Increasing the Efficiency of Distributed Goal-Filling Algorithms for Self-Reconfigurable Hexagonal Metamorphic Robots

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1 Increasing the Efficiency of Distributed Goal-Filling lgorithms for Self-Reconfigurable Hexagonal Metamorphic Robots J. ateau,. Clark, K. McEachern, E. Schutze, and J. Walter Computer Science Department, Vassar College, Poughkeepsie, NY, US bstract The problem addressed is the distributed reconfiguration of a system of two-dimensional, hexagonal mobile robots (modules), from an initial straight chain into an arbitrary shaped, connected goal configuration that satisfies a simple admissibility condition. We present algorithms that improve the efficiency of the deterministic algorithms presented in [] and []. In this paper, we first present a new algorithm for optimally filling a chain of cells that bisect the goal configuration. Then we present reconfiguration algorithms that combine techniques used in the goal-filling algorithm papers cited above combined with the bridging algorithms presented in [] and []. We compare the performance of our new algorithms to existing goal-filling algorithms via simulation using a discrete event simulator. The results of our simulation are presented and discussed. Keywords: Metamorphic robots, distributed reconfiguration, self-reconfiguration. Introduction self-reconfigurable robotic system [] is a collection of independently controlled, mobile robots, each of which has the ability to connect, disconnect, and move around adjacent robots. Metamorphic robotic systems [] are a subset of self-reconfigurable systems. In metamorphic systems, each module is identical in structure, motion constraints, and computing capabilities. The modules have a regular symmetry so that they can be assembled with no gaps between adjacent modules. In metamorphic robotic systems, robots achieve locomotion by moving over a substrate composed of one or more other robots. The mechanics of locomotion depend on the hardware and can include module deformation to crawl over neighboring modules [], [] or to expand and contract to slide over neighbors []. lternatively, moving robots may be constrained to rigidly maintain their original shape, requiring them to roll over neighboring robots [0], [9], [0]. Shape changing in these composite systems is envisioned as a means to accomplish various tasks, such as bridge building, structural support, satellite recovery, tumor excision [], and automated movement of two-dimensional arrays of solar collectors or shields. The complete interchangeability of the robots provides a high degree of system fault tolerance. Self-reconfiguring robotic systems may be potentially useful in environments that are not amenable to direct human observation and control (e.g., interplanetary space, undersea depths) or for tasks that are monotonous for humans. The motion planning problem for a metamorphic robotic system is to determine a sequence of robot motions required to go from a given initial configuration (I) to a desired goal configuration (G). Many existing motion planning strategies rely on centralized algorithms to plan and supervise the motion of the system components [], [], [], [], [8]. Others use totally distributed approaches which rely on heuristic approximations or require communication between robots in each step of the reconfiguration process [], [0], [], [9], [0]. We focus on a system composed of planar, hexagonal robotic modules as described by Chirikjian []. We present a motion planning strategy that assumes knowledge of all initial coordinates of cells in G. We start with a centralized phase in which modules match themselves to a position in G. This is followed by a distributed phase when as many robots as possible move in parallel. Our distributed approach offers the benefits of localized decision making, the potential for greater system fault tolerance, and less communication between modules than other approaches. We have previously applied this approach to the problem of reconfiguring a straight chain to an intersecting straight chain [] and a straight chain to a goal configuration that satisfies a general admissibility condition [], []. We modify the admissibility requirement on G in this paper.. Related work Chirikjian [] and Pamecha [] discuss centralized algorithms for planar hexagonal modules that use the distance between all modules in I and the coordinates of each goal position to accomplish the reconfiguration of the system, moving a single module in each time step. Pamecha et al. [] define the distance between configurations as a metric and apply this metric to system self-reconfiguration using a simulated annealing technique to drive the process towards completion. In [], motion planning time is shown to be in O(n) for n modules when particular motion

2 constraints are used on hexagonal metamorphic robots. They do not address the reconfiguration time as we do in this paper. Centralized motion planning strategies for systems of two dimensional robotic modules are also examined by Nguyen et al. [] and analysis is presented for the number of moves necessary for specific reconfigurations. centralized motion planning strategy for three dimensional cubic robots is presented by Rus and Vona []. set of distributed motion planning algorithms for a system of cubic robots is presented by utler et al. in []. In another paper [], utler et al. present a rule set that can be run by vertical "layers" of cubic modules and a distributed control algorithm for locomotion is described that will work in any system composed of cubic modules. Distributed approaches are taken by Murata, et al. to reconfigure a system of two dimensional hexagonal modules [0], and a system of three dimensional cubic modules []. Yim et al. [9] and Zhang et al. [0] present distributed algorithms to reconfigure three dimensional rhombic docecahedral modules. In [9], Miao et al. present algorithms to reconfigure two dimensional hexagonal modules to envelop obstacles. Unlike the planning algorithms presented in this paper, these algorithms are probabilistic and require message passing between neighboring modules.. Our approach This paper examines distributed motion planning strategies for a planar metamorphic robotic system undergoing a reconfiguration from a straight chain to a goal configuration satisfying certain simple properties. In our algorithms, robots are identical but act as independent agents making decisions based on their current position and the sensory data obtained from physical contacts with adjacent robots. We have shown that collision-free reconfiguration in certain scenarios, like those presented in our earlier papers [], [], [], can be accomplished using algorithms that do not require any message passing. Our long term goal is to seek an understanding of the necessary building blocks for reconfiguration, starting with algorithms in which no algorithm messages need to be passed between participating robots during reconfiguration. Therefore, our algorithms are more communication efficient than the distributed approaches of [], [0], [9] and [0]. Our proposed scheme uses a classification of robot types based on connected edges similar to the classification used by Murata et al. [0] for connected vertices. In the algorithms presented in this paper, each robot independently determines whether it is in a movable state based on the cell it occupies in the plane, the locations of cells in the goal configuration, and which of its sides are adjacent to occupied cells. Robots move from cell to cell and modify their states as they change position. Since the robots know the coordinates of the goal cells, we show that each of them can independently choose a motion plan that avoids module collision. One of the contributions of this paper is the presentation of algorithms that allow modules to move with only a single space between them while also ensuring that moving modules do not come into contact in acute angle corners. To accomplish this, we use ideas developed in our earlier work on bridging algorithms [], []. In these algorithms, certain modules temporarily halt during reconfiguration, forming bridges for other modules to cross. fter all modules have passed over a bridge module, the bridge module resumes motion. We use a similar technique to avoid module collision in this paper.. System Model The plane is partitioned into equal-sized hexagonal cells and labeled using the same coordinate system as described by Chirikjian []. Our model provides an abstraction of the hardware features and the interface between the hardware and the application layer.. ssumptions about modules M e S g f (a) C C Fig. : efore (a) and after (b) module movement: M is moving module, S is substrate, and shaded cells are unoccupied. - Each module is identical in computing capability and runs the same program. - Each module is a hexagon of the same size as the cells of the plane and always occupies exactly one of the cells. - Each module knows at all times: its location (the coordinates of the cell that it currently occupies), the location of the cells in G, its orientation (which edge is facing in which direction), and which of its neighboring cells is occupied by another module. Modules move according to the following rules: ) Modules move in lockstep rounds. ) In a round, a module M is capable of moving to an adjacent cell, C, iff (see Fig. for an example) (a) cell C is currently empty, C S e f (b) C M g

3 (b) module M has a neighbor S (called the substrate) that is also adjacent to cell C and does not move in the round, and (c) the neighboring cell to M on the other side of C from S, C, is empty. ) Only one module tries to move into a particular cell in each round. Note that the modules may be deformable, in which case each module moves by changing joint angles to crawl over an unmoving substrate. lternately, the modules may be rigid, using sliding movements as specified in [9] to move over the substrate. If the algorithm does not ensure that each moving module has an immobile substrate, as specified in rule (b), then the results of the round are unpredictable and can lead to deadlock. Likewise, collision may result if the algorithm does not ensure rule.. Centralized Pre-processing Phase Our objective is to design a deterministic distributed algorithm that will cause the modules to follow a collisionfree plan from an initial straight chain configuration, I, to an admissible goal configuration, G. This algorithm should ensure that modules do not collide with each other, and the reconfiguration should be accomplished in the most efficient way possible (in terms of time of reconfiguration). We assume G is oriented such that cells have flat surfaces facing north (N) and south (S). This way we can unambiguously describe the eastmost, westmost, and inner columns of G. Definition : n admissible goal configuration is connected and has no vertical gaps within columns. For simplicity, we require that the straight chain I consists of n modules and that it initially intersects G in one cell. The module in the cell that overlaps G does not move during reconfiguration. In a centralized pre-processing phase, modules determine their positions in straight chain I based on their distance from the cell in I that overlaps G. Module numbering proceeds from (at the greatest distance from the overlapping cell) to n (the cell adjacent to the overlapping cell). t the start of the reconfiguration, there are n empty cells in G. Our previous strategy, presented in [], to fill G while avoiding collision, was to find a contiguous path (what we call the substrate path, or SP) of goal cells that most evenly bisects G. fter the SP cells are filled, the cells to the N and S of this path are filled in parallel, with minimal intermodule spacing. Filling the SP first guarantees that no module on the N will collide with a module on the S, allowing both segments to be filled in parallel. We designed our algorithms to further ensure that no pair of moving modules becomes adjacent throughout the reconfiguration, and for that reason, our earlier algorithms required two unoccupied spaces between each pair of modules traversing the same surface. This spacing was also applied to the modules filling the SP. Figure shows the result of -cell versus -cell separation when modules move through an acute angle. (a) (b) (c) (d) Substrate cell (e) Moving module Fig. : Modules rotating clockwise over a substrate surface. Parts (a)-(c) show the outcome for modules with -cell separation. Parts (d) and (e) show how moving modules come in contact in acute angle corners if there is only -cell separation. In part (e), the modules are in a configuration in which the choice of substrate could jeopardize the reconfiguration if a moving module chooses another moving module as a substrate (we call this a deadlock situation). In this paper, we place a restriction on the SP so that modules can move with a single unoccupied cell separation in both the clockwise (CW) and counter clockwise (CCW) directions. Definition : n admissible SP is contiguous and has no vertical sections. Our procedure for finding a SP in an admissible goal configuration is given below:. Find the midpoint cell in each column of the goal configuration and assign that cell to be a SP cell. If a column length is even, choose the path cell north or south (based on the distribution of cells in the next column) of where the midpoint would be if the column length was odd.. Check if SP found in step is contiguous. If not, use algorithms presented in [] that test every possible path in G from left to right. If any SP with no vertical segments is found, continue with reconfiguration algorithms given in this paper.. If the only SP found contains vertical segments, use reconfiguration algorithms described in [] to complete reconfiguration. These algorithms use a more conservative cell inter-module spacing to avoid collision and deadlock. Since we are assuming that G is oriented in columns with cell sides normal to N and S, the slope of I (i.e., the direction from module to the cell in I that overlaps

4 G) must be either NE or SE. If the SP has no vertical segments, there is an algorithm to fill the path using single cell spacing between each pair of moving modules. The FILLSUSTRTEPTH algorithm is given in Fig.. With one exception, to be discussed in Sect., all modules in I begin moving in the round they become free, based on the cell in G to which they have been mapped. free module is one that has one of the FREE contact patterns shown in Fig.. It can be seen from this figure that a FREE contact pattern is one in which a module i has at least sides that are not occupied by another module and in which module i s movement (in i s local view) will not cause the system to become disconnected. TRPPED FREE OTHER Indicates non contact edge Indicates contact edge Fig. : Contact patterns possible in algorithm. From Fig. (a) we can see that initially, the only free module in I is in position, furthest from G. Modules become free in order of increasing position number, with module i becoming free after the second rotation of module i. Modules running the FILLSUSTRTEPTH algorithm in positions i and i + (where i and i + < length of SP) will be in the same N-S column as they move toward G. When modules move with this orientation and spacing, they are guaranteed not to collide in a contiguous SP that contains no vertical sections and that spans all columns of G. To see why, consider that exactly one of the N or S modules, x, in a moving pair of modules will always be closer, in terms of grid distance, to a particular SP cell because of the restriction on either NE or SE SP slope, and the closer modules x will therefore reach the path first. Variables used in algorithm FILLSUSTRTEPTH: num: Length of SP, not counting initial overlapping cell. initialslope: Slope of I, SW to NE (called NE in algorithm) or NW to SE (called SE in algorithm). lastslope: Slope of last two cells on SP (NE or SE). s shown in Fig., in most cases the first num modules fill the SP (although not always sequentially, as explained below). However, when num is odd and initialslope lastslope, the module in position num+ reaches the SP before the module in position num. Once the SP cells are calculated, the remainder of the modules are matched to positions on the N and S of the SP. Figure shows the order the modules fill the SP when the length of the path is odd and the initialslope lastslope. lgorithm FILLSUSTRTEPTH(). if (initialslope == NE):. module rotates CW. else: // initialslope == SE. module rotates CCW. if (num is even) or (lastslope == initialslope):. modules...num alternate directions, starting in direction opposite of module with no delay after the round in which they become FREE.. else: // (num is odd) and (lastslope!= initialslope) 8. modules...num+ alternate directions, starting in direction opposite of module with no delay after the round in which they become FREE. Fig. : lgorithm for modules on substrate path. (i) unoccupied goal cell # (a) (c) (e) (g) (j) (o) (p) (q) (r) module in non-goal cell (b) (d) (f) (h) # module in goal cell (k) (l) (m) (n) module initially in goal Fig. : Illustration of module placement when the path length is odd and initialslope lastslope. Configurations are labeled sequentially (a) through (r). Modules and would fill cells to the south of the SP (not shown). To see why this algorithm works, observe in Fig. how pairs of moving modules are in the same vertical column at each step. Every time the SP has a bend (which must be either to the NE or SE), the relative ordering of the module position numbers in the SP may change. So, for example, suppose the odd numbered modules start filling the path, each followed by an even numbered module. fter the path bends, the even modules may begin filling the path before the odd ones. lgorithm FILLSUSTRTEPTH assumes the first num (or num+) modules will be the first to enter cells on

5 the SP. Every obtuse bend followed by a straight chain of two or more modules in the SP changes the sequence of module positions on the path from odd/even to even/odd (and vice versa). If num is even, the first num modules will enter the path, although not necessarily in position order. If num is odd and the last path direction equals the initial path direction, the final module on the path (an odd numbered module) will reach the path before its even numbered counterpart. The most complicated case is handled by lines 8 of FILLSUSTRTEPTH. In this case, the final bend causes the even numbered module in position num+ to reach the SP before the module in position num. In this case, the module in position num will fill a goal cell on either the N or S side of the SP (see modules and in Fig., which will fill cells to the S).. Distributed Reconfiguration Phase Each module calculates its rotation direction and delay before moving, after it determines its position in I. Modules that will end the reconfiguration on the SP do not have to choose final goal destinations because the first module to enter a SP goal cell stops in that cell. However, the algorithm matches the remainder of the modules to goal cells on the N and S of the SP, alternating between CW and CCW rotation. Modules fill the N and S columns from right to left and from the SP northward and southward, like the algorithms presented in []. These modules choose their final destination from a list of goal cells that is generated by and is the same at every module. Single-cell spacing between moving modules is the most efficient movement pattern since a module can t move into a neighboring occupied cell. However, this spacing may cause deadlock problems when the SP forms an acute angle with a goal column (see Fig. ). To solve this problem, we use a technique like that presented in [], [8] and [] to either permanently or temporarily halt particular modules in areas where collision or deadlock may occur. Each module runs a mapping algorithm in which certain goal cells are marked as WIT-INDICTOR (WI) or STOP- INDICTOR () cells. Each WI cell has an adjacent empty cell marked as a WITCELL () and each cell has an adjacent goal cell marked as a STOPCELL(). Modules mapped to cells choose the cell as their final destination. module mapped to a cell temporarily halts in the cell when the adjacent WI cell is occupied (this only occurs once per WI per reconfiguration). The modules that temporarily stop in cells are matched to the extreme N or S cell in the column directly to their right. These modules temporarily stop until all cells but the extreme N or S cell in the column to the right are filled, when they resume movement. To prevent trapping modules in cells, particular goal cells are marked as DELYSET (DS). ny module mapped to a DS goal cell will delay steps before it begins moving out of the initial chain after the first round in which it becomes FREE. Note that and WI markers are only needed when modules will collide in acute angle corners, which cannot happen if the column immediately to the right is short enough (see, for example, the th goal column from left side of goal configuration in Fig. ). - stop indicator WI - wait indicator - stop cell - wait cell DS - delay set WI DS WI DS WI Initial Line cell Goal cell Fig. : Placement of markers in each module s map of G to avoid deadlock in acute angle corners. The modules in s or s prevent moving modules from making contact in acute angle corners by filling the corner with a module, either permanently () or temporarily (). Filling this corner cell allows the rest of the modules in the column immediately to the right to move over the corner with no deadlock. elow we describe scenarios in which and cells are used. For the examples given in Figs. and 8, we will concentrate on the columns of G on the north side of the SP (the south side is symmetric). ) The first time a module i enters an in column when there is an occupied adjacent cell in column (as shown in Fig. ), module i permanently stops in the. Column must be non-empty in this case. ) The first time a module i enters a in column when there is an occupied adjacent WI cell in column (as shown in Fig. 8), module i delays further movement until all but one goal cell in column (the N-most cell in this case) is filled. The module in the then starts to move again to fill the N- most cell in column. Column is empty in this case. When there is a non-empty goal column to the left of empty column with an associated cell, the module matched to the DS goal cell waits for steps after it becomes FREE in the initial line before starting to move. Note that a goal cell may be marked as both a WI and DS. fter the proper cells are marked, each module creates a list of the goal cells, in the order they are to be filled.

6 Column: (a) (b) (c) (d) (e) (f) (g) (h) Module in final position of goal Empty goal cell Moving module Fig. : Segment of N side and SP with cells marked as and. Figure (a) shows a segment of the map at each module after cells are marked initially; figures (b) (h) show the consecutive states after modules rotating CW begin filling column from the left in a S to N fashion. t this point, all N columns to the right of column are filled. Once the cell in column is occupied, the next module to enter the cell in column stops and column continues to be filled. Column: Since modules know their position in I before the start of motion planning, once each module has created the list, each chooses the goal destination that corresponds to their position number. Final destinations and take-off delays are chosen prior to the distributed reconfiguration phase and the timing of temporary module halt periods are determined by each module according to its local environment during distributed reconfiguration.. Simulation In this section, we briefly describe the results of running our new and old algorithms on the same configurations, both for filling the SP and for filling the rest of the goal configuration, using a discrete event simulator. The main difference between the algorithms presented in this paper and the original algorithms is the amount of intermodule spacing between moving modules. In order to avoid deadlock in acute angle corners, the original algorithm required there to be at least empty spaces between moving modules. The new algorithm requires only a single space between moving modules. We are still in the process of testing the performance of the new versus the old algorithms, but in every trial the number of rounds for the reconfiguration was lower for the new algorithms. Table shows the number of rounds needed for reconfiguration on square goal configurations of increasing size (for example, see Fig. 9). For each configuration of the same size, identical SPs were used. DS WI WI DS WI DS (a) (b) (c) (d) (e) (f) Table : Comparison of algorithms on square configurations. Number of Modules Number of Rounds Old Number of Rounds New (X) 00 (0X0) 8 00 (0X0) 00 8 (X) 8 DS DS DS DS (g) (h) Module in final position of goal (i) Empty goal cell (j) (k) Moving module (l) Waiting module Fig. 8: Segment of N side and SP with cells marked as WI,, and DS. Fig. (a) shows the initial map at each module; figures (b) (l) show the consecutive states after modules rotating CW begin filling column from the left in a S to N fashion. t the time step shown in part (b), all N columns to the right of column are filled. Once the WI cell in column is occupied, the next module to enter the cell in column waits until column is filled except for the N-most cell. Then the module in the begins moving to the N-most cell in column. Parts (k) (l) show the delay between the module ending in the DS cell and the module in the. Table : Comparison of algorithms on comb configurations. Number of Modules Number of Rounds Old Number of Rounds New stop indicator - stop cell Module initially in goal Initial Line cell Goal cell Fig. 9: Square configuration with modules.

7 When no modules experience a temporary delay, as in the square configurations listed in Table, the new algorithms are more efficient than the older ones. The number of rounds used by the new algorithms remained consistent at just over twice the number of modules in the square configurations as the number of modules increased, making the running time linear in the number of modules. The number of rounds used by the old algorithms grows faster than those used by the new algorithms as the number of modules increases. WI - wait indicator - wait cell - WI and DS Module initially in goal Initial Line cell WI Goal cell Fig. 0: Comb configuration with 9 modules. Larger combs had same length vertical columns but a longer horizontal backbone. When some modules experience a temporary delay and the SP does not bisect the goal, as in the comb configuration shown in Fig. 0, the reconfiguration time of the new algorithms is more than double the number of modules, but is still better than the performance of the old algorithms as the number of modules increases (see Table ). Even though the new algorithms we present in this paper are more efficient than our previous algorithms, they do not work for as many goal configurations as do the older algorithms. ecause they use the conservative two cell inter-module spacing, the older algorithms require fewer constraints on the SP. In particular, the older algorithms work as long as the SP is right-monotone. Moreover, they result in a collision and deadlock free reconfiguration even when the SP contains vertical sections, as long as the vertical sections are separated by at least columns []. We do not feel that this difference is a major problem considering that the goal configuration can be separated into segments that end at the first vertical column in the substrate path and the new algorithms can be applied to each such segment, from right to left.. Conclusions and future work We have presented algorithms for filling an arbitrary goal configuration in a column-wise fashion. These algorithms guarantee that collision and deadlock do not occur when run on an admissible goal configuration. The algorithms were shown via simulation to be more efficient (although not asymptotically so) than our older algorithms. For future work, we are continuing to write algorithms to comprise a complete, deterministic planner for the reconfiguration of hexagonal, metamorphic robots. s part of this complete planner, we are concentrating on deterministic algorithms to reconfigure an arbitrary but admissible shape initial configuration into a straight chain. References [] Z. utler, K. 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