(θ) ξ tanφcosθ/l+ξ sn(θ) ], () wth ξ and ξ two ntegrators added to the system. It can be seen that the dynamc controller takes the form v = ξ v = ξ cosφ tanφ/ξ Lr cosφ snθ/ξ +Lr cosφ cosθ/ξ ξ = ξ ξ = ξ tanφ /L +r cosθ +r snθ, () where the feedback terms r ( =, ) are gven by r =... z d, +k a, ( z d, z )+k v, (ż d, ż )+k p, (z d, z ), () where z d, ż d, z d and... z d are computed for the desred trajectory to track (see Secton IV-B). The feedback gans are such that the polynomals λ +k a, λ +k v, λ+k p,, =,, () are Hurwtz (all roots of the polynomal are real negatve).

3 Maxmum trackng error for v = m/s ; φ = o Maxmum trackng error for v = m/s ; φ = o Maxmum trackng error for v = m/s ; φ = o.. v d,y [m/s] v d,y [m/s] v d,y [m/s]... v [m/s] d,x v [m/s] d,x v [m/s] d,x Fg.. Maxmum trackng errors n [m] for a desred trajectory gven by v d V and saturated at m. From left to rght, the ntal drvng velocty v and the steerng angle φ vary from (v, φ) = ( m/s, ),( m/s, ) to (m/s, ). Images best vewed n color. Furthermore, recall [] that ths controller allows for parkng maneuvers. Constrants n maxmum steerng angle, drvng and steerng velocty nputs and drvng acceleraton are drectly added by saturatng the respectve varables (φ, ξ, v and ξ ). B. Trackng of constant-speed straght-lne trajectory Due to poston and rotaton nvarance, consder a car ntally centered at the orgn (p() = ) and wth orentaton θ() =. Consder a desred straght-lne trajectory gven by a constant velocty v d and passng through p(). Denote v d = v d and θ d = atan(v d ). The feedback terms of Equaton () are then gven by r (t) = k a, z (t)+k v, (v d cosθ d ż (t)) +k p, ((v d t s L/)cosθ d z (t)) r (t) = k a, z (t)+k v, (v d snθ d ż (t)) +k p, ((v d t s L/)snθ d z (t)), (6) where s = f the car to be tracked s consdered to move forward and s = otherwse. Ths ambguty appears because the trajectory to be tracked s gven wth respect to the center of the robot, whlst the controller s desgned for rear-wheel trackng. The ntal condtons of the varables are gven by [ L z() = cosθ() L snθ() ] ; ξ() = [ v a ] where v = v () and a = v () are the drvng velocty and acceleraton respectvely. In our mplementaton, we choose s = sgn(cosθ d cosθ ICR +snθ d snθ ICR ) where θ ICR = sgn(φ)(π/ atan(/ tanφ ) ) + θ() s the angle between the abscssa and the perpendcular to the lne formed by the nstantaneous center of rotaton (ICR) and the mddle pont p() of the vehcle at ntal tme. As a further smplfcaton, n our experments, we consder a =, thus guaranteeng contnuty n velocty but not n acceleraton. Despte the ntalzaton, t may occur that the trackng robot and the tracked vrtual car move wth opposte orentatons,.e. one forward and one backward. Ths would (7) lead to perfect trackng of the rear wheel but large error n the trackng of the reference robot center pont. In order to compensate, f ths stuaton s detected (cosθ d cosθ(t) + snθ d snθ(t) < ), the velocty of the tracked pont z d s temporally ncreased, or decreased respectvely, untl the orentaton of the reference car s reversed. Note that the center pont of the reference car always moves at speed v d. C. Achevable veloctes Gven the ntal condtons of the robot (ntal drvng velocty v and steerng angle φ) and the desred velocty v d V R, ts trajectory subject to the controller presented n ths secton s smulated and the maxmum trackng error n the robot center pont s computed. For gven φ and v, the set of precomputed trackng errors for v d V s denoted by E φ,v. Consder V φ,v,ε = {v d V E φ,v (v d ) ε}, (8) the subset of V of veloctes that can be tracked wth an error lower than ε (computed wth respect to the robot center pont). We consder the dscretzatons V = [ v max : v : v max ], φ Φ = [ φ max : φ : φ max ] and v V = [ v max : v : v max ]. For φ Φ, v V and v d V, the trajectores of the car-lke robot are smulated, and the maxmum trackng errors precomputed and stored n a lookup table. Note that ths computaton s expensve, but s done off-lne and only once for the knematcs of a gven robot. In our smulatons, the feedback gans of Equaton () are computed such that all roots equal to - (MagneBke) and -. (fast car). In Fgure, the maxmum trackng errors obtaned for the knematcs of the fast car are vsualzed for (v, φ) = ( m/s, ),( m/s, ) and(m/s, ). Note that due to symmetry, the trackng errors only need to be computed for one half of the full range of steerng angles φ, e.g. φ [ φ max, ]. However, the same does not hold true for the drvng veloctes. Fgure shows the trackng errors for the MagneBke robot. Here contnuty n speed s not mposed, whch results

4 v d,y [m] Maxmum trackng error for o steerng angle v [m] d,x Fg.. Maxmum trackng errors n [m] for a desred trajectory gven by v d V, here shown for φ =. In the case of MagneBke, v s set to s v d, and thus precomputaton ncludes varyng steerng angles only. Fg.. Constrants n velocty space generated by ORCA τ from multple j robots. The regon of collson-free veloctes ORCA τ s hghlghted and v s dsplayed. n sets that are clearly dfferent from the sets of the car llustrated n Fgure (see also Secton IV-D below). It s observed that the areas of best trackng are strongly related to the steerng angle of the front wheel, whch has an mpact n the maneuverablty of the robot. D. Parameters for the smulated vehcles The parameters for the smulated vehcles are as follows. ) Car: φ max = o, v max = m/s, v max = o /s, a max = m/s, L = m, φ = o, v =.m/s. ) MagneBke: φ max = 8 o, v max =.m/s, v max = o /s, L =.m, φ = o, v =.m/s. For the MagneBke, unconstraned acceleraton s consdered. To allow for dscontnutes n drvng velocty, n Equaton (7), the ntal condtons may be rewrtten as ξ() = [s v d,]. V. RECIPROCAL COLLISION AVOIDANCE FOR HOLONOMIC ROBOTS B-ORCA reles on the concept of Optmal Recprocal Collson Avodance (ORCA) for holonomc robots presented by []. In ths secton an overvew of ORCA s gven. Consder a group of N dsk-shaped robots wth radus r at poston p and wth current velocty v. Further consder each robot has a preferred velocty v pref towards ts goal poston. The velocty obstacle for robot [,N] nduced by any other robot j s defned as the set of relatve veloctes v = v v j leadng to a collson wthn a tme-horzon τ VO j τ = { v t [,τ], t v D(p j p, r +r j ) }, (9) whered(p,r) = {q q p < r} s the open ball of radus r centered at p. The set of collson-free veloctes ORCA τ j for robot wth respect to robot j can be geometrcally constructed from VO j τ. Frst, the mnmum change n velocty u = (argmn v (v opt v opt j ) ) (v opt v opt j ), () v VO τ j whch needs to be added to v to avod a collson, s computed. v opt s the optmzaton velocty, set to the current velocty v of the robot. From our experence wth ORCA, ths choce gves good results. It can further be seen that ORCA τ j = {v (v (v opt +λ,j u)) n } () where n denotes the outward normal of the boundary of VO j τ at (vopt v opt j ) + u, and λ,j defnes how much each robot gets nvolved n avodng a collson (where λ,j + λ j, = ). λ,j = λ j, =. means both robots help to equal amounts to avod colldng wth each other; λ,j = means robot fully avods collsons wth a dynamc obstacle j. Lkewse, the velocty obstacle can be computed for statc obstacles []. ORCA τ, the set of collson-free veloctes for robot s then gven by ORCA τ = D(,V max ) j ORCA τ j, () Fgure shows the set ORCA τ for a confguraton wth multple robots. The optmal collson-free velocty for robot s gven by v = argmn v v pref. () v ORCA τ Ths optmzaton wth lnear constrants can be effcently solved, returnng a convex and compact set ORCA τ and a collson-free velocty v. In order to avod recprocal dances, one of the sdes of VO j τ may slghtly be enlarged to avod the symmetry. In our case, VO j τ s enlarged by.m/s to one sde. VI. THE B-ORCA ALGORITHM The B-ORCA method frst of all precomputes the trackng errors E φ,v wth respect to the straght-lne trajectores defned by the velocty vectors v d V for all possble ntal steerng angles φ Φ and ntal veloctes v V followng Secton IV-B. In ths step the knematcs of the robot are taken nto account. As the veloctes to be tracked are consdered relatve to a robot s orentaton, the

5 Fg.. Trajectores of ten car-lke robots exchangng antpodal postons on a crcle. Left: Experment wth car-lke vehcles. Mddle: Experment wth MagneBkes. Rght: Experment, where one car s non-reactve (straght-lne trajectory n red), thus gnorng the other robots. prevously obtaned trackng errors are not only nvarant to the poston of the robot but also to ts current orentaton. In the followng,e φ,v s expressed n a relatve frame orented wth θ, whlst ORCA τ s computed n the global reference frame. In every teraton of the collson avodance stage, each robot reads out ts sensors and gans knowledge about ts nternal state, gven by ts poston p, orentaton θ, steerng angle φ, current velocty v, preferred velocty, current drvng velocty v = v, radusr and desred radus extenson ˆε. Furthermore, each robot obtans from ts neghbors va communcaton or sensng ther poston p j, current velocty (or velocty estmate) v j and extended radus r j +ε j. Gven a group of N robots, wth known aggressveness λ,j, fxed maxmum tme to collson τ max and sensng range d max, assume a known fxed update rate of the controller of dt c and of the sensng of dt s, wth dt c << dt s. The followng steps are computed ndependently by each robot n every teraton: v pref ) A preferred velocty v pref towards the goal s obtaned. ) The extended radus r + ε s set to r + mn j (ˆε, (d(,j) r r j )/), where d(,j) denotes the dstance (mddle ponts) from robot to robot j. ) All robots (ncludng robot ) wthn d max are consdered as holonomc robots of radus r j +ε j. Followng Secton V the set ORCA τ s computed. ) A new collson-free velocty v s computed, such that t s closest to v pref and such that t verfes v ORCA τ V φ,v,ε. Thus, v = argmn v v pref v ORCA τ V. () φ,v,ε ) The trajectory gven by v s tracked wth control update rate dt c, as descrbed n Secton IV. If ORCA τ V φ,v,ε =, the tme to collson τ max s reduced (τ max = τ max /), and steps ) and ) are repeated. If τ max reaches a mnmum admssble value τmax mn v max /a max, the problem s consdered unfeasble and robot decelerates at maxmum acceleraton. If ths s the case for robot, all other robots must fully avod collsons wth t n the comng tme steps whle ts optmzaton remans unfeasble; ths s acheved by temporally settng λ j, = for every other robot j. A. Implementaton detals on step ) of B-ORCA Dependng on the complexty of V φ,v,ε, two optons are dscussed below. ) Polygonal approxmaton of V φ,v,ε : Lkewse to [], the set V φ,v,ε may be approxmated by a convex polygon P φ,v,ε V φ,v,ε (or by two convex polygons respectvely). If the approxmaton s accurate, step ) of B- ORCA can be effcently computed as an optmzaton wth lnear constrants gven by P φ,v,ε and ORCA τ. Ths s the case for the sets depcted n Fgure. ) Samplng of V φ,v,ε : For complex sets V φ,v,ε where a convex polygonal approxmaton s over-restrctve, the optmzaton can be solved by samplng. Ths s the case for the sets depcted n Fgure. As a nave approach, startng from the velocty v pref and searchng the dscrete space ORCA τ V φ,v,ε for the closest velocty v could computatonally be expensve. Nevertheless, v can be effcently computed. Frst, v s obtaned solvng the optmzaton wth lnear constrants gven by Equaton (). Then, the procedure n step ) of the algorthm contnues wth a constraned wave expanson from v as follows: An ordered lst s ntalzed wth v V as the closest velocty to v accordng to a gven dstance metrc, and all ts neghbors are added keepng ascendng order n dstance. Whle the lst s non-empty the frst velocty v of the lst (wth mnmum dstance to v ) s checked. If v verfes a set of lnear constrants,.e. v ORCA τ, the lst s expanded wth the neghbor veloctes of v. If v further verfes the precomputed E φ,v (v) ε,.e. v V φ,v,ε, then v s drectly returned as the collson-free velocty v. Ths search method s bounded to the convex polygon gven by ORCA τ, and thus the optmal velocty s found n a few steps unless ORCA τ V φ,v ε =, where no soluton exsts. B. Remarks on the B-ORCA algorthm Remark (Collson-free): B-ORCA guarantees collson -free trajectores. In each tme-step, the planned straght-

6 lne trajectores gven by v are collson-free for holonomc robots of radus r +ε. Further, the trajectory of each carlke robot stays wthn ε of the planned straght lne. Ths guarantees that the dstance between two robots s greater than the sum of ther rad, thus requrng step ) of B- ORCA. After each tme-step a new collson-free trajectory s computed, leadng to more complex global paths. Remark (Knematc contnuty): B-ORCA guarantees trajectores wth contnuty n (at least) velocty and steerng angle, and fully respects the knematc constrants and lmts n actuators, veloctes and acceleratons. Ths propertes follow from the controller presented n Secton IV. Remark (Convergence): Convergence to goal destnatons s not fully guaranteed n a reasonable tme. Deadlock stuatons may result when the robot s collson-free velocty closest to ts preferred velocty tends to zero or V φ,v ε s over-restrcted. Ths can be resolved by choosng a new preferred velocty gven by a global path planner. VII. SIMULATION RESULTS A set of smulated experments has been conducted to show the performance of the proposed B-ORCA algorthm. The smulated bcycles and car-lke vehcles are governed by the knematcs and parameters of Secton III and Secton IV. Furthermore, the followng parameters are chosen for the smulatons: ) Car: τ max = s, τmax mn = s, d max = m, dt c =.s, dt s =.s and ˆε = m. ) MagneBke: τ max = s, τmax mn = s, d max = m, dt c =.s, dt s = s and ˆε =.m. The desred extenson ˆǫ of the robots rad s selected as a value that presents a good trade-off between radus enlargement and maneuverablty for the consdered robots. Although the aggressvenessλ,j can be varable, t s chosen as λ,j =. for every par of robots n the presented smulatons, and thus all robots take the same responsblty n avodng collsons. Three experments are presented n ths work, all of them performed wth ten smulated vehcles of both types (cars and MagneBkes), as follows: Experment : Exchange of antpodal postons on a crcle. Experment : Exchange of antpodal postons on a crcle; one robot acts as dynamc obstacle and does not perform any collson avodance. The remanng nne robots take full responsblty (λ,j = ) n avodng t. Experment : All robots start from random postons, orentatons and steerng angles and move to random goal postons. In all of the experments, unform nose n poston of ampltude.m for the cars and.m for the MagneBkes s added. In the left of Fgure the trajectores of all ten smulated cars, and n the mddle of Fgure the trajectores of all ten smulated MagneBkes are dsplayed for the frst experment. Fnally, on the rght of Fgure the trajectores of the cars are shown for the second experment, where one of the cars Fg. 6. Trajectores of ten car-lke robots startng from a random confguraton and movng to random goal postons. The straght lne and dashed lne trajectores represent the mddle and rear-wheel ponts of the cars, respectvely. The robots are dsplayed n ther ntal confguratons and goal postons are represented by red crcles. s non-reactve and follows a straght-lne trajectory towards ts goal. These experments all present extreme symmetry and are thus challengng. B-ORCA performs best n more natural scenaros, where robots are n any poston wth any orentaton and steerng angle, and the velocty-based local collson avodance provdes a smple soluton. In Fgure 6 the trajectores of the thrd experment are shown. In ths case, the ten cars start from a random confguraton and evolve towards a set of random goals. The paths are agan smooth. The robots are stopped n the proxmty of ther goals because the controller of Secton IV s desgned for trajectory trackng. In order to have perfect convergence, a poston controller must be appled when reachng the neghborhood of the goals. In the accompanyng vdeo, all three experments are presented n full length for both vehcle types, where for each robot three arrows are plotted, representng v pref (red), v (blue) and v (black). We have further mplemented the B-ORCA algorthm under ROS, and are currently expermentng on collson avodance wth several real MagneBke robots [], []. VIII. CONCLUSION AND FUTURE WORK In ths work, a dstrbuted method for recprocal local collson avodance among bcycle or car-lke robots, socalled B-ORCA, s presented, where each ndvdual robot does not need nformaton about the knematcs of other robots. The method guarantees collson-free motons and acheves smooth trajectores as shown n smulated experments wth ten MagneBke and ten car robots. The method

7 reles on the ORCA algorthm that computes a collsonfree velocty as f the robots were holonomc. The method further reles on a trajectory trackng controller for car-lke vehcles, whch could essentally be substtuted by any other trackng controller for knematc constrants dfferent than those presented n ths paper. Furthermore, recprocal collson-free motons are guaranteed n heterogeneous groups of robots wth car-lke robots runnng B-ORCA, navgatng n an envronment wth dfferentally-drven robots runnng NH-ORCA [] and holonomc robots runnng ORCA []. Moreover, collsons wth both dynamc and statc obstacles are avoded, except n the cases of unfeasblty when due to the knematc constrants of the robot, no soluton exsts. Nevertheless, n order to avod deadlocks n a scenaro wth statc obstacles, a global path planner s requred. Further research s needed n solvng deadlock stuatons n extremely crowded stuatons. For less controlled envronments, or a full ntegraton of sensng and actuaton, the method must also be extended to compensate for uncertantes and communcaton delays. REFERENCES [] A. Bretenmoser, F. Tâche, G. Caprar, R. Segwart and R. Moser, MagneBke - Toward Mult Clmbng Robots for Power Plant Inspecton, n Proc. of The 9th Int. Conf. on Autonomous Agents and Multagent Systems,. [] J. Borensten, Y. Koren, The vector feld hstogram - fast obstacle avodance for moble robots, n IEEE Trans. Robot. Autom., (7), 78 88, 99. [] P. Forn, Z. Shller, Moton plannng n dynamc envronments usng velocty obstacles, n Int. J. Robot. Res. (7)(7), 76 77, 998. [] O. Khatb, Real-tme obstacle avodance for manpulators and moble robots, n Int. J. Robot. Res. (), 9 98, 986. [] T. Sméon, S. Leroy, J.-P. Laumond, Path coordnaton for multple moble robots: a resoluton complete algorthm, n IEEE Trans. Robot. Autom. (8)(),. [6] J. Peng, S. Akella, Coordnatng Multple Robots wth Knodynamc Constrants Along Specfed Paths, n Int. J. Robot. Res., vol. no. 9-,. [7] D.E. Chang, S. Shadden, J.E. Marsden, R. Olfat Saber, Collson Avodance for Multple Agent Systems, n Proc. IEEE Conf. Dec. Contr.,. [8] D.M. Stpanovć, P.F. Hokayem, M.W. Spong, D.D. Šljak, Cooperatve Avodance Control for Multagent Systems, n ASME J. Dyn. Sys. Meas. Control, (9)(), , 7. [9] C. Pradaler, J. Hermosllo, C. Koke, C. Brallon, P. Bessre, C. Lauger, The CyCab: a car-lke robot navgatng autonomously and safely among pedestrans, n Robotcs and Autonomous Systems vol., no., pp. -67,. [] O. Brock, O. Khatb, Hgh-speed navgaton usng the global dynamc wndow approach, n Proc. IEEE Int. Conf. Robot. Autom., 999. [] J. van den Berg, S. J. Guy, M. Ln and D. Manocha, Recprocal n- body Collson Avodance, n Int. Symp. on Robotcs Research, 9. [] A. De Luca, G. Orolo and C. Samson, Feedback control of a nonholonomc car-lke robot, n Robot Moton Plannng and Control, chapter, Sprnger, 998. [] J. van den Berg, M.C. Ln, D. Manocha, Recprocal Velocty Obstacles for real-tme mult-agent navgaton, n Proc. IEEE Int. Conf. Robot. Autom., 8. [] J. Snape, J. van den Berg, S.J. Guy, D. Manocha, Independent navgaton of multple moble robots wth hybrd recprocal velocty obstacles, n Proc. IEEE Int. Conf. Intell. Rob. Syst., 97 9, 9. [] J. Alonso-Mora, A. Bretenmoser, M. Rufl, P. Beardsley, R. Segwart, Optmal Recprocal Collson Avodance for Multple Non- Holonomc Robots, n Proc. Int. Symp. on Dstrbuted Autonomous Robotcs Systems,. [6] J. Snape, S.J. Guy, D. Manocha, Navgatng Multple Smple- Arplanes n D Workspace, n Proc. IEEE Int. Conf. Robot. Autom.,. [7] J. van den Berg, J. Snape, S. J. Guy, D. Manocha, Recprocal Collson Avodance wth Acceleraton-Velocty Obstacles, n Proc. IEEE Int. Conf. on Robots and Automaton,. [8] B. Kluge and E. Prassler, Recursve Agent Modelng wth Probablstc Velocty Obstacles for Moble Robot Navgaton Among Humans, n Sprnger Tracts n Adv. Robotcs, (), -, 7. [9] P. Trautman and A. Krause. Unfreezng the Robot: Navgaton n Dense, Interactng Crowds, n Proc. IEEE Int. Conf. Intell. Rob. Syst.,. [] F. Tâche, W. Fscher, G. Caprar, R. Moser, F. Mondada and R. Segwart, Magnebke: A Magnetc Wheeled Robot wth Hgh Moblty for Inspectng Complex Shaped Structures, n Journal of Feld Robotcs, (6), 76, 9.