A New Meta-Module for Controlling Large Sheets of ATRON Modules

Transcription

1 A New Meta-Module for Controlling Large Sheets of ATRON Modules David Brandt and David Johan Christensen The Adaptronics Group, The Maersk Institute, University of Southern Denmark {david.brandt, Abstract In this paper we present a 2D meta-module for the ATRON robot, which simplifies the motion constraints significantly. The motion capabilities of the new meta-module is similar to that of previous sliding cube style modules with the addition of one extra action, which is shown to improve the motion capabilities of the modules greatly. In general our work shows that if three simple actions, which is implemented by our proposed meta-module, is implemented by a self-reconfigurable robot module or meta-module then the control of the robot will be simple. The improved motion capabilities allow us to use offline planning for large groups consisting of hundreds of metamodules. The simple motion constraints of the meta-module further allows us to implement a distributed cluster flow locomotion gait, with the interesting property that the group of meta-modules self-organizes into a shape that makes the locomotion efficient. I. INTRODUCTION A self-reconfigurable robot is a modular robot consisting of a number of homogeneous or heterogeneous modules. The modules can be connected to form many different configurations. Further, the modules are able to autonomously rearrange themselves in the group to change the configuration in order to solve a given task or adapt to the environment. The research of self-reconfigurable robots began more than two decades ago with the groundbreaking work by Fukuda and Nakagawa [1]. Since then many self-reconfigurable robots have been both proposed and built including but not limited to [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13]. The main reason for the growing interest in selfreconfigurable robots is that they have a number of unique and interesting features. Firstly, the ability to self-reconfigure allows the robot to change its morphology in order to adapt to the environment or to the task at hand in very versatile ways. Even small groups of self-reconfigurable robot modules can form many usefull configurations such as walkers, vehicles, snakes, manipulators etc. Secondly, due to the redundancy of the system larger groups of modules have the potential to be tolerant to some degree of failure since faulty modules can simply be replaced by spare working modules from other parts of the group. This paper considers a method for reducing the complexity of the motion constraints of the ATRON self-reconfigurable robot, but first we need to informally define what we mean by the notions of motion constraints and motion capabilities. The motion constraints and the motion capabilities of a self-reconfigurable robot is basically each other s inverse. Meaning, if a robot has good motion capabilities then the motion constraints are simple and vice versa. The motion constraints of a self-reconfigurable robot define how difficult it is to move a module in a cluster of modules. The motion constraints for the ATRON modules [11] are quite complex since much intermodule cooperation is needed to move a module due to the simple 1 DOF module kinematics. The focus of this paper is the control of self-reconfigurable robots or, more specifically, the control of two dimensional sheets build from modules of the ATRON self-reconfigurable robot, which is further described in Section III. So far both offline planning methods [14] as well as distributed controllers [15], [16] for the ATRON robot have been proposed. In all attempts to control the ATRON robot the limiting factor on the success has been the complex motion constraints arising from the limited motion capabilities of the individual modules. The motion constraints were found to limit either the precision of the control [16], the scalability [14] or make the construction of the controllers very tedious [15]. One of the promising methods for reducing the complexity of the motion constraints of a self-reconfigurable robot is the notion of meta-modules where a small group of modules is regarded as a single entity. The idea behind the meta-module approach is that the motion capabilities of a meta-module should be far greater than that of a single module, such that the meta-module is less dependent upon the help of other modules when moving in the cluster. In this paper we introduce a new symmetric two dimensional 4 module meta-module for the ATRON selfreconfigurable robot. This meta-module allows us to selfreconfigure large surfaces of ATRON modules in an efficient way. It is demonstrated that both centralized as well as distributed control is feasible using these meta-modules. II. RELATED WORK Meta-module approaches for reducing the complexity of the control or for simplifying reconfiguration planning for self-reconfigurable robots have been proposed several times. In [7] a meta-cube for the I-Cubes robot is proposed consisting of 8 cubes and 16 links. A 3-layered planned is applied and problems involving up to 6 meta-cubes is solved in the order of minutes on a 600 MHz Pentium III computer. Meta-modules have previously been used to control the ATRON robot in [16], where a distributed approach using artificial evolution and neural networks was used for failure tolerant control of 3 module meta-modules. The experiments included test trails with more than 3000 ATRON modules. For the Molecule robot meta-modules have been used for centralized planning as described in [4]. The approach is

2 Fig. 1. A small picture sequence illustrating the basic principles of how one ATRON module is dependent upon the others in order to move within the cluster. The connections between modules are not shown. to divide the planning problem into three layers. Trajectory planning, configuration planning and task-level planning. Algorithms for both trajectory planning and configuration planning is presented. The algorithm proposed for planning the trajectory of a single Molecule on the surface of a group of Molecules is a variation of Dijkstra s shortest path algorithm where the surface of the structure is seen as a graph. The configuration planning problem is solved by introducing meta-modules, which allow modules to pass through the structure, such that the configuration planning problem is reduced to a matching problem, of deciding which meta-module should move where, and a simple 3D manhattan routing problem. In [10] Rus and Vona use 16 module meta-modules for the Crystalline robot as the basis for a centralized motion planning algorithm for the general self-reconfiguration problem. The algorithm is based on a melt-grow approach where the initial configuration is first melted into an intermediate configuration and from this intermediate configuration the goal configuration is grown. The intermediate configuration used is a simple line configuration. We will later show that the proposed meta-module effectively reduces the motion constraints of the ATRON robot to be quite similar to a 2D version of the sliding cube style modules used by Butler in [17] and Støy in [18]. This allows these algorithms to be transferred to the ATRON robot using the proposed meta-module. The distributed control algorithm presented in this paper uses some inspiration from the work on the Fracta robot by Murata et al. In [2] a completely distributed control algorithm is presented. In this algorithm each module decides whether to move or not based completely on local information. If a module decides to move it moves randomly. The results show that the algorithm is able to reconfigure small groups of Fracta modules, however, deadlocks did occur in some trails. III. ATRON The robot platform used for this work is the ATRON robot which is a lattice-based modular self-reconfigurable robot system consisting of homogeneous modules. The basic module design is based on two hemispheres connected by a rotational joint. Further, each hemisphere is equipped with four connectors, two male and two female; the male/female connector design is chosen for mechanical reasons. The modules are equipped with infra red transmitters and receivers used for both local communication and sensing obstacles in close proximity of the module. Figure 1 shows how a module Fig. 2. Left: Picture of a single meta-module. The meta-module consists of four ATRON modules placed in a square. Right: Screen shot from the simulator showing the square grid structure of the meta-modules. Fig. 3. The three basic actions of a meta-module. There is no constraints upon the presence or absence of meta-modules in the light gray squares. It should be noted that in the move illustrated in the rightmost figure the two non-moving meta-modules are connected by the moving module throughout the entire move. can move another module by connecting to it and rotating. The design is such that one module is capable of lifting two other modules. A complete description of the module design can be found in [11]. A. ATRON Simulator The work presented in this paper involves the control of more ATRON modules than the 100 we currently have built. To explore scalability and speed up the time used for experiments a simulator for the ATRON system is used. The simulator is an efficient transition based simulator, which is able to simulate self-reconfiguration with more than 5000 ATRON modules. The different timing issues are simulated by running the simulator in time steps, and at every time step the controllers of the modules are simulated one at a time in random order, described as the D1 activation scheme in [17]. The simulator contains routines for collision testing which handles both module/module collisions and module/environment collisions. The simulator contains no simulation of real-world physics such as friction, stability etc. IV. META-MODULE In this paper we propose a new meta-module for the ATRON robot. The purpose of the meta-module is to reduce the complexity of self-reconfiguration control for the ATRON robot by reducing the complexity of the motion constraints of the modules. The proposed meta-module consists of four ATRON modules connected in a square configuration. A picture of a single meta-module can be seen in Figure 2 along with an illustration of the orthogonal 4-connected 2D grid formed by the meta-modules. One of the key features of the proposed meta-module is its high degree of symmetry and simple motion constraints which reduces the complexity of the self-reconfiguration

3 Number of meta-modules Successrate TABLE I THE SUCCESSRATE FOR RANDOM CONFIGURATION TO LINE PLANNING PROBLEMS FOR THE SIMPLE PLANNER WHEN THE DIAGONAL MOVE DEPICTED IN THE RIGHTMOST PICTURE IN FIGURE 3 IS DISALLOWED. Fig. 4. Pictures of how a configuration of 12 ATRON modules can perform a meta-action. The acting meta-module consists of the four white modules while the blue modules are passive structure modules. The time needed to perform the shown meta-action is approximately 1 minute. problem. Due to the high degree of symmetry only three different actions are defined for a meta-module. The three actions are illustrated in Figure 3. The execution of a single meta-module action requires the execution of single module actions i.e. connect/disconnect/rotate some of which can be performed simultaneously. All movements of metamodules are based on rotation and mirror symmetries of these three basic actions. If we compare the proposed meta-module with other selfreconfigurable robots we realize that the motion capabilities of the meta-module is very similar to that of the square meta-morphic robot module described by Pamecha et al. in [3] and a 2D version of the simulated cube style modules considered by Butler et al. in [17] and Støy in [18]. There is, however, one difference. The capability of moving diagonally in between two neighbouring modules, as depicted in the rightmost picture of Figure 3, is unique to the proposed metamodule. Figure 4 shows four pictures of how a small group of ATRON modules can perform the meta-action shown in the middle picture of Figure 3. V. CENTRALIZED CONTROL In order to evaluate the motion constraints of the metamodules a very simple planner was utilized. It is important to note that the planner is deliberately kept very simple such that one would expect it to get stuck in local minima easily. The reason for this choice is that the focus of this paper is not the planner but more importantly the capabilities of the meta-modules. The planner is greedy, in each time step it evaluates all possible meta-module movements in random order, the first legal move which decreases the distance from the current configuration to the goal configuration is performed. This is repeated until the goal is reached. If at some point all possible moves are examined without an improving move being found then the planner is unable to find a solution and terminates. It is clear that this simple planner is incapable of escaping from a local minima, since an increase in distance is never accepted. An important part of the planning is the metric used for finding the distance from the current configuration to the goal. The motion capabilities of the meta-modules resembles that of the square meta-morphic module, so we have adopted the optimal assignment metric presented by Pamecha et al. in [19]. When measuring the distance between two configurations the metric assigns each module in one configuration to a module in the other and calculates the sum of module to module distances as the result. The assignment is performed such that the sum of module to module distances is minimal, which can be computed using the Hungarian method. The module to module distances are calculated as the maximum of the two axis aligned distances. dist(a, b) = max( b x a x, b y a y ) (1) The motion capabilities of the meta-modules implies that performing one meta-module move in a given configuration can only change the distance from the configuration to another configuration by 1, 0, or -1. This, in turn, implies that the metric also serves as a lower bound on the number of meta-module moves needed to reconfigure [19]. If we consider the greedy planner, where each action must decrease the distance to be accepted, and the used metric, which is also a lower bound on the number of moves required for the reconfiguration, together with the fact that each module move can only change the distance by one, it is clear that if a solution is found by the planner then it must be the optimal solution. The asymptotic running time of the planner can be found by a few simple considerations. In worst-case the planner must consider 8 moves for each of the n meta-modules in each time step. For each move the new distance must be calculated using the optimal assignment metric which run in O(n 3 ) due to the use of the Hungarian method. The final consideration is that in each time step the distance to the goal configuration must decrease by one so the maximum number of time steps to consider is equal to the initial distance d. These considerations yields that the total asymptotic running time is O(dn 4 ). A. Results To get a more quantitative measure of the existence of local minima we analyzed a simple case using a probably

4 approximately correct, PAC, approach. First, we will give a brief introduction to the PAC method. The method provides a bound on the number of correct experiments necessary to bound the size of the error region with some confidence. The relation between the number of experiments n, the size of the error region ε and the confidence δ is given by: n > 1 ε ln ( 1 δ In order to utilize the PAC method we conducted a series of experiments. In each experiment the simple greedy planner was used for planning the reconfiguration from a random initial configuration to a line configuration. Having a line configuration as a goal eliminates the possibility of local minima due to the shape of the goal configuration, so any local minima would be due to the motion constraints of the meta-modules or the planner itself. Four series of experiments were conducted with 10, 25, 50 and 100 meta-modules respectively. In each series 1000 planning trials were conducted. The planner successfully solved all of the 4000 planning problems. Applying equation 2 yields that with 99% confidence the region of planning problems, from a random configuration to a line, that the planner would fail to solve is no more than 0.46% of the total number of such planning problems. A unique feature of the proposed meta-module is that it is capable of moving diagonally between two other metamodules while keeping them connected throughout the move. This is illustrated in the right picture of Figure 3. In order to investigate the importance of this special move we conducted a series of similar experiments, with the diagonal move disabled, while we measured the rate of success of the planner. We conducted experiments with 5, 10, 15 and 20 meta-modules and each series consisted of 1000 trails. The results of the experiment is summarized in Table I. The results show that new diagonal move greatly improves the performance of the planner, with the move disallowed the planner does no longer solve 100% of the given problems, in fact the successrate drops to 0% for problems with 20 or more meta-modules. ) VI. DISTRIBUTED CONTROL A simple distributed control algorithm for performing locomotion for the meta-modules is proposed. The algorithm is completely distributed and relies only on local communication. In order to describe the algorithm we first define the notion of a legal direction for a given meta-module. A direction is legal for a meta-module if it can move in that direction without moving to a position that is already occupied by another meta-module or an obstacle, and without disconnecting the group of modules. Based on the notion of legal directions it is simple to describe the proposed algorithm. A meta-module moves in the legal direction which reduces its distance to the global goal the most. If no legal directions exist or none of them reduces the distance to the goal it does not move. (2) (a) (b) (c) Fig. 5. Three patterns used for determining if a given direction of motion is legal. (a) In this pattern both directions are legal since no disconnection is possible. (b) Here all directions are illegal since it is not possible to be sure that the structure will not disconnect from moving. (c) Here it is legal to move to the upper left whereas the other two directions are illegal. Note that the move is only legal since that particular action maintains connectivity throughout the move. One of the crucial points of the algorithm is how the function for testing if directions are legal is to be implemented. The function consists of two parts, one for testing if positions are occupied and one for maintaining connectedness. Testing if a neighbouring position is occupied is done simply by using the proximity sensors of the ATRON modules and through communication with the neighbours to see if they are present. Testing if performing a given move would break the connectivity of the entire group is not trivial when based only on local information. Hence a conservative approximation approach is used. The approach is inspired by the classification of local connectivity on the Fracta robot presented in [2]. The idea is that if the local connectedness is preserved then so is the global connectedness. This leads to a set of 32 configuration patterns of the meta-module neighbours for which the local connectedness preserving moves are easily identified. Each time a meta-module wants to move, it identifies how the local configuration of meta-modules is by communicating with its neighbours. Then the correct pattern is found in the list of 32 patterns which then tells which directions are legal. Three examples of local patterns is shown in Figure 5. The final function to implement is the improvement function which defines which of the legal moves that should be performed if any. In this work the improvement function is very simple, in order to perform locomotion in a given direction a distant point in that direction is set as the goal point and the improvement function simply returns the decrease in distance to the goal point that would be the result of moving in the given direction. This assumes that all metamodules know their initial position and keep track of their current position as they move. A. Results In order to investigate the properties of the distributed locomotion algorithm we conducted a series of experiments where we measured the speed of motion of the center of mass for groups of ATRON meta-modules. In each experiment the meta-modules started in a random configuration and the goal

5 Fig. 6. The measured speed of the center of mass for clusters of metamodules with sizes from meta-modules. It is seen that the speed of the large clusters containing meta-modules increase in the beginning of the experiments. oblong until it has a snake like shape. The result of this change of shape is that the ratio of moveable meta-modules increase as the ratio of the number of meta-modules in the periphery of the configuration compared to the total number of modules increase. Two pictures from one of the experiments with 100 meta-modules can be seen in Figure 7. The results of the previous experiment lead us to investigate how the initial speed of the clusters depended upon the number of meta-modules in the clusters. Since the speed is proportional to the ratio of moving meta-modules relative to the total number of meta-modules we used the following approximation for the ratio of moving modules. We observed that the initial configurations were quite compact and could roughly be approximated by a circle. The modules able to move in such a configuration is placed in the periphery of the circle. The ratio between the length of the periphery P and the area A of a circle is given by: P A = 2πr πr 2 = 2 π r π = 2 π (3) A In order to verify our assumptions this can be rewritten: V init = c P A = c A c = V init A (4) Fig. 7. Two configurations of 100 meta-modules from the experiments. Left: The initial configuration from one of the experiments. Right: The configuration after time steps. Note that the configuration now is stretched such that many more meta-modules are capable of moving simultaneously. was chosen to be a distant point. Since the meta-modules are arranged in a square lattice the efficiency of the locomotion might depend on the direction of the locomotion relative to the axes of the lattice. For this reason we conducted experiments with locomotion along one of the axes of the lattice, at an angle of 22.5 relative to the lattice axes and at an angle of 45. Further, we conducted the experiments with groups of meta-modules of varying size from 2 metamodules to 200 meta-modules. The experiments showed a number of interesting properties of the proposed locomotion algorithm. As one would expect small clusters were faster than large clusters, however, the largest clusters with meta-modules increased their speed during the beginning of the experiments. The speed of the large clusters converged towards the same speed, which was slightly lower than that of the 10 metamodule cluster. In Figure 6 the speed of the center of mass of the clusters is shown. The explanation for this increase in speed is found by looking at the configuration during the experiments. Initially the configuration is random, however, the randomly generated configurations are usually quite compact with only a few holes. During the beginning of the experiments the configuration becomes more and more Result from evaluating this expression for the entire range of the experiments from meta-modules varied less that a factor of 2 leading us to believed that the assumptions are correct. We also conducted experiments for investigating how the speed of the cluster depended on the direction of the movement relative to the principal axes of the lattice. Again, we conducted experiments with cluster from metamodules moving at angles of 0, 22.5, and 45 relative to the lattice axes. It was found that the speed of motion was fastest at an angle of 45 relative to the lattice axes. The speed was approximately 6.3% lower at an angle of 22.5 and 18.8% lower at 0. The reason for the slightly faster motion in diagonal directions is that the diagonal moves move a meta-module a distance of 2 lattice units, as opposed to the 1 lattice unit for the non-diagonal move, while only taking around 20 25% longer to execute. VII. DISCUSSION In the previous sections we investigated the properties of the proposed meta-module in a number of experiments in both a centralized reconfiguration planning setup as well as for distributed cluster flow locomotion. In both setups the meta-module showed great potential to simplify the control of 2 dimensional surfaces of ATRON modules. In the centralized planning experiments a very simple planner, which was unable to escape from local minima, was used. It was shown that for reconfiguring from any configuration to a line configuration it is very unlikely to encounter local minima. In fact, optimal solutions were found in all of 4000 randomly generated test cases.

6 Using the planner to plan between arbitrary configurations would be quite straightforward by first planning from the initial configuration to a line and, secondly, planning from the goal configuration to the same line and then simply reversing the sequence found in the second part. This is possible since all meta-module moves are completely reversible. Of course, the sequence of moves found by this method would not necessarily be optimal since the intermediate line configuration might not be part of an optimal solution. One of the very useful findings of the planning experiments was that the existence of a special diagonal move was very important to the effectiveness of the planner. If the move was disallowed the planner could only solve small problems and it often got stuck in local minima. The unique feature of this special diagonal move is that the moving meta-module can maintain its connection to two other meta-modules while moving. The reason for this moves great impact on the planner performance is probably that it makes it possible for meta-modules, placed inside chains of meta-modules, to move, which is otherwise not possible. The usefulness of this special move also suggests that adding a similar move to the sliding cube style modules used by Butler [17] and Støy [18] would make them easier to control. In the second part of the experiments we demonstrated how locomotion of a cluster of meta-modules could be achieved by a simple distributed control algorithm where all meta-modules would simply try to minimize their distance to a distant goal in a greedy manner. Again the simple motion constraints of the meta-modules made this approach feasible since local minima did not occur. As a side effect of the locomotion algorithm the efficiency of locomotion for large clusters increased during the beginning of the experiments. This effect was due to the fact that the meta-modules self-organized into a more efficient shape, where a larger percentage of the modules were able to move simultaneously, than the initial random configuration. VIII. CONCLUSIONS AND FUTURE WORK In this paper we have proposed a new 2D four module meta-module for the ATRON robot. We have shown that the motion constraints of the meta-module are very simple such that we can achieve efficient control of the modules using simple algorithms. This has been demonstrated in two ways. First, we showed that a simple greedy planner, with asymptotically polynomial running time, could solve reconfiguration planning problems of interesting sizes optimally. Further, we demonstrated how a simple distributed algorithm could achieve efficient cluster flow locomotion. The cluster flow algorithm had a side effect that caused the metamodules to self-organize into a shape that gradually resulted in a more efficient locomotion. Finally, we have shown that the addition of a single action to the sliding cube style type of modules improves the motion capabilities greatly making them much easier to control. Future work could include investigation of more elaborate distributed algorithms to achieve locomotion in environments containing obstacles. Also a possible generalization to the three dimensional case holds unaddressed problems. REFERENCES [1] T. Fukuda and S. Nakagawa, Approach to the dynamically reconfigurable robotic system, Journal of Intelligent and Robotic Systems, vol. 1, no. 1, pp , March [2] S. Murata, H. Kurokawa, and S. Kokaji, Self-configurable machine, Transactions of the Society of Instrument and Control Engineers, vol. E-1, no. 1, pp , 2001, reprinted/translated from Trans. SICE, Vol. 31, No. 2, , [3] A. Pamecha, C.-J. Chiang, D. Stein, and G. Chirikjian, Design and implementation of metamorphic robots, in Proceedings of the 1996 ASME Design Engineering Technical Conference and Computers in Engineering Conference, Irvine, California, August [4] K. D. Kotay and D. L. Rus, Algorithms for self-reconfiguring molecule motion planning, in Proceedings of IEEE International Conference on Intelligent Robots and Systems (IROS2000). IEEE Press, [5] E. Yoshida, S. Murata, H. Kurokawa, K. Tomita, and S. Kokaji, A distributed method for reconfiguration of a three-dimensional homogeneous structure, Advanced Robotics, vol. 13, no. 4, pp , [6] P. Will, A. Castano, and W.-M. Shen, Robot modularity for selfreconfiguration, in Proceedings of SPIE Sensor Fusion and Decentralized Control II, Boston, MA, USA, 1999, pp [7] K. C. Prevas, C. Ünsal, M. O. Efe, and P. K. Khosla, A hierarchical motion planning strategy for a uniform self-reconfigurable modular robotic system, in Proceedings of IEEE International Conference on Robotics and Automation (ICRA2002). IEEE Press, 2002, pp [8] S. Murata, K. Tomita, E. Yoshida, H. Kurokawa, and S. Kokaji, Selfreconfigurable robot - module design and simulation, in Proceedings of 6th International Conference on Intelligent Autonomous Systems (IAS-6), 1999, pp [9] M. Yim, D. G. Duff, and K. D. Roufas, Polybot: a modular reconfigurable robot, in Proceedings of the 2000 IEEE International Conference on Robotics and Automation (ICRA 00), San Francisco, CA, USA, April 2000, pp [10] D. Rus and M. Vona, Crystalline robots: Self-reconfiguration with compressible unit modules, Autonomous Robots, vol. 10, no. 1, pp , January [11] E. H. Østergaard, K. Kassow, R. Beck, and H. H. Lund, Design of the ATRON lattice-based self-reconfigurable robot, Autonomous Robots, vol. 21, no. 2, pp , September [12] S. C. Goldstein and T. Mowry, Claytronics: A scalable basis for future robots, in Proceedings of Robosphere 2004, Moffett Field, California, USA, November [13] W.-M. Shen, M. Krivokon, M. Rubenstein, C. H. Chiu, J. Everst, and J. B. Venkatesh, Multimode locomotion via self-reconfigurable robots, Autonomous Robots, vol. 20, no. 2, pp , April [14] D. Brandt, Comparison of a* and rrt-connect motion planning techniques for self-reconfiguration planning, in Proceedings of the 2006 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 06), Beijing, China, October 2006, pp [15] E. H. Østergaard and H. H. Lund, Distributed cluster walk for the ATRON self-reconfigurable robot, in Proceedings of the The 8th Conference on Intelligent Autonomous Systems (IAS-8), Amsterdam, Holland, March 2004, pp [16] D. J. Christensen, Evolution of shape-changing and self-repairing control for the atron self-reconfigurable robot, in Proceedings of the IEEE Int. Conference on Robotics and Automation (ICRA2006), Orlando, Florida, May 2006, p. to appear. [17] Z. Butler, K. Kotay, D. Rus, and K. Tomita, Generic decentralized control for a class of self-reconfigurable robots, in Proceedings, IEEE International Conference on Robotics and Automation (ICRA 02). Washington, DC, USA: IEEE Press, 2002, pp [18] K. Støy, Using cellular automata and gradients to control selfreconfiguration, Robotics and Autonomous Systems, vol. 54, pp , [19] A. Pamecha, I. Ebert-Uphoff, and G. S. Chirikjian, Useful metrics for modular robot motion planning, IEEE Transactions on Robotics and Automation, vol. 13, no. 4, pp , 1997.