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Collaborative exploration for a team of mobile robots: Centralized and decentralized strategies
COLLABORATIVE EXPLORATION FOR A TEAM OF MOBILE ROBOTS: CENTRALIZED AND DECENTRALIZED STRATEGIES Lucilla Giannetti, Stefano Pagnottelli, Paolo Valigi
Dipartimento di Ingegneria Elettronica e dell'Injormazione Universita di Perugia Via G. Duranti, 93 - 06125 Perugia - Italy [giannetti,pagnottelli,valigi]~diei.unipg.it
Abstract: In this paper, the problem of collaborative exploration of a wholly unknown environment by means of a team of mobile robots is studied. To allow for a quantitative comparison of different strategies, some performance indexes are introduced. Two different strategies are proposed: a decentralized strategy, based on free-market concepts, and a strategy based on the presence of a coordinating robot. The proposed strategies will be evaluated by means of simulation models. The proposed strategies are of interest also in underwater environments. Copyright
©
2004 IFAC
Keywords: Mobile robots, multi robot exploration, robot coordination.
1. INTRODUCTION
The goal of underwater robotic is the design and realization of autonomous agents able to move in the depths, to recognize its position and to interact with the environment. Applications of actual interest cover , among many others, sea science (e.g., explorations concerning ocean fauna, flora or geology), off-shore (the industry of hydrocarbon mining), archeology and environment protection, monitoring and maintenance of deep pipelines and deep ocean telecommunication cable operations. Multiple-vehicle operations for underwater missions using autonomous robots is expected to receive increasing attention (see, e.g, Kuroda et al. (1994); Chappell et al. (1994)): a robot team may be widely dispersed to get a high probability of detecting some interesting object (like archeological finds) or localizing the position of missing people. The coordination problem is studied also for the
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case of three-dimensional space motion (Smith et al. (2001)). Working with a multi-robot system , collaborative strategies are key issues, e.g., to optimize the completion time of assigned task, and resource usage. In this case, communication among agents is a relevant aspect, which is especially crucial in underwater applications (Chappell et al. (1994)). The collaborative exploration problem has received large attention in the literature, and several different approaches have been proposed. In Yamauchi (1999) a collaborative exploration problem is studied, for a team of mobile robots. The collaboration is based on the sharing of a single map, while goal planning for each mobile robot is carried out in a decentralized and independent manner. Each robot has its own global occupancy map, which is used to plan the local exploration. To increase the environment knowledge, robots navigate on jrontiers, that is, regions on
465
the boundary between unexplored and explored space.
periments showing the performance of the proposed schemes.
The approach in Yamauchi (1999) is related to Yamauchi et al. (1998) , where the ARIEL architecture is presented. ARIEL realizes a frontierbased exploration, and an autonomous localization, based on a continuous localization integrated with odometry. In Burgard et al. (2000, 2002) a probabilistic approach to the exploration of collaborative multiple robots is proposed. As in Yamauchi et al. (1998), the exploration is based on the concept of frontier-cell, and the selection of the goal for each robot is carried out in a coordinated manner: different frontier cells are assigned to different robots. The choice of frontier cells is based on a trade-off between the cost of reaching the target point and its utility. While the path cost is proportional to occupancy probability of crossed cells, the utility is used to coordinate robots avoiding to choose the same frontier cell. As a matter of fact, when a target point is selected for a robot, the utility of adjacent cells is reduced according to their visibility probability stated by available sensors. In Simmons et al. (2000) a central mapper is used to coordinate exploration of robot . It receives sensors information from each robot and builds a single global map based on maximum likelihood estimate: such a global map will then be transmitted to all the robots. A central coordination is adopted to assign exploration tasks to individual robots. To organize the exploration, robots sent "bid" to the central executive that tries to maximize the total expected utility of the robots by assigning them tasks. Among all " bids" , the executive selects the "bid" with the highest net utility, using a simple greedy algorithm, and assigns that task to the proposer robot. Net utility is equivalent to informa tion gain minus cost of visiting the frontier cell. While information gain is the number of unknown cells that fall within sensor range of the frontier cell, the cost is estimate computing the optimal path from the robot 's current position. Whenever the best task is assigned, the bids of remaining robot are updated and the central executive repeats the procedure choosing the highest remaining net utility and assigning tasks to the relative robots until each robot have a target point to reach .
An integrated approach to the communications and navigation problem for Autonomous Underwater Vehicles (AUVs) has been studied in Singh et al. (1996) and Chappell et al. (1994), while Williams et al. (2000) a solution is proposed to the SLAM problem for underwater autonomous vehicles, which is further discussed , in a general setting, in Williams et al. (2002). The exploration problem is quite often studied together with the localization one, in the framework of multi-robot SLAM (see, among others, Liu and Thrun (2003) and references therein), and in other approaches, such as Markov localization (see, among others, Fox et al. (2000) and references therein). A relevant issue addressed in underwater application is distributed control, studied in Stilwell (2002) for team of robots .
The approach proposed in Cohen (1996) is based on the use of a navigator and a set of cartographers. The cartographers move within the environment in random order. Upon detection of the target location by means of a cartographer, the navigator is notified, and it starts moving toward the target position. The problem of task allocation and coordination among a team of mobile robots is studied in Mataric and Sukatme (2001), with extensive ex-
466
In this paper, the exploration problem will be addressed, for a team of autonomous robots as well as the simultaneous exploration and localization problem. The major contributions of the paper are a discussion of the problem , a set of performance indexes to numerically evaluate alternative strategies, a heuristic solutions to the problem, and, finally, simulation results to compare these two strategies.
2. PROBLEM FORMULATION In this papers is studied a collaborative exploration problem for a team of N autonomous robots. In particular, we consider the case of a group of mobiles whose task is the exploration (and map building) of an unknown, unstructured environment. We seek for collaborative solutions to the above problem , assuming a team of identical robots. To carry out in a satisfactory manner the exploration task , the robots have also to solve a localization problem. In this paper , the focus is on the exploration problem alone, therefore assuming each robot can perfectly locate itself. The collaborative localization problem is subject of on going research, as well as the cooperative simultaneous localization and mapping one. The reference scenario for the exploration problem considered here comprises an unknown flat environment, containing fixed obstacles placed at unknown locations. The exploration task consists in scanning the whole environment, identifying all the obstacles and therefore building a complete map. To cope with this problem, the environment is cell-decomposed, according to a regular grid map. Each cell is identified by the integer (x, y) coordinate of its center. Each robot is assumed to be equipped with a suitable sensing system
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allowing the exploration of the set A(x, y) containing all the cells adjacent to his current (x, y) location, i.e., all the cells that can be reached by one step move along the x and/or y coordinate: A(x,y) = {(x,y) : x = x ± 1,y = y ± 1}. Cells along the border of the environment have reduced adjacent set. A cell (x, y) is termed visited whenever at least one robot went through the cell, whereas a cell (x, y) is termed explored whenever at least one robot visited a cell in its adjacent set. To compare different exploration strategies, we introduced some performance indexes, measuring the redundancy of exploration, the number of visited cell, and the total exploration time. To formally introduce the problem, let X denote the environment to be explored, and let Ei and Vi denote the portion of environment explored and visited, respectively, by the i-th robot , i = 1, ... , N. Then, the condition under which the exploration problem is solved is given by: (1)
which implies that the whole environment has to be explored. The performance of strategies aimed at solving the above problem can be evaluated by means of the following indices. Let nij = Ei n Ej , Vi,j = 1, . . . , N be the portion of environment (i.e., the set of cells) explored by both the i-th and the j-th robot . A good exploration algorithm minimizes
(2) where 1nl denote the number of cell in set n. Let Aij = Vi n Vj , Vi,j = 1, . . . , N , be set of cells visited by both the i-th and the j-th robot. A second performance index is (3)
Finally, let Ti be the exploration time of the ith robot. The time required to explore the whole environment is then given by: P3
= maXTi· i
(4)
3. EXPLORATION STRATEGIES The exploration problem will be tackled by means of two heuristic strategies. The first one is based on a decentralized approach, based on free-market concepts. The second strategy, on the contrary, is based on the key role of a coordinator, taking decision affecting the behavior of the whole team.
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3.1 Free-market collaborative exploration strategy In the free-market policy for collaborative exploration, the team of robots is not hierarchically structured, instead every robot has the same competence and ability. The exchange of information and the subsequent path planning process are based on the so called free-market concept, as in Dias and Stentz (2000) . The free-market approach defines revenue and cost functions across the possible plans for executing a specified task. Free-market approach does not use centralized planning: robots are free to exchange information and services and enter into contracts among themselves, as they see fit. Here the use of a cost function C is proposed, based on distance: the waiting cost is the distance between the current location of a robot and the goal location. The revenue function R is based on the growth of information that the visit of the cell would determine. To represent the state of the system, for each step of exploration, an occupancy map is used. A codification is used to mark a cell like empty, occupied by obstacles, occupied by other robot or unexplored. The gain of exploration is determined by the growth of information risen by the visit of a cell, in particular we use the number of unexplored cells adjacent to the current position. The utility of each cell is calculated as the different between cost and revenue. We did a weight mean, to give advantage to cells with a lower revenue together with limited costs. The goal of each robot is to explore a larger amount of cells with less cost. The first question, which must be answered to design a satisfactory team, is the strategy to select the target cell . In this paper we propose a greedy strategy, by which the cell, which is in the center of the nearest unexplored zone, is chosen as the target point. The rationale for this choice is that this method is simpler to implement and easier to manage with respect to other methods with far away target cells. After target cell selection, the assignment of a cost is needed, to quantify revenue after visit that cell. Furthermore, it's important to take care to choose different target for each robot, to avoid collision. With a decentralized team the problem has to be faced by the local agents near the contested cell, with the choice of "local" priority. Two rules are used in this paper: rule 1. The cell at the intersection between two trajectories is reserved on a FIFO basis. rule 2. The transit through the same cell is allowed one robot at a time.
467
3.2 Coordinated collaborative exploration strategy
The centralized strategies is based on the assumption that the team of mobile robots comprises a coordinator, which is fixed (e.g., in the origin of the global reference frame), and a number of mobile agent. The agents can alternate between two roles: the role of explorer and the role of marker. The explorers moves around the environment, according to navigation policies that will be described subsequently, while the markers allow for the explorers to localize based on measurements of relative position and orientation. The coordinator also plays the role of marker. Localization is based on triangulation, and makes use of observability results (Conticelli et al. (2000)) and a related observability index. Thus, the exploration problem is here integrated with the localization one (see Leonard and Smith. (1997); Carpenter and Medeiros (2001) for issues concerning localization in underwater environment, and Chappell et al. (1994) for issues concerning inter-robot communication). Upon the occurrence of role-assignment events, (among which also the completion of an exploration period of predefined length), each robot undergoes a verification, and role re-assignment may be carried out for pair of robots. Within two such events, exploration proceeds according to a centralized policy. The destination cell for each walker, which is always a frontier cell (see Yamauchi (1999)) , is computed on the basis of cost (namely, distance from current location) and revenues (namely, number of un-explored cell encountered along the path) associated to each possible choice. The revenue of each cell takes into account the increase in information associated with a visit to the cell. The cost of each frontier cell is computed via a slightly modified version of the A' algorithm. Simulation experience shows that the best choice for the destination to assign each walker is a frontier cell with an average value for the revenue. As a matter of facts, selecting the cell with the highest revenue will increase the probability of neglecting small sets of unexplored cells: turning back to these sets later on would increase overall completion time. Similarly, selection among destination cells with the same average revenue is carried out in order to keep high the distance among explorers. This is similar to the utility concept in Burgard et al. (2000) . Additional heuristic rules are used to deal with potential collision among explorers (based on priority concepts), with specific configuration of partly identified obstacles, and other situations of interest.
468
4. PERFORMANCE EVALUATION The exploration strategies proposed in the previous sections have been studied by means of simulation models . The simulation tool is Matlab-based, and allows for easy implementation of exploration strategies, together with the computation of the performance indexes. The tool also allow for an easy definition of the features of the environment to be explored , such as obstacle positions and dimensions. An alternative simulation tool is under consideration, based on Vaughan (2002), which is a public domain application. The behavior of the decentralized strategy is described in the following Figures 1 through 3, where the evolution of the exploration process is illustrated. In all the figures, obstacles are in black, while in Figure 3, the lines indicate the actual paths covered by the three robots . The scenario is characterized by sparse obstacles, and the exploration is carried out by a team comprising three robots. As the exploration proceeds, the three robots distribute over the region in an autonomous manner, exploring different portions of the environment. By following in detail the evolution of the exploration task, it is possible to appreciate the cooperation among robots, namely each robot directs himself toward non-explored regions, and remains away from other robots, thus improving the efficiency of the strategy. Simulation of other environments, such as a space with wall-like barriers, hence characterized by a number of sub-regions, shows that each robot of the team moves toward a different sub-region. The characterization of the strategy in terms of completion time indicate that the whole is explored within 103 time unit, with a moderate value of index PI (i.e. , cells explored by more than one robot) and a quite negligible value for P2 (i .e. , cells visited more than once). The centralized strategy has been tested by using the same scenario considered above, i.e., an open space, with sparse obstacles. The exploration is carried out by using a team comprising four robots, two of them acting as explorers and the other as markers, according to the policy proposed in the previous section. The exploration progress is illustrated in Figure 4, where the circles indicate the robot acting as markers, while the triangles indicate robots acting as explorers. The performance indexes for the centralized strategy are reported in the following table, for the environment considered above, and for additional environments. Notice that environments with complex obstacles yield a strong increment in the indexes P2 e P3 .
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Table 1. Comparing performance indexes Notice that, in terms of completion time (index
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5. CONCLUSIONS Two heuristic strategies have been proposed, to tackle the problem of collaborative exploration for unknown environment, by mea ns of a team of mobile robots. The first strategy is decentralized, and uses free-market concepts to allow robot to negotiate for exploration. The second strategy is based on the presence of a coordinator, and on a team of robots that may behave as markers or explorers, according to coordinator indications.
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Simulation results illustrate the performance of the two strategies. Current research activity is considering a formal statement of the collaborative exploration problem, and other exploration policies. The design and realization of a team of mobile robots is an additional ongoing activity (of interest in indoor environments). Acknowledgment This work has been supported by ASI under project T.E.M.A., contract I/R/124/02. The authors thanks Dr. Agostino Miele, who contributed to the discussion on the free-market strategy, and Dr. Massimo Apostolico, who contributed to the discussion on the coordinated strategy.
REFERENCES Burgard, W., M. Moors, D. Fox, R Simmons and S. Thrun. Collaborative multi-robot exploration. In Proceeding of the 2000 IEEE International Conference on Robotic and Automation San Francisco, CA, April 2000. Burgard, W., M . Moors, and F. Schneider. Collaborative exploration of unknown environments with teams of mobile robots. In M. Beetz, J . Hertzberg, M. Ghallab and M.E. Pollack, editors, Plan-Based Control of Robotic Agents, volume 2466 of Lecture Notes in Computer Science. Springer Verlag, 2002. Carpenter, RN . and M.R. Medeiros. Concurrent mapping and localization and map matching on autonomous underwater vehicles. In Proc. of MTSjIEEE Conference and Exhibition OCEAN, volume 1, pages 380- 389, 2001. Chappell, S.G ., J.C. Jalbert, P. Pietryka and J. Duchesney. Acoustic communication between two autonomous underwater vehicles. In Proceedings of the 1994 Symposium on Autonomous Underwater Vehicle Technology, pages 462- 469, 1994. Cohen. W. Adaptive mapping and navigation by teams of simple robots. Journal of Robotics and A utonomous Systems, 18:411- 434 , 1996. Conticelli, F., A. Bicchi and A. Balestrino. Observability and nonlinear observers for mobile robot localization. In In IFAC Int. Symp . on Robot Control, SyRoCo 2000, Wien, 2000. Dias, M. B . and A. Stentz. A free market architecture for distributed control of a multirobot system. In In Sixth Int'l Conf. on Intelligent Autonomous Systems, Venice, Italy, pages 115122, 2000. Fox, D., W. Burgard, H. Kruppa and S. Thrun. A probabilistic approach to collaborative multirobot localization. In Special Issue of Autonomous Systems on Multi-Robot Systems, 2000. Kuroda, Y., T. Ura and K Aramaki . Vehicle control architecture for operating multiple ve-
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hicles. In Proceedings of the 1994 Symposium on Autonomous Underwater Vehicle Technology, pages 323 - 329, 1994. Leonard, J. J. and C. M. Smith. Sensor data fusion in marine robotics. In Proc. of the International Society of Offshore and Polar Engineering, Honolulu, HI" pages 100-106, 1997. Liu Y. and Sebastian Thrun. Gaussian multirobot SLAM. Sumitted to NIPS 2003, 2003. Mataric, M.J. and G. Sukatme. Task-allocation and coordination of multiple robots for planetary exploration. In Proceedings of the International Conference on Advanced Robotics, 2001. Simmons, R, D. Apfelbaum, W. Burgard, D. Fox, M. Moors, S. Thrun and H. Younes. Coordination for multi-robot exploration and mapping. In Proceedings National Conference on Artificial Intelligence, Austin TX, August 2000. Singh, H. , J. Catipovic, R Eastwood, L. Freitag, H. Henriksen, F. Hover, D. Yoerger, J.G. Bellingham and B.A. Moran. An integrated approach to multiple AUV communications, navigation and docking. In Proc. IEEE Oceanic Engineering Society OCEANS, pages 59-94, 1996. Smith, T.R, H. HanBmann, and N.E. Leonard. Orientation control of multiple underwater vehicles with symmetry-breaking potentials. In Proc. of 40th IEEE Conference on Decision and Control, Orlando, FL, USA, pages 4598- 4603 , 2001. Stilwell. D. J. Decentralized control synthesis for a platoon of autonomous vehicles . In IEEE Int. Conf. on Robotics and Automation, Washington, DC, pages 744-749, 2002. Stage a multiple robot Vaugha n, R T. simulator. Technical report, Institute for Robotics and Intelligent Systems, http//playerstage.sourceforge.net , Technical Report IRIS-00-393 , University of Southern California, 2002. Williams, S. B. G. Dissanayake and H. DurrantWhyte. Autonomous underwater simultaneous localisation and map building. In IEEE Int. Conf. on Robotics and Automation, volume 2, pages 1793- 1798, 2000. Williams, S. B . G. Dissanayake and H. DurrantWhyte . An efficient approach to the simultaneous localisation and mapping problem. In IEEE Int. Conf. on Robotics and Automation, Washington, DC, volume 2, pages 2743- 2748, 2002. Yamauchi, B. Decentralized coordination for multirobot exploration. Robotics and Autonomous Systems, 29:111-118, 1999. Yamauchi, B., A. Schultz, and W . Adams. Mobile robot exploration and map-building with continuous localization. In Proceedings of the 1998 IEEE International Conference on Robotics and Automation Leuven, Belgium, May 1998.
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Dipartimento di Ingegneria Elettronica e dell'Injormazione Universita di Perugia Via G. Duranti, 93 - 06125 Perugia - Italy [giannetti,pagnottelli,valigi]~diei.unipg.it
Abstract: In this paper, the problem of collaborative exploration of a wholly unknown environment by means of a team of mobile robots is studied. To allow for a quantitative comparison of different strategies, some performance indexes are introduced. Two different strategies are proposed: a decentralized strategy, based on free-market concepts, and a strategy based on the presence of a coordinating robot. The proposed strategies will be evaluated by means of simulation models. The proposed strategies are of interest also in underwater environments. Copyright
©
2004 IFAC
Keywords: Mobile robots, multi robot exploration, robot coordination.
1. INTRODUCTION
The goal of underwater robotic is the design and realization of autonomous agents able to move in the depths, to recognize its position and to interact with the environment. Applications of actual interest cover , among many others, sea science (e.g., explorations concerning ocean fauna, flora or geology), off-shore (the industry of hydrocarbon mining), archeology and environment protection, monitoring and maintenance of deep pipelines and deep ocean telecommunication cable operations. Multiple-vehicle operations for underwater missions using autonomous robots is expected to receive increasing attention (see, e.g, Kuroda et al. (1994); Chappell et al. (1994)): a robot team may be widely dispersed to get a high probability of detecting some interesting object (like archeological finds) or localizing the position of missing people. The coordination problem is studied also for the
CAMS 2004
case of three-dimensional space motion (Smith et al. (2001)). Working with a multi-robot system , collaborative strategies are key issues, e.g., to optimize the completion time of assigned task, and resource usage. In this case, communication among agents is a relevant aspect, which is especially crucial in underwater applications (Chappell et al. (1994)). The collaborative exploration problem has received large attention in the literature, and several different approaches have been proposed. In Yamauchi (1999) a collaborative exploration problem is studied, for a team of mobile robots. The collaboration is based on the sharing of a single map, while goal planning for each mobile robot is carried out in a decentralized and independent manner. Each robot has its own global occupancy map, which is used to plan the local exploration. To increase the environment knowledge, robots navigate on jrontiers, that is, regions on
465
the boundary between unexplored and explored space.
periments showing the performance of the proposed schemes.
The approach in Yamauchi (1999) is related to Yamauchi et al. (1998) , where the ARIEL architecture is presented. ARIEL realizes a frontierbased exploration, and an autonomous localization, based on a continuous localization integrated with odometry. In Burgard et al. (2000, 2002) a probabilistic approach to the exploration of collaborative multiple robots is proposed. As in Yamauchi et al. (1998), the exploration is based on the concept of frontier-cell, and the selection of the goal for each robot is carried out in a coordinated manner: different frontier cells are assigned to different robots. The choice of frontier cells is based on a trade-off between the cost of reaching the target point and its utility. While the path cost is proportional to occupancy probability of crossed cells, the utility is used to coordinate robots avoiding to choose the same frontier cell. As a matter of fact, when a target point is selected for a robot, the utility of adjacent cells is reduced according to their visibility probability stated by available sensors. In Simmons et al. (2000) a central mapper is used to coordinate exploration of robot . It receives sensors information from each robot and builds a single global map based on maximum likelihood estimate: such a global map will then be transmitted to all the robots. A central coordination is adopted to assign exploration tasks to individual robots. To organize the exploration, robots sent "bid" to the central executive that tries to maximize the total expected utility of the robots by assigning them tasks. Among all " bids" , the executive selects the "bid" with the highest net utility, using a simple greedy algorithm, and assigns that task to the proposer robot. Net utility is equivalent to informa tion gain minus cost of visiting the frontier cell. While information gain is the number of unknown cells that fall within sensor range of the frontier cell, the cost is estimate computing the optimal path from the robot 's current position. Whenever the best task is assigned, the bids of remaining robot are updated and the central executive repeats the procedure choosing the highest remaining net utility and assigning tasks to the relative robots until each robot have a target point to reach .
An integrated approach to the communications and navigation problem for Autonomous Underwater Vehicles (AUVs) has been studied in Singh et al. (1996) and Chappell et al. (1994), while Williams et al. (2000) a solution is proposed to the SLAM problem for underwater autonomous vehicles, which is further discussed , in a general setting, in Williams et al. (2002). The exploration problem is quite often studied together with the localization one, in the framework of multi-robot SLAM (see, among others, Liu and Thrun (2003) and references therein), and in other approaches, such as Markov localization (see, among others, Fox et al. (2000) and references therein). A relevant issue addressed in underwater application is distributed control, studied in Stilwell (2002) for team of robots .
The approach proposed in Cohen (1996) is based on the use of a navigator and a set of cartographers. The cartographers move within the environment in random order. Upon detection of the target location by means of a cartographer, the navigator is notified, and it starts moving toward the target position. The problem of task allocation and coordination among a team of mobile robots is studied in Mataric and Sukatme (2001), with extensive ex-
466
In this paper, the exploration problem will be addressed, for a team of autonomous robots as well as the simultaneous exploration and localization problem. The major contributions of the paper are a discussion of the problem , a set of performance indexes to numerically evaluate alternative strategies, a heuristic solutions to the problem, and, finally, simulation results to compare these two strategies.
2. PROBLEM FORMULATION In this papers is studied a collaborative exploration problem for a team of N autonomous robots. In particular, we consider the case of a group of mobiles whose task is the exploration (and map building) of an unknown, unstructured environment. We seek for collaborative solutions to the above problem , assuming a team of identical robots. To carry out in a satisfactory manner the exploration task , the robots have also to solve a localization problem. In this paper , the focus is on the exploration problem alone, therefore assuming each robot can perfectly locate itself. The collaborative localization problem is subject of on going research, as well as the cooperative simultaneous localization and mapping one. The reference scenario for the exploration problem considered here comprises an unknown flat environment, containing fixed obstacles placed at unknown locations. The exploration task consists in scanning the whole environment, identifying all the obstacles and therefore building a complete map. To cope with this problem, the environment is cell-decomposed, according to a regular grid map. Each cell is identified by the integer (x, y) coordinate of its center. Each robot is assumed to be equipped with a suitable sensing system
CAMS 2004
allowing the exploration of the set A(x, y) containing all the cells adjacent to his current (x, y) location, i.e., all the cells that can be reached by one step move along the x and/or y coordinate: A(x,y) = {(x,y) : x = x ± 1,y = y ± 1}. Cells along the border of the environment have reduced adjacent set. A cell (x, y) is termed visited whenever at least one robot went through the cell, whereas a cell (x, y) is termed explored whenever at least one robot visited a cell in its adjacent set. To compare different exploration strategies, we introduced some performance indexes, measuring the redundancy of exploration, the number of visited cell, and the total exploration time. To formally introduce the problem, let X denote the environment to be explored, and let Ei and Vi denote the portion of environment explored and visited, respectively, by the i-th robot , i = 1, ... , N. Then, the condition under which the exploration problem is solved is given by: (1)
which implies that the whole environment has to be explored. The performance of strategies aimed at solving the above problem can be evaluated by means of the following indices. Let nij = Ei n Ej , Vi,j = 1, . . . , N be the portion of environment (i.e., the set of cells) explored by both the i-th and the j-th robot . A good exploration algorithm minimizes
(2) where 1nl denote the number of cell in set n. Let Aij = Vi n Vj , Vi,j = 1, . . . , N , be set of cells visited by both the i-th and the j-th robot. A second performance index is (3)
Finally, let Ti be the exploration time of the ith robot. The time required to explore the whole environment is then given by: P3
= maXTi· i
(4)
3. EXPLORATION STRATEGIES The exploration problem will be tackled by means of two heuristic strategies. The first one is based on a decentralized approach, based on free-market concepts. The second strategy, on the contrary, is based on the key role of a coordinator, taking decision affecting the behavior of the whole team.
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3.1 Free-market collaborative exploration strategy In the free-market policy for collaborative exploration, the team of robots is not hierarchically structured, instead every robot has the same competence and ability. The exchange of information and the subsequent path planning process are based on the so called free-market concept, as in Dias and Stentz (2000) . The free-market approach defines revenue and cost functions across the possible plans for executing a specified task. Free-market approach does not use centralized planning: robots are free to exchange information and services and enter into contracts among themselves, as they see fit. Here the use of a cost function C is proposed, based on distance: the waiting cost is the distance between the current location of a robot and the goal location. The revenue function R is based on the growth of information that the visit of the cell would determine. To represent the state of the system, for each step of exploration, an occupancy map is used. A codification is used to mark a cell like empty, occupied by obstacles, occupied by other robot or unexplored. The gain of exploration is determined by the growth of information risen by the visit of a cell, in particular we use the number of unexplored cells adjacent to the current position. The utility of each cell is calculated as the different between cost and revenue. We did a weight mean, to give advantage to cells with a lower revenue together with limited costs. The goal of each robot is to explore a larger amount of cells with less cost. The first question, which must be answered to design a satisfactory team, is the strategy to select the target cell . In this paper we propose a greedy strategy, by which the cell, which is in the center of the nearest unexplored zone, is chosen as the target point. The rationale for this choice is that this method is simpler to implement and easier to manage with respect to other methods with far away target cells. After target cell selection, the assignment of a cost is needed, to quantify revenue after visit that cell. Furthermore, it's important to take care to choose different target for each robot, to avoid collision. With a decentralized team the problem has to be faced by the local agents near the contested cell, with the choice of "local" priority. Two rules are used in this paper: rule 1. The cell at the intersection between two trajectories is reserved on a FIFO basis. rule 2. The transit through the same cell is allowed one robot at a time.
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3.2 Coordinated collaborative exploration strategy
The centralized strategies is based on the assumption that the team of mobile robots comprises a coordinator, which is fixed (e.g., in the origin of the global reference frame), and a number of mobile agent. The agents can alternate between two roles: the role of explorer and the role of marker. The explorers moves around the environment, according to navigation policies that will be described subsequently, while the markers allow for the explorers to localize based on measurements of relative position and orientation. The coordinator also plays the role of marker. Localization is based on triangulation, and makes use of observability results (Conticelli et al. (2000)) and a related observability index. Thus, the exploration problem is here integrated with the localization one (see Leonard and Smith. (1997); Carpenter and Medeiros (2001) for issues concerning localization in underwater environment, and Chappell et al. (1994) for issues concerning inter-robot communication). Upon the occurrence of role-assignment events, (among which also the completion of an exploration period of predefined length), each robot undergoes a verification, and role re-assignment may be carried out for pair of robots. Within two such events, exploration proceeds according to a centralized policy. The destination cell for each walker, which is always a frontier cell (see Yamauchi (1999)) , is computed on the basis of cost (namely, distance from current location) and revenues (namely, number of un-explored cell encountered along the path) associated to each possible choice. The revenue of each cell takes into account the increase in information associated with a visit to the cell. The cost of each frontier cell is computed via a slightly modified version of the A' algorithm. Simulation experience shows that the best choice for the destination to assign each walker is a frontier cell with an average value for the revenue. As a matter of facts, selecting the cell with the highest revenue will increase the probability of neglecting small sets of unexplored cells: turning back to these sets later on would increase overall completion time. Similarly, selection among destination cells with the same average revenue is carried out in order to keep high the distance among explorers. This is similar to the utility concept in Burgard et al. (2000) . Additional heuristic rules are used to deal with potential collision among explorers (based on priority concepts), with specific configuration of partly identified obstacles, and other situations of interest.
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4. PERFORMANCE EVALUATION The exploration strategies proposed in the previous sections have been studied by means of simulation models . The simulation tool is Matlab-based, and allows for easy implementation of exploration strategies, together with the computation of the performance indexes. The tool also allow for an easy definition of the features of the environment to be explored , such as obstacle positions and dimensions. An alternative simulation tool is under consideration, based on Vaughan (2002), which is a public domain application. The behavior of the decentralized strategy is described in the following Figures 1 through 3, where the evolution of the exploration process is illustrated. In all the figures, obstacles are in black, while in Figure 3, the lines indicate the actual paths covered by the three robots . The scenario is characterized by sparse obstacles, and the exploration is carried out by a team comprising three robots. As the exploration proceeds, the three robots distribute over the region in an autonomous manner, exploring different portions of the environment. By following in detail the evolution of the exploration task, it is possible to appreciate the cooperation among robots, namely each robot directs himself toward non-explored regions, and remains away from other robots, thus improving the efficiency of the strategy. Simulation of other environments, such as a space with wall-like barriers, hence characterized by a number of sub-regions, shows that each robot of the team moves toward a different sub-region. The characterization of the strategy in terms of completion time indicate that the whole is explored within 103 time unit, with a moderate value of index PI (i.e. , cells explored by more than one robot) and a quite negligible value for P2 (i .e. , cells visited more than once). The centralized strategy has been tested by using the same scenario considered above, i.e., an open space, with sparse obstacles. The exploration is carried out by using a team comprising four robots, two of them acting as explorers and the other as markers, according to the policy proposed in the previous section. The exploration progress is illustrated in Figure 4, where the circles indicate the robot acting as markers, while the triangles indicate robots acting as explorers. The performance indexes for the centralized strategy are reported in the following table, for the environment considered above, and for additional environments. Notice that environments with complex obstacles yield a strong increment in the indexes P2 e P3 .
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5. CONCLUSIONS Two heuristic strategies have been proposed, to tackle the problem of collaborative exploration for unknown environment, by mea ns of a team of mobile robots. The first strategy is decentralized, and uses free-market concepts to allow robot to negotiate for exploration. The second strategy is based on the presence of a coordinator, and on a team of robots that may behave as markers or explorers, according to coordinator indications.
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Simulation results illustrate the performance of the two strategies. Current research activity is considering a formal statement of the collaborative exploration problem, and other exploration policies. The design and realization of a team of mobile robots is an additional ongoing activity (of interest in indoor environments). Acknowledgment This work has been supported by ASI under project T.E.M.A., contract I/R/124/02. The authors thanks Dr. Agostino Miele, who contributed to the discussion on the free-market strategy, and Dr. Massimo Apostolico, who contributed to the discussion on the coordinated strategy.
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hicles. In Proceedings of the 1994 Symposium on Autonomous Underwater Vehicle Technology, pages 323 - 329, 1994. Leonard, J. J. and C. M. Smith. Sensor data fusion in marine robotics. In Proc. of the International Society of Offshore and Polar Engineering, Honolulu, HI" pages 100-106, 1997. Liu Y. and Sebastian Thrun. Gaussian multirobot SLAM. Sumitted to NIPS 2003, 2003. Mataric, M.J. and G. Sukatme. Task-allocation and coordination of multiple robots for planetary exploration. In Proceedings of the International Conference on Advanced Robotics, 2001. Simmons, R, D. Apfelbaum, W. Burgard, D. Fox, M. Moors, S. Thrun and H. Younes. Coordination for multi-robot exploration and mapping. In Proceedings National Conference on Artificial Intelligence, Austin TX, August 2000. Singh, H. , J. Catipovic, R Eastwood, L. Freitag, H. Henriksen, F. Hover, D. Yoerger, J.G. Bellingham and B.A. Moran. An integrated approach to multiple AUV communications, navigation and docking. In Proc. IEEE Oceanic Engineering Society OCEANS, pages 59-94, 1996. Smith, T.R, H. HanBmann, and N.E. Leonard. Orientation control of multiple underwater vehicles with symmetry-breaking potentials. In Proc. of 40th IEEE Conference on Decision and Control, Orlando, FL, USA, pages 4598- 4603 , 2001. Stilwell. D. J. Decentralized control synthesis for a platoon of autonomous vehicles . In IEEE Int. Conf. on Robotics and Automation, Washington, DC, pages 744-749, 2002. Stage a multiple robot Vaugha n, R T. simulator. Technical report, Institute for Robotics and Intelligent Systems, http//playerstage.sourceforge.net , Technical Report IRIS-00-393 , University of Southern California, 2002. Williams, S. B. G. Dissanayake and H. DurrantWhyte. Autonomous underwater simultaneous localisation and map building. In IEEE Int. Conf. on Robotics and Automation, volume 2, pages 1793- 1798, 2000. Williams, S. B . G. Dissanayake and H. DurrantWhyte . An efficient approach to the simultaneous localisation and mapping problem. In IEEE Int. Conf. on Robotics and Automation, Washington, DC, volume 2, pages 2743- 2748, 2002. Yamauchi, B. Decentralized coordination for multirobot exploration. Robotics and Autonomous Systems, 29:111-118, 1999. Yamauchi, B., A. Schultz, and W . Adams. Mobile robot exploration and map-building with continuous localization. In Proceedings of the 1998 IEEE International Conference on Robotics and Automation Leuven, Belgium, May 1998.
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