Autonomous multi-vehicle formations for cooperative airfield snow shoveling

Transcription

1 1 Autonomous mult-vehcle formatons for cooperatve arfeld snow shovelng Martn Hess Martn Saska Klaus Schllng Unversty of Wuerzburg, Computer Scence VII: Robotcs and Telematcs, Germany Abstract In ths paper an approach for snow shovelng of arfelds usng formatons of snowploughs s ntroduced. The vehcle formatons splt and reunte dependng on the varyng wdth of the roads to clean. An optmal schedule s derved for the vehcles guaranteeng complete coverage of the graph whch s used to model the arfeld. Based on the schedule a smple path s generated for the formaton centers. Even durng turns the generated moton for the car-lke robots wthn the formaton ensures proper breadthways coverage. Fnally the method s applcablty s verfed by smulatons as well as hardware experments. Index Terms Cooperatve sweepng, autonomous robot formatons, optmal coverage I. INTRODUCTION Wth ncreasng ar traffc n the whole world, man nternatonal arports are compelled to serve the utmost amount of flghts also durng contrary weather condtons. Better sensors, communcaton and navgaton systems enable to launch and land even durng heavy snowfall. But the condton of the runways s a lmtng factor for the unnterrupted full-load runnng of the arport. The novel arport snow shovelng approach ntroduced n ths paper offers a perodcal autonomous system based on formaton drvng of car-lke moble robots. The unnterrupted arport operatons are ensured by dvdng all servce roads to protectve zones and by swtchng the runways between actve and shovelng mode. Due to the formaton drvng, all roads are cleaned always completely, whch s mportant for unexpected rescue operatons or even for an emergency landng. In the presented algorthm mnmal cleanng tme n each zone of the arport and complete cleanng of the whole arport durng one loop s guaranteed. The short executon tme s accomplshed by splttng formatons for shovelng smaller roads and by reuntng to a sgnfcant number for the man runways. The developed method has been ntensvely tested n varous smulatons as well as n laboratory hardware experments. As a testng scenaro for the proposed task schedulng algorthm Frankfurt arport was chosen, whch s one of the largest arports n Europe. Bgger arports n the rest of world may dffer n the number of runways but the complexty of the servce roads s comparable. Due to the decomposton of the optmzaton problem to subproblems focusng on each runway and ts envronment, we suppose that our approach s useful n general. For the hardware experments and for the snow shovelng smulator a smaller scenaro wth one runway and two servce roads was arranged, that contans all sgnfcant parts of a larger arport map (splttng and reuntng of formatons, straght movement and curves wth splt-up and jont formatons). Ths restrcton s necessary due to the lmted workspace n our laboratory and also due to the computng power needed for smulatng the snow partcles movement. The calculatons requred for communcaton, localzaton, moton plannng and trajectory trackng used n the real world experment as well need only an nconsderable part of processor tme. The rest of the paper s organzed as followed: Related work s descrbed n Secton II. Secton III provdes the descrpton of the task plannng algorthm. In subsecton III-A a novel approach for optmal route schedulng s proposed, whle subsecton III-B descrbes, how the fnal path and the desred nputs for the formaton movement are derved. In Secton IV t s explaned how the robots are controlled wthn the formaton and how a desred shape of the formaton can be obtaned. After ths the results of accomplshed smulatons and hardware experments are presented n Secton V followed by concludng words and plans for future work n Secton VI. II. RELATED WORK In the recent years, ssues nvolvng mult-robot coverage have been addressed by many researchers. Most of the research n ths area can be classfed nto two categores. On one hand s the statc coverage, whch ams to fnd a confguraton of the nodes n order to optmally cover or observe a predefned area. Theoretcal results for optmal statc coverage are well descrbed n [3] and [12]. More related to the work n ths paper are approaches concernng dynamc coverage, whose goal t s to vst each place n a predefned area by at least one node n order to cover t completely. Often t s desred to coordnate the agents n a way that tme optmalty s acheved. E.g. n [9], dstrbuted coverage s carred out by sharng an ncrementally created adjacency graph of the decomposed area between the robots. The work n ths paper s mostly related to cooperatve sweepng, whch s a well-known subfeld of dynamc coverage. The goal of sweepng s to fnd a specfc moton for a robot n order to cover a 2-dmensonal area by ts effecter. Fundamental research n the feld of cooperatve sweepng was done by Kurabayash et al. [11, 10]. A decentralzed approach usng a negotaton mechansm was proposed n [15]. Later, these basc off-lne methods were extended for real tme applcatons wth an obstacle avodance algorthm by Luo et al. [13, 14]. In contrast to these papers whch manly

2 2 concentrate on dstrbutng a gven area to the agents, our work consders also nonholonomc vehcle knematcs as well as the fact that the effecter of snowploughs s dsplaced from the vehcles barycenter. III. TASK PLANNING In ths secton a novel path plannng algorthm for snow shovelng of an arport by formatons of moble robots s ntroduced. The man dea of the approach s to descrbe the roads n a map of the arport by edges of a graph. Furthermore the nodes represent the centers of all crossngs. An example of such graph s depcted n Fg. 4. The frst subtask of the path plannng s to fnd a tme optmal sequence of the edges coverng all roads that need to be cleaned. In the second phase the graph edges are replaced by a path assgned wth a certan speed functon that acts as a reference trajectory for the vehcle formatons. The path has to be feasble for the car-lke robots and should be optmal wth respect to maxmum breadthways coverage of the road by the formaton. A. Route schedulng A state n the presented method s descrbed by the actual postons of all robots and the hstory of ther movements. Addtonally t conssts the tme needed for the edge executons, and the set of edges that reman to be cleaned. The smplest way to fnd an optmal schedule for all vehcles s to evaluate the complete space of states, where the state wth mnmum executon tme that contans no more edges to clean s the optmal soluton. The followng base rules for the schedulng were fxed after dscussons wth a specalst from an arport: 1) Each road should always be entrely cleaned. The reason s that partly cleaned paths wth snow hlls remanng on the surface could be dangerous n emergency stuatons. For rescue and fre fghtng vehcles and for planes makng a forced landng, roads wth constant layer of snow are safer than those wth a cluttered surface. 2) A formaton s allowed to turn about only n dead ends (nodes 14, 43 and 79 n Fg. 4). Ths rule prevents complcatons wth snow remanng after the turnng. One possblty to fnd the optmal soluton s the breadth frst search algorthm, that systematcally evaluates the whole space. It s nterestng to thnk about the total number of states that need to be expanded, where expandng means to evaluate all sons of a state n the state tree. In the shovelng scenaro the mnmum number of crossng lnes n each node s three (nodes wth only two edges can be replaced by one long edge between t s neghbors). Thus, each vehcle stuated n an expandng node has to choose one out of at least three possble actons. The snowploughs are not allowed to use the same edge agan (due to the second rule), but they may stay n place and wat for a suffcent number of vehcles (the frst rule). One state, provdng that the suffcent number of vehcles for each adjacent edge of the current node s one (or the roads are already cleaned) can be expanded to S n = (n + 1)!, (1)!(n 1)! states, where denotes the number of vehcles and n s equal to the adjacent edges n the current node. To estmate a lower bound for the number of states that need to be evaluated we assume that all nodes have 3 adjacent edges and 1 snowplough s appled for shovelng. When applyng 100 transtons,.e. movements from a node to an adjacent node, we get (S 3 1) 100 = (2) states to evaluate. It s easy to see, that such an amount cannot be evaluated by conventonal computers n approprate tme. Fortunately not all of the states are feasble. Usually each arport has several parallel runways wth approprate servce roads. In order to prevent collsons between snowploughs and the arport traffc, such parts should be cleaned separately. Durng the cleanng phase arport traffc s not allowed to enter the cleanng area and conversely the other parts of the arport are forbdden for the ploughs. Frankfurt arport s usually usng two parallel runways labeled A and B. For the schedulng algorthm we dvded all roads nto the three non-overlappng sets A, B and AB (see Fg. 4). The edges of set A (resp. B) need to be cleaned up completely whle runway A (resp. B) s n shovelng mode, because these roads are used by arplanes, when the approprate runway s n actve mode. Edges of set AB can be cleaned wth set A or B, but together at least wth one of them. These auxlary roads are only used by the ground crew, that are able to avod the snowploughs. Therefore they do not need to be closed for the robots. Resultng from the decomposton, the searched space s lmted to about 40 edges for each area. Therefore the lower bound for the total number of states correspondng to the assumptons for (2) that need to be expanded durng the breadth frst algorthm s equal to (S 3 1) 40 = (3) whch s stll unsatsfactory. If more than one vehcle s utlzed for the shovelng task the number n (3) s even hgher. To get around ths another smplfcaton of the search space, whch results from the specfc character of the arport, s made. The servce roads are almost two tmes narrower than the man runways. Due to ths we can expect that the optmal soluton wll be to frstly clean the runway by the whole formaton and to dvde t nto two equal parts afterwards n order to clean up the remanng roads. Hence n the fnal soluton there s no tme wasted for robots watng for others to form up. To mprove the algorthm we also changed the selecton rule that determnes the state that wll be expanded n the next step. The breadth frst algorthm usually chooses the frstly found state that s yet unexpanded. But f nstead the unexpanded state wth the lowest waste tme (tme lost by usng a path that does not need to be cleaned or by watng for the second formaton at the end of the plan) wll be expanded, the frstly found soluton s the optmal one. Ths s because all other states wth possble lower waste tme are already evaluated. The complete cleanng loop for Frankfurt arport s descrbed n 6 steps:

3 3 Fg. 2. Fg. 1. Paths prepared for the testng scenaro. Ths smplfed scenaro s solved by two snowploughs. The lnes should be followed by the vertcal center of the vehcle formaton. 1) All robots clean up runway A n a bg formaton. The schedule s a sequence of the edges from node 22 to node 43. 2) Two smaller formatons clean the remanng edges of A. Edges of AB can be also used. 3) Both formatons move to node 19. The shortest schedule s found by the Djkstra algorthm [8]. 4) The reunted formaton cleans runway B from node 19 to node 79. 5) The dvded formatons clean all edges B + AB that were not cleaned n steps 2) - 4). The already cleaned edges B + AB can be used durng ths step, too. 6) Both formatons move to node 22. The sequence s restarted wth step 1). The optmal soluton for Area A, that s depcted n Fg. 5 was obtaned by the algorthm wth state evaluatons. B. Path plannng The nput for the path plannng s a sequence of edges found by the route schedulng algorthm. To obtan a path, every two neghborng edges are replaced by two lnes and one crcle n between. The crcle should have mnmum radus wth respect to feasblty for the whole formaton,.e. the curve s feasble for the robot that has to do the sharpest turn. The fnal path s adjusted for the vertcal center of the formaton and not for the sngle robots. Thus problems can occur when the formaton s dvded or unted. Ths was solved by splttng the vertcal center of formaton to two or more ponts or by untng multple ones. The new centers need to be chosen n terms of the new formatons n order to keep a smooth path for each robot. Fg. 1 shows a smple example where one formaton s splt nto two and later agan reuntng. IV. MOTION COORDINATION In ths secton we wll descrbe, how the moton of the robots s coordnated n order to fulfl the desred snowcleanng task. A. Moton generaton for the vehcle formatons On parts of the arfeld, e.g. on the wde runways t s desred to remove the snow n a preferably short tme nterval. Formaton consstng of 3 snowploughs durng a left turn. A reasonable soluton to ths can be acheved by utlzng a formaton of multple snowplough vehcles for cleanng these areas. Snce we assume car-lke knematcs for the autonomous snowplough robots, a moton plannng method applcable for ths type of vehcles s needed. It should generate feasble trajectores for all vehcles, whch result n a proper collectve coverage of the ploughs shovels on the arfeld. We developed a sutable approach for the trajectory generaton based on [1] and [2], whch wll be shortly descrbed below. A formaton conssts of N robots arranged around a reference pont C, whch follows the planned path wth predefned speed (see Fg. 2). C was chosen on the vertcal centerlne n front of the robots, so the formaton can deal wth the swtchng mechansm mentoned n secton III-B. A reference trajectory s fully defned by speed vc (dc ) and curvature Kc (dc ), where dc denotes the traveled dstance of pont C. The poston of robot, where {1,..., N } s descrbed by the relatve coordnates [p, q ] wth respect to the reference pont. The horzontal component p s measured n curvelnear coordnates wth the curvature of the reference trajectory at the orgn, whch s equal to the pont C. In ths descrpton [p, q ] wll stay constant, durng perods when the formaton s arrangement remans statc. The eucldan dstance between vehcles vares wth changes of curvature and thus can lead to a collson between vehcles, f [p, q ] for {1,..., N } are not chosen properly. The control nputs for vehcle are computed as v (s ) = Qvc (s ) (4) and K (s ) = S Kc (s ) + Q Kc (s ) 2 dq ds (s ) Q2 2 dq dkc 1 q (s )Kc (s ) ddsq2 (s ) + q (s ) ds (s ) (s ) ds, + Q2 (5) where sgn(1 q (s )Kc (s )) s 2 2 dq (s ) + 1 q (s )Kc (s ), Q = ds S = (6) (7) and s = dc + p denotes the momentary traveled dstance of vehcle along the reference trajectory. To alter the formaton

4 4 n order to avod obstacles or due to splttng or reuntng of the group t s possble to modfy p and q contnuously wth the traveled dstance along C1-contnuous parts of the reference trajectory. Detaled nformaton on the dervaton of these equatons and a smulatve experment can be found n [6]. The descrbed approach results n optmal breadthways coverage for the snow shovelng task, because even n curves the vehcles are orented n parallel at equally traveled dstances along the reference trajectory. When a formaton s dvded the moment when splttng the reference pont s tme synchronzed. On the other hand the moment of untng formatons s trggered by the arrval of the last subgroup, where groups that arrve earler have to wat. Ths makes the system more robust aganst devatons from the desred trajectory. (a) B. Trajectory trackng In order to make the moton of the sngle snowplough robust aganst uncertantes and perturbatons t s useful to apply a trajectory trackng control. For ths work an approach for car-lke robots utlzng full-state lnearzaton va dynamc feedback was used. Wth approprate gans for the controller, the barycenter of the applyng vehcle located at the center of ts rear axle wll be stablzed along the desred trajectory whch should be 3 tmes dfferentable almost everywhere to receve accurate results. When followng a path of the form descrbed n III-B, small neglgble devatons occur n the connecton ponts of crcles and lnes. After passng such a pont, the actual state wll converge exponentally back to the desred state. To track a trajectory the robots always need to know ther own poston n a fxed coordnate system. Ths can be acheved by utlzng an accurate postonng system (e.g. DGPS) whch s nstalled at the arfeld. More nformaton as well as mathematcal descrpton and analyss can be found n [4]. V. EXPERIMENTAL RESULTS The functonalty of the route schedulng algorthm has been verfed by an experment prepared for Frankfurt arport. The schedule depcted n Fg. 5 s the optmal plan for shovelng the servce roads of A. In ths part of cleanng the bg formaton s splt nto two parts wth an equal number of robots (see step 2) of the cleanng loop n secton III-A). The two sequences show order and tme n whch each node should be vsted. The schedules for both subformatons begn and end n the same node, whch are the ones where the formatons are dvded and unted (Nodes 41 and 24). As mentoned before all sgnfcant parts of the arport snow shovelng scenaro have been tested n smaller scenaros. Fgure 3 shows two snapshots from the smulatons performed. The sze of the scenaro s adjusted to the hardware experment descrbed later as we expect smlar performance for largescale vehcles and envronment. The two snowploughs start to remove the snow from the man runway n a formaton. After shovelng the frst turn cooperatvely the formaton s splt to clean the smaller roads separately (see Fg. 3(a)). Before the (b) Fg. 3. Smulaton snapshots (unts n meter): (a) Splttng of formaton. (b) One cleanng cycle complete. snowploughs reunte the vehcle on the center road has to wat for the other one to pass by. Otherwse the robot comng from the south would leave an amount of snow n the turnoff. Once the robots are reunted, they splt vertcally durng the next turn n order to proceed to the startng pont along the man runway (see Fg. 3(b)). It can be observed that the snow s not optmally removed durng turns of the snowploughs. Ths results form the fact that the path s followed by the robots barycenter and not by the center of ts shovel. Snce the robots dynamcs are left out completely n the smulaton, we carred out hardware experments wth dentcal expermental setup. The snow s made of small peces of polystyrene and the shovels are smply straght bars mounted transversely to the bumper of the car-lke MERLIN robots [17]. Fg. 6 depcts three dfferent states of one shovelng cycle. The poston determnaton of the robots n the experment reles on dead reckonng whch s based on nformaton obtaned from wheel encoders and a gyroscope. Therefore the poston error accumulates wth tme, whch explans why the shovels of the two robots do not overlap anymore after one shovelng cycle (see Fg. 6(c)). Furthermore t s not possble to correct the ntal poston and orentaton error. To avod ths n a real arfeld scenaro an accurate postonng system should be utlzed to obtan external feedback for the robots absolute poston. The phenomenon that snow partcles reman n the ntersecton of the roads results from the smple desgn of the shovels used and wll therefore not occur when utlzng modern snowploughs.

5 5 Fg. 4. Map of Frankfurt Arport wth roads and runways parttoned nto 3 sets. The satellte map was obtaned from [7]. Fg. 5. Tme schedule for cleanng the servce roads n area A. The sequences for two formatons show order and tme n whch each node should be vsted.

6 6 to the ones n our prevous work [5]. The suboptmal snow cleanng durng turns that was observed n the snow shovelng smulaton also needs to be mproved. It mght be possble to derve an approprate path for the barycenter of the robot by applyng a transformaton to the optmal path of ts shovel or by applyng a trajectory trackng controller wth respect to the center of the shovel. (a) ACKNOWLEDGMENT Ths work was supported by the Eltenetwork of Bavara through the program Identfcaton, Optmzaton and Control wth Applcatons n Modern Technologes. Fg. 6. Snapshots from the hardware experments: (a) Intal State. (b) The robot on the mddle road wats for the other one passng by. Ths pcture was taken from the opposng sde. (c) One cleanng cycle complete. (b) (c) VI. CONCLUSIONS In ths paper a novel snow shovelng method for arports was presented. The approach utlzes snowploughs drvng n a formaton and can be used for cleanng arports of dfferent szes. The autonomous car-lke robots can be splt and unted to varously szed formatons dependng on the wdth of the roads to clean. The ntroduced algorthm offers optmal and complete shovelng durng unnterrupted arport operatons. All parts of the method as well as the whole complex have been ntensvely tested n varous smulaton and hardware experments. The results showed abltes and qualtes of the method, but also emphaszed some defcences and possble tasks for future work. At the moment the approach s lackng robustness wth respect to unexpected events. Therefore we plan to enhance the vehcle coordnaton such that t can deal wth robot falures as well as wth unforeseen obstacles. On the other hand the task schedulng could be extended for fndng a global optmal sequence for the whole arport. At the moment we can only verfy a suboptmal soluton for each regon, but there may exst a better dstrbuton of the unclassfed paths (area AB). Ths task mght be solvable by multple travelng salesman strateges [16]. In the task plannng the composton of the fnal path should be mproved to avod the small devatons n the robots movement n the crcle-lne connecton. Ths can be prevented by usng splne paths smlar REFERENCES [1] T. D. Barfoot and C. M. Clark. Moton plannng for formatons of moble robots. Robotcs and Autonomous Systems, 46:65 78, February [2] T. D. Barfoot, C. M. Clark, and S. M. Rock. Knematc path-plannng for formatons of moble robots wth a nonholonomc constrant. In IEEE Internatonal Conference on Intellgent Robots and Systems, October [3] J. Cortes, S. Martnez, Karatas T., and F. Bullo. Coverage control for moble sensng networks. IEEE Transactons on Robotcs and Automaton, 20(2): , [4] A. De Luca, G. Orolo, and C. Samson. Feedback control of a nonholonomc car-lke robot. In J.-P. Laumond, edtor, Plannng robot moton. Sprnger-Verlag, [5] M. Hess, M. Saska, and K. Schllng. Formaton drvng usng partcle swarm optmzaton and reactve obstacle avodance. In Proceedngs Frst IFAC Workshop on Multvehcle Systems (MVS 06), Salvador, Brazl, [6] M. Hess, M. Saska, and K. Schllng. Enhanced moton plannng for dynamc formatons of nonholonomc moble robots. In Proceedngs of the 6th IFAC Symposum on Intellgent Autonomous Vehcles (IAV2007), September [7] [ct ]. Google maps. [8] B.J. Jorgen and G. Gutn. Dgraphs: Theory, Algorthms and Applcatons. Elsever North Holland, [9] C.S. Kong, N.A. Peng, and I. Reklets. Dstrbuted coverage wth multrobot system. In Proc. of the Conf. on Robotcs and Automaton, pages , Orlando, Florda, [10] D. Kurabayash, J. Ota, T. Ara, S. Ichkawa, S. Koga, H. Asama, and I. Endo. Cooperatve sweepng by multple moble robots wth relocatng portable obstacles. In Proc. of the Conf. on Intellgent Robots and Systems, pages , Osaka, Japan, [11] D. Kurabayash, J. Ota, T. Ara, and E. Yoshda. Cooperatve sweepng by multple moble robots. In Proc. of the Conf. on Robotcs and Automaton, Mnneapols, Mnnesota, [12] W. L and C. G. Cassandras. Dstrbuted cooperatve coverage control of sensor networks. In 44th IEEE Conference on Decson and Control and 2005 European Control Conference, pages , December [13] C. Luo and X. Yang. A real-tme cooperatve sweepng strategy for multple cleanng robots. In Proc. of the Symp. on Intellgent Control, Vancouver, Canada, [14] C. Luo, X. Yang, and D. Stacey. Real-tme path plannng wth deadlock avodance of multple cleanng robots. In Proc. of the Conf. on Robotcs and Automaton, Tape, Tawan, [15] T.W. Mn and H.K. Yn. A decentralzed approach for cooperatve sweepng by multple cleanng robots. In Proc. of the Int. Conf. on Intellgent Robots and Systems, Vctora, Canada, [16] S. Mtrovc-Mnc and R. Krshnamurt. m-tsptw: Bounds for the number of vehcles. In Proc. of the INFORMS Annual Meetng, November [17] K. Schllng and Q. Meng. The merln vehcles for outdoor applcatons. In G. R. Gerhart, C. M. Shoemaker, and D. W. Gage, edtors, Unmaned Ground Vehcle Technology IV, Proceedngs of SPIE, volume 4715, pages 43 49, 2002.