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1 SPECIAL ISSUE OF IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION: MULTI-ROBOT SSTEMS, 00 Distributed reonfiguration of hexagonal metamorphi robots Jennifer E. Walter, Jennifer L. Welh, and Nany M. Amato Abstrat The problem addressed is the distributed reonfiguration of a metamorphi roboti system omposed of any number oftwo dimensional hexagonal robots (modules) from speifi initial to speifi goal onfigurations. The initial onfiguration onsidered is a straight hain of roboti modules, while the goal onfigurations onsidered satisfy a more general admissibility" ondition. A entralized algorithm is desribed for determining whether an arbitrary goal onfiguration is admissible. We prove this algorithm orretly identifies admissible goal onfigurations and finds a reonfiguration surfae, or substrate path" within any admissible goal onfiguration. The main result of the paper is a distributed algorithm for reonfiguring a straight hain into an admissible goal onfiguration. Different heuristis are proposed to improve the performane of the reonfiguration algorithm and simulation results demonstrate the use of these heuristis. Keywords Metamorphi robots, distributed reonfiguration I. INTRODUCTION A topi of reent interest in the field of robotis is the development of motion planning algorithms for roboti systems omposed of a set of robots (modules) that hange their position relative to one another, thereby reshaping the system. A roboti system that hanges its shape due to individual roboti motion has been alled self-reonfigurable [5] or metamorphi []. A self-reonfigurable roboti system is a olletion of independently ontrolled, mobile robots, eah of whih has the ability to onnet, disonnet, and move around adjaent robots. Metamorphi roboti systems, a subset of self-reonfigurable systems, are further limited by requiring eah module to be idential in struture, motion onstraints, and omputing apabilities. Typially, the modules have a regular symmetry so that they an be paked densely, i.e., paked so that gaps between adjaent modules in a onfiguration that densely paks the plane are as small as possible. In these systems, robots ahieve loomotion by moving over a substrate omposed of one or more other robots. The mehanis of loomotion depends on the hardware and an inlude module deformation to rawl over neighboring modules [3], [9] or to expand and ontrat to slide over neighbors [0]. Alternatively, moving robots may be onstrained to rigidly maintain their original shape, requiring them to roll over neighboring robots [6], [3], [4]. Shape hanging in these omposite systems is envisioned Department of Computer Siene, Texas A&M University, College Station, T FA: Walter (ontating author): jennyw@s.tamu.edu, Phone: (979) Welh: welh@s.tamu.edu, Phone: (979) Amato: amato@s.tamu.edu, Phone: (979)86-75 as a means to aomplish various tasks, suh as bridge building, satellite reovery, or tumor exision [9]. The omplete interhangeability of the robots provides a high degree of system fault tolerane. Also, self-reonfiguring roboti systems are potentially useful in environments that are not amenable to diret human observation and ontrol (e.g., interplanetary spae, undersea depths). The motion planning problem for a metamorphi roboti system is to determine a sequene of robot motions required to go from a given initial onfiguration (I) to a desired goal onfiguration (). Many developers of self-reonfigurable roboti systems [5], [6], [7], [9], [0], [], and [3] have devised motion planning strategies speifi to the hardware onstraints of their prototype robots. Most of the existing motion planning strategies rely on entralized algorithms to plan and supervise the motion of the system omponents [], [3], [5], [9], [0], []. Others use distributed approahes whih rely on heuristi approximations and require ommuniation between robots in eah step of the reonfiguration proess [6], [7], [3], [4]. We fous on a system omposed of planar, hexagonal roboti modules as desribed by Chirikjian [3]. We onsider a distributed motion planning strategy, given the assumption of initial global knowledge of. Our distributed approah offers the benefits of loalized deision making and the potential for greater system fault tolerane. Additionally, our strategy requires less ommuniation between modules than other approahes. We have previously applied this approah to the problem of reonfiguring a straight hain to an interseting straight hain []. In this paper we address the problem of distributed reonfiguration from a straighthain of robots to goal onfigurations that satisfy a more general admissibility" ondition. A entralized algorithm is desribed for determining whether an arbitrary goal onfiguration is admissible, and if so, finding a path with ertain properties. The main result of the paper is a distributed algorithm for reonfiguring a straight hain into an admissible goal onfiguration, whih uses the path found by the previous algorithm. Different heuristis for hoosing the path are proposed to improve the performane of the reonfiguration algorithm and the performane of these heuristis is explored through simulation. II. RELATED WORK Chirikjian [3] and Pameha [9] disuss entralized algorithms for planar hexagonal modules that use the distane between all modules in I and the oordinates of eah goal

2 SPECIAL ISSUE OF IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION: MULTI-ROBOT SSTEMS, 00 position to aomplish the reonfiguration of the system. Pameha et al. [9] define the distane between onfigurations as a metri and apply this metri to system selfreonfiguration using a simulated annealing tehnique to drive the proess towards ompletion. Upper and lower bounds on the number of moves for reonfiguration between general shapes are given by Chirikjian [3]. Lower bounds for the general ase are obtained by finding a perfet mathing between modules in I and positions in suh that the sum of the distanes between pairs is minimized. Centralized motion planning strategies for systems of two dimensional roboti modules are examined by Nguyen et al. [8] and analysis is presented for the number of moves neessary for speifi reonfigurations. The authors show that the absene of a single exluded lass of initial onfigurations is suffiient to guarantee the feasibility of motion planning for a system omposed of a single onneted omponent. A entralized motion planning strategy for three dimensional ubi robots is presented by Rus and Vona [0]. In this paper, the proposed modules inorporate an atuator mehanism that auses module expansion and ontration, resulting in the sliding movement of a module over its neighbors. A entralized algorithm whih takes O(n ) time to reonfigure a system of n modules is presented. Centralized algorithms for deomposing a system of modules into a hierarhy oftwo dimensional substrutures are presented by Casal and im []. Reonfiguration of the system involves onnetivity hanges within and between these substrutures, along with substruture reloation. The paper onentrates on the deomposition algorithms and does not present algorithms for motion planning within substrutures. A distributed approah is taken by Murata et al. to reonfigure a system of two dimensional hexagonal modules [6], and a system of three dimensional ubi modules [7]. In these approahes, eah module senses its own onnetion type and lassifies itself by the number of modules that it physially ontats. Modules use a formula that relates their onnetion type to the set of onnetion types in the goal onfiguration to quantify their fitness to move. Modules ommuniate with physial neighbors to ensure that only the modules that have fitness greater than the loal fitness average move in the same time step, hoosing a diretion at random. These distributed algorithms use random loal motions to onverge toward the goal onfiguration, a slow proess that appears impratial for large onfigurations. These shemes also ignore the onsequenes of module ollision and do not distinguish the relative loation of modules in the plane, i.e., two onfigurations are the same if the modules omposing them have the same onnetions. Another distributed reonfiguration algorithm, for three dimensional rhombi dodeahedral modules, is presented by im et al. [3] In this strategy, eah module uses loal information about its own state (the number and loation of its urrent neighbors) and information about the state of its neighbors obtained through inter-module ommuniation to heuristially hoose moves that lower its distane to the goal onfiguration. Several heuristi approximation algorithms for distributed motion planning of three dimensional rhombi dodeahedral robots are presented by hang et al. [4] In this two phase approah, modules use neighbor-to-neighbor ommuniation in the first phase to ahieve a semi-global view of the initial onfiguration, using as many rounds as neessary to avoid violation of module motion onstraints prior to eah phase of movement. III. Our Approah This paper will examine distributed motion planning strategies for a planar metamorphi roboti system undergoing a reonfiguration from a straight hain to a goal onfiguration satisfying ertain properties. In our algorithms, robots are idential, but at as independent agents, making deisions based on their urrent position and the sensory data obtained from physial ontats with adjaent robots. Our purpose is to seek an understanding of the neessary building bloks for reonfiguration, starting with algorithms in whih no messages need to be passed between partiipating robots during reonfiguration. Reonfiguration in ertain senarios, like the ones presented in this and our earlier paper [], an be aomplished using algorithms that do not require any message passing. Therefore, our algorithms are more ommuniation effiient than the distributed approahes of [6], [3] and [4]. Another ontribution of our work is how our system model abstrats from speifi hardware details about the robots. In this paper, we onsider two dimensional, hexagonal robots like those desribed by Chirikjian [], using the same definition of lattie distane between robots in the plane. Our proposed sheme uses a new lassifiation of robot types based on onneted edges similar to the lassifiation used by Murata et al. [6] for onneted verties. In the algorithms presented in this paper, eah robot independently determines whether it is in a movable state based on the ell it oupies in the plane, the loations of ells in the goal onfiguration, and on whih sides it ontats neighbors. Robots move from ell to ell and modify their states as they hange position. Sine the robots know the oordinates of the goal ells, we show that eah of them an independently hoose a motion plan that avoids module ollision. In this paper we also present preise onditions for admissible goal onfigurations based on the motion onstraints of our robots. We present an algorithm that ensures these admissibility onditions and prove that this algorithm orretly identifies admissible goal onfigurations. The admissibility onditions presented in this paper differ from those presented by Rus and Vona [0] and Nguyen et al. [8]. In the first ase, this differene is due to module shape and motion onstraints, and, in the seond ase, the differene is due to assumptions on module motion, as will be explained in Setion V. In Set. IV we desribe the system assumptions and the problem definition. Setion V ontains a entralized al-

3 SPECIAL ISSUE OF IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION: MULTI-ROBOT SSTEMS, 00 3 gorithm that determines whether or not a given onfiguration is admissible. Setion VI presents and analyzes a distributed algorithm for reonfiguring a straight hain to an admissible goal onfiguration. In Set. VII we present simulation results omparing the performane of our algorithm using different heuristis. Setion VIII provides a disussion of our results and future work. module M has a neighbor S that does not move in the round (alled the substrate) and S is also adjaent to ell C, and () the neighboring ell to M on the other side of C from S, C, is empty. 3. Only one module tries to move into a partiular ell in eah round. A. Coordinate System IV. SSTEM MODEL The plane is partitioned into equal-sized hexagonal ells and labeled using the oordinate system shown in Fig., as in Chirikjian []. g f e M S C C C 3 S e f C M g Fig.. (-3,0) (-,0) (0,) (0,) (,0) (-,) (,0) (0,0) (-,0) (,-) (0,-) (0,-) (3,0) Coordinates in a system of hexagonal ells. iven the oordinates of two ells, =(x ;y ) and = (x ;y ), we define the lattie distane, LD, between them as follows: Let x = x x and y = y y. Then ρ max(j xj; j yj) if x y <0; LD( ; )= j xj+j yj otherwise: The lattie distane desribes the minimum number of ells a module would need to move through to go from ell to ell. B. Assumptions About the Modules Our model provides an abstration of the hardware features and the interfae between the hardware and the appliation layer. - Eah module is idential in omputing apability and runs the same program. - Eah module is a hexagon of the same size as the ells of the plane and always oupies exatly one of the ells. - Eah module knows at all times: ffl its loation (the oordinates of the ell that it urrently oupies), ffl its orientation (whih edge is faing in whih diretion), and whih of its neighboring ells is oupied by another module. Modules move aording to the following rules. ffl. Modules move in lokstep rounds.. In a round, a module M is apable of moving to an adjaent ell, C, iff (see Fig. for an example) ell C is urrently empty, Fig.. Before and after module movement: M is moving, S is substrate, C, C, and C 3 are empty ells. If the algorithm does not ensure that eah moving module has an immobile substrate, as speified in rule, then the results of the round are unpreditable. Likewise, the results of the round are unpreditable if the algorithm does not ensure rule 3. C. Problem Definition We want a distributed algorithm that will ause the modules to move from an initial onfiguration, I, in the plane to a known goal onfiguration,. V. ADMISSIBLE CONFIURATIONS In this setion we define admissible onfigurations and desribe a entralized algorithm that tests whether a given onfiguration is admissible. A. Definition of Admissible Configuration Without loss of generality, assume I is a straight hain oriented north-south, no goal ell is to the west of I, and I and interset in the southernmost module of I and nowhere else. The number of modules in I and the number of ells in is n. Figure 3 gives examples of orientations of I and that satisfy these assumptions in whih n =6. In this figure, ells in I are numbered with solid borders and goal ells are shaded. The assumptions onerning the orientation of I and an be made without loss of generality beause, if I is a straight hain that is not oriented in this way, the algorithms presented in [] for straight hain to straight hain reonfiguration an be used to reorient I in relation to Fig. 3. Example orientations of I and

4 SPECIAL ISSUE OF IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION: MULTI-ROBOT SSTEMS, 00 4 Let ; ;:::; m be the olumns of from west to east suh that eah olumn is oriented north-south and eah is omposed of a ontiguous hain of goal ells. Figure 4 shows how the olumns of are labeled and gives an example of a onfiguration of in whih eah olumn is a ontiguous hain of goal ells. Figure 4 gives an example of a onfiguration of in whih olumns 3 and 5 are omposed of a non-ontiguous hain of goal ells. Note that it is easy to hek that eah olumn of is omposed of a ontiguous hain of goal ells by sanning eah olumn of in a preproessing step. In the remainder of this paper, north and south segments of p may be referred to as vertial segments when speifi diretion of the segment is not important. Segments direted to the east may be referred to as horizontal or easterly segments when speifi diretion is not important. Conditions and 3 of Definition 3 speify where avertial edge may be added to p relative to goal ells in the three olumns to the east. Conditions and 3 say that any vertial segment ofpmust be separated from any vertial segment inthe opposite diretion and to the east by at least 3 olumns. Definition 4: is an admissible goal onfiguration if there exists an admissible substrate path in Fig. 4. Two onfigurations of : eah olumn is omposed of a ontiguous hain of goal ells, and olumns 3 and 5 are omposed of non-ontiguous hains of goal ells. Let p be a ontiguous sequene of distint ells, ; ;:::; k. Then Definition : p is a substrate path if ffl p begins with the ells in I, from north to south, ffl subsequent ells are all in, and ffl the last ell is in the easternmost olumn of ( m ). i i- i i i i- i i i Fig. 5. Labels for north segment ending in i and south segment ending in i (ells that must not be goal ells are shaded). i Intuitively, an admissible substrate path is a hain of goal ells whose surfae allows the movement of modules without ollision or deadlok, provided the hoies of module rotation and delay are appropriate. That is, provided the motion planning algorithm allows for adequate spae between moving modules, there are no pokets or orners on the surfae of the substrate path in whih modules will beome trapped. The admissibility onditions for a substrate path are diretly related to the degree of parallelism desired, i.e., how losely moving modules an be spaed. If moving modules are separated by only a single empty ell, they will beome deadloked in aute angle orners when running our algorithms []. However, aute angle intersetions are very ommonplae in onfigurations of hexagonal robots. Thus, wehose to make our algorithms appliable to a wide range of goal onfigurations by separating moving modules by two empty ells. Our definition of admissibility is therefore based on onfiguration surfaes over whih moving modules with two empty ells between them an move without beoming deadloked. Definition : A segment of p is a ontiguous subsequene of p of length. In a south segment, eah ell is south of the previous and analogously for a north segment. Definition 3: p is an admissible path if. eah ell in p is adjaent to the previous, but not to the west (i.e., onseutive higher numbered ells may not be on the northwest or southwest side of a given ell),. for eah north segment ofpending with i, ells i, i, and i are not goal ells (see Figure 5) and i+, i+, and i+3 do not form any south segments, and 3. for eah south segment ofpending with i, ells i, i, and i are not goal ells (see Figure 5) and i+, i+, and i+3 do not form any north segments Fig. 6. Example admissible and inadmissible (ells in I have solid borders and ells in are shaded). Figure 6 depits an example of an admissible onfiguration of, where the line through I and is an admissible substrate path. Figure 6 depits a onfiguration of that violates admissibility ondition. The substrate path shown is inadmissible, as is every other possible substrate path for this onfiguration. Our definition of admissible lasses of goal onfigurations differs from that presented by Rus and Vona[0] beause

5 SPECIAL ISSUE OF IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION: MULTI-ROBOT SSTEMS, 00 5 the modules used by these authors were ubi, with a different set of motion onstraints and mode of loomotion. Even though our modules are two dimensional and hexagonal, like those of Nguyen et al. [8], our definition of admissible lasses of goal onfigurations is different than theirs beause our assumptions about module motion are different. Nguyen et al. assume that a module moves by rigid rotation around a vertex it shares with another module. Our motion onstraints are similar to those presented by Chirikjian[], where loomotion is aomplished by a ombined rigid body rotation and shape transformation produed by hanging joint angles. B. Algorithms to Detet Admissible Configurations and Find Substrate Paths Condition for determining the admissibility ofan be easily aomplished by sanning in olumns from north to south, northwest to southeast, and northeast to southwest, to determine if there exists an orientation in whih eah i is ontiguous. If there is no orientation in whih eah i is ontiguous, then is not admissible. Our proedure for finding an admissible substrate path in (ondition for the admissibility of ) proeeds by first onstruting a direted graph H as follows: ffl Label the olumns of as desribed in Set. V-A, with the ells in eah i labeled i;, i;,:::, from north to south. Then ell ; is also in I, but no other goal ells are in I. ffl Represent eah goal ell as a node in the graph H. Add an extra node to the graph in the ell diretly north of ell ; and all this node ;0. Initially there is an undireted edge between eah pair of adjaent goal ells. ffl The ells to the north, south, northeast, and southeast of i;j are labeled N i;j,s i;j,ne i;j, and SE i;j, respetively (note that some of these ells might not be goal ells and thus are not represented in the graph). Fig. 7. m Direted graph H formed by algorithm. The first phase direts edges in the undireted graph and marks the nodes that are determined to have an admissible path to a goal ell in the easternmost olumn. The olumns are proessed from east to west. First, every node in olumn m is marked. As shown in Fig. 7, eah olumn west of olumn m onsists of three segments: (A) the north segment of nodes with no goal ells to the east (shaded light gray), (B) the entral segment of nodes that have goal ells to the east (unshaded), and (C) the south segment of nodes that have no goal ells to the east (shaded dark gray). Segment (A) is initially skipped. Eah node in segment (B) is given an outgoing edge to eah ofitsmarked east neighbors, with the exeption of the situation where a NE edge would be direted toward a neighbor with an outgoing S edge or where a SE edge would be direted toward a neighbor with an outgoing N edge. Nodes in segment (C) are proessed north to south. Eah node is marked and given a direted edge to its north neighbor if the north neighbor is marked and if the goal ells in a loal neighborhood satisfy a ertain admissibility" ondition (disussed below). Finally, nodes in segment (A) are proessed south to north. Similarly to segment (C), eah node is marked and given a direted edge to its south neighbor if the south neighbor is marked and if the goal ells in a loal neighborhood satisfy a ertain admissibility" ondition (disussed below). The arrows in Fig. 7 show the edges that are direted and the diretion given to the edges. The ross-hathed ells are those that remain unmarked after the algorithm has been run. The ell on the north and the two ells on the south of olumn do not satisfy the admissibility" ondition, so no edges are direted from these ells in the algorithm. The Diret Edges algorithm (see Figures 8 and 9) direts some of the edges in the graph, as desribed above. The variables used in the pseudoode are as follows: onpath i;j : Boolean variable. Initially, onpath i;j is false for all goal ells in olumns» i» m and true for all nodes in olumn m. At a partiular node i, the status of the onpath i;j variable at the nodes N i;j, S i;j, NE i;j, and SE i;j is onpath N i;j, onpath S i;j, onpath NE i;j, ffl and onpath SE i;j. ffl x: Variable used to save the position of the southernmost ell that has not been heked by the algorithm. ffl d: Diretion to be heked, either N or S. remove: Set ontaining at most goal ell oordinate. Initially, remove = f;g at all nodes. path: List of oordinates of goal ells that are added to the substrate path. The labels used in the IsAdmissible proedure are depited in Figures 0 and. From the isadmissible proedure, we an see that if any edges at a node are direted to the east, then no edges at that node will be direted to the north or south. Also, sine the ells in eah olumn are ontiguous, if an edge at a node is direted to the north, then no edge at that node will be direted to the south and vie versa. In Setion V-C we will show that, after onstruting H, ifonpath ;0 = true, then there exists an admissible substrate path from ;0 to some ell in m beause of the way H is onstruted. In the next setion, we show that if the algorithm fails to find an admissible substrate path with respet to ;0, then does not ontain suh a path. To find an admissible substrate path, we begin at node ;0 and move in any allowable diretion (i.e., over any ffl ffl direted edge to a goal ell for whih onpath is true) until reahing some goal ell in olumn m. If a node has a

6 SPECIAL ISSUE OF IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION: MULTI-ROBOT SSTEMS, 00 6 For eah olumn i := m downto do:. x :=. j := 3. while (j»j i j) 4. while ( i;j in north setion) 5. j++ 6. end while 7. x := j 8. while ((j»j i j) and ( i;j has one adjaent node to the east)) 9. if (( i;j has node to NE) and (onpath NEi;j )) 0. if (no edge is direted S from NE i;j ). onpath i;j := true. diret edge to NE 3. end if 4. end if 5. if (( i;j has node to SE) and (onpath SEi;j )) 6. if (no edge is direted N from SE i;j ) 7. onpath i;j := true 8. diret edge to SE 9. end if 0. end if. j++. end while 3. while (j»j i j) 4. if ((onpath Ni;j ) and (isadmissible(s; i; j))) 5. onpath i;j := true 6. diret edge to N 7. end if 8. j++ 9. end while 30. while (x > 0) 3. if ((onpath Si;x ) and (isadmissible(n; i; x))) 3. onpath i;x := true 33. diret edge to S 34. end if 35. x 36. end while 37. end while Fig. 8. Pseudoode for algorithm Diret Edges. Proedure isadmissible(d; i; j) returns boolean. if (,,oris a goal ell) //Case. return false 3. end if 4. if ((A and C are goal ells) and 9 an edge direted d out of Q)) 5. if (B is not a goal ell) 6. return false //Case 7. else 8. remove i;j = fp g //Case 3 9. end if 0. end if. return true Fig. 9. Pseudoode for Proedure isadmissible. N i,j i,j A P B Q C i,j S i,j Fig. 0. Labels used in IsAdmissible proedure for d = S and d = N. A B P C Q direted edge to only one neighbor for whih onpath is true (either N, S, NE, or SE), then we go in that diretion. The only other possibility is that a node has two neighbors for whih onpath is true, NE and SE. In this ase, a heuristi is used to deide whether to go NE or SE. If the deision to go N or S is taken in olumn i (i<m), then a partiular ell in the graph two olumns to the east may haveonpath set to false (the remove" ell alulated in IsAdmissible, Case 3). The hoie of this edge may mean that ertain later hoies are no longer available. Algorithm Find Path, shown in Figure, is used to find and onstrut an admissible substrate path. Initially, i := and j := 0 and path := h ;0 i. while ((onpath i;j ) and (i <m)). if i;j has an edge direted to east neighbor with onpath = true 3. update i and j to index one suh neighbor (heuristi hoie) 4. append i;j to path 5. else if i;j has an edge direted to a north or south neighbor with onpath = true 6. update i and j to index that neighbor 7. append i;j to path 8. for the goal ell in remove i;j 9. onpath := false 0. end if. end while Fig.. Pseudoode for Algorithm Find Path. C. Analysis of Algorithms Diret Edges and Find Path The running time of the algorithm to find the graph H and to find an admissible substrate path is O(n), sine eah node has a onstant number of (undireted) neighbors. Algorithm Diret Edges is orret if it marks every ell in that is on any admissible substrate path from ell ;0 to olumn m and direts the edges properly. We require that algorithm Find Path returns a path that ends in olumn m if and only if is admissible. To prove orretness, we start with some observations and laims regarding the performane of the Diret Edges algorithm: Observation : If a goal ell is marked at time t by algorithm Diret Edges, then has either one or two neighboring goal ells that were marked before t and an edge is direted from toward at least one neighbor that was marked before t. Observation : If line 0 of Diret Edges returns false for some goal ell in the entral setion, then line 6 will return true (and vie versa). To see why Observation is true, onsider a ell i with a marked neighbor l to the NE (see Figure ). If l has an edge direted S, then i must have a marked neighbor k to the SE. It annot be the ase that k has an edge direted to the N, sine k was marked before l, by the ation of Diret Edges. Therefore, line 6 of Diret Edges will return true and an edge will be direted from i to k. An analogous argument an be made for a node i that has a marked neighbor l to the SE when l has a marked neighbor to the N.

7 SPECIAL ISSUE OF IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION: MULTI-ROBOT SSTEMS, 00 7 i l k l i k preeded only by a N or NE edge. We proeed with the proof of orretness starting with the following theorem. Fig.. Senarios for Observation and Claim. Observation 3: After exeuting Diret Edges, the following is true of the goal ells in eah setion of olumns of H: Central setion: Eah goal ell will either be marked with at least one edge direted to the east or it will not be marked. In this setion, only goal ells with no marked neighbors remain unmarked. North setion: Eah goal ell will either be marked with one edge direted to the south or it will not be marked. () South setion: Eah goal ell will either be marked with one edge direted to the north or it will not be marked. Claim : After exeuting Diret Edges, no aute angle turns an be formed by any direted path from ell ;0 to a ell in olumn m. Proof: The only possible aute angle turns on a direted path from ell ;0 to a ell in olumn m our when. a N (S) edge is followed immediately by a SE (NE) edge, or. a NE (SE) edge is followed immediately by a S (N) edge. Suppose, in ontradition, ase is allowed by algorithm Diret Edges and after the exeution of Diret Edges there exists a ell i that has an outgoing edge to the N, toward ell l, and l has an outgoing edge to the SE, toward ell k (see Figure ). Then ell k must be a marked goal ell and it must be the NE neighbor of ell i. But then i would be in the entral setion of its olumn and would not have an edge direted to the N, a ontradition. An analogous argument an be made for a S edge immediately followed by a NE edge. Case is not possible beause of the ation of Diret Edges, as stated in Observation. Claim refers to a S edge on a direted path followed to the east by anedge. An analogous argument an be made for a N edge followed by a S edge. Claim : If S and N edges our on the same direted path in H after Diret Edges finishes exeution, then eah S edge must be separated from the next N edge on the path by at least easterly edges, a SE and then a NE edge. Proof: From the ation of Diret Edges, we an see that a S edge annot immediately follow a N edge on any direted path in (or vie versa), beause a single goal ell annot be in both the north and south setions of a olumn. By Claim, any S edge an be immediately followed only by a S or SE edge and any N edge an be immediately Theorem : If is admissible, then Find Path returns a path that ends in m. Proof: We begin by showing, in Lemma, that Diret Edges will mark ells on all admissible paths leading to any ell in olumn m. Lemma : For every goal ell, if there is an admissible path from to a ell in m, then algorithm Diret Edges marks. Proof: The proof is by indution on the order in whih ells are sanned by algorithm Diret Edges. Label all goal ells in from ; ;:::; k, where k = n as follows:. Start with the ells in olumn m, labeling them from ; ;:::; q, where q = j m j and q» k, in inreasing order from north to south.. For the ells in olumn j, where» j < m (if m>), ontinue labeling the ells in inreasing order from q+ ; q+ ;:::; k, in the order they are sanned by algorithm Diret Edges. For the basis of the indution, node is in olumn m. The lemma holds vauously for all goal ells in olumn m sine all goal ells in m are initially marked. For the indutive hypothesis, assume the lemma holds for all ells ;:::; i ( < i < k). We will show the lemma also holds for ell i. We assume i is a prefix of an admissible path ending in olumn m. We will show that i must be marked. For the remainder of this proof, refer to Figure 0, where i = i;j and l = N i;j. If i is in olumn m, then the lemma holds vauously, sine all ells in m are marked. Suppose i is not in olumn m. We have the following ases: Case : i is in the south setion of its olumn. Sine we assume that there exists an admissible path from i to olumn m,any suh path must go through i 's north neighbor l, whih was already sanned by algorithm Diret Edges (l <i). So l must be on some admissible path to m and must be marked by the indutive hypothesis. If i is not marked in lines 4 7 of Diret Edges, itmust be that IsAdmissible returned false in line 4, meaning that either ase or of IsAdmissible was violated. It is easy to see that ase of IsAdmissible ensures that no north edge that violates ondition of Definition 3 is direted. We need to show that if Case of IsAdmissible returns false, then there are no paths from i to m that satisfy ondition. Referring to Figure 3, onsider the pattern of segments that must exist at the start of a path from l to m if line 6 of IsAdmissible (Figure 9) returns false. If ase of IsAdmissible is exeuted, then A, Q, and C are goal ells, ells B,,, and are not goal ells, and Q must have an edge direted south. Sine Q has an edge direted south,

8 SPECIAL ISSUE OF IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION: MULTI-ROBOT SSTEMS, 00 8 Fig. 3. l i A P B Segment patterns following l (non-goal ells are shaded). C was marked before Q by Observation. oal ells A and C annot have edges direted to the south or ase of IsAdmissible would not have been passed. Sine A is a goal ell, l must have an edge direted east to A, and A must be marked or l would not have been marked, by Observation 3. oal ell A annot have an edge direted north, or ase of IsAdmissible would not have been passed for that edge beause Q and C are goal ells. SoAmust have an edge direted east to P. Therefore, P must be a marked goal ell and P must have an edge direted east to marked goal ell Q, by Observation 3. Then every path from i to m must violate ondition, sine the edge from Q to C is a south edge and every path must inlude this edge within the third segment onany path from i to m. Thus, if i is not marked, it must be that the north segment formed by i and l violates Definition 3, ondition or for every possible admissible path from i to m. But then i does not form a prefix of an admissible path ending in m, a ontradition. Therefore, IsAdmissible must return true when i is sanned, and i will be marked with an edge direted toward l,by Observation. Case : i is in the north setion of its olumn. The argument is analogous to ase. Case 3: i is in the entral setion of its olumn. Let l (l < i) be an east neighbor of i through whih the admissible path from i to m goes. By the indutive hypothesis, l is marked. Therefore, by Observation 3, so is i. Therefore, if i is the prefix of an admissible path that ends in olumn m, i will be marked by algorithm Diret Edges and, by Observation, edges from i will be direted toward the beginning of any admissible path for whih it is a prefix. We now ontinue with the proof of Theorem to show that if is admissible, then Find Path returns a path that ends in m. In ontradition, suppose the theorem is false and that is admissible but Find Path does not return a path that ends in m. Then at some point in its exeution, Find Path must get stuk", i.e., it must add a ell that has no marked neighbors to path. Sine is admissible, there is at least one admissible path starting with ell ;0 and ending in olumn m. Thus, ell ;0 is marked, by Lemma. Let ; ;:::; j be the ells that are added to path during the exeution of Find Path on, where ell j is in some olumn i, Q C» i<m, and ell j has no marked neighbors. Sine j was added to path,itmust have been marked. So j must have had at least one marked neighbor to the northeast, southeast, north, or south, in olumn i or i+,at the time Find Path started exeution. Thus, at least one neighbor to the northeast, southeast, north or south of ell j that was marked by algorithm Diret Edges must have been unmarked during the exeution of Find Path, prior to the addition of j to path. We an see from the ell oordinates that are added to remove in proedure IsAdmissible that sine j lost a marked neighbor to the north, south, or east, j 's position must be within the two olumns to the east of the olumn of for whih line 9 of Find Path was exeuted. In other words, j must have lost all its marked neighboring goal ells when a north or south segment was added to path in olumn i or i. Lemmas and 3 provide useful information about the result of removing ells from an admissible path in Find Path. These lemmas refer to a north segment but an analogous argument an be made for a south segment. Lemma : For any north segment formed by goal ells i;j+, i;j,ifremove i;j+ 6= ; when i;j is appended to path in line 7 of Find Path, then there are no edges direted north out of ell i;j or out of any ell in olumns i+, i+,or i+3. Proof: The proof is by examination of the onfiguration of goal ells that must exist in the three olumns to the east of i;j when the north segment formed by i;j+, i;j is added to path, ausing the ell in remove i;j+ to be unmarked. If remove i;j+ is not empty, then Case 3 of IsAdmissible was exeuted when the edge was direted from i;j+ to i;j in algorithm Diret Edges. Refer to Figure 4 for an explanation of the labels used in this proof. Fig. 4. i,j i,j+ A P B Q C Senarios for Lemmas and 3 (non-goal ells are shaded). Sine Case 3 of IsAdmissible was exeuted at the time the edge was direted from i;j+ to i;j in algorithm Diret Edges, ells A, B, C, and Q are goal ells,, and are not goal ells, P is the ell unmarked in line 9 of Find Path, and Q has an edge direted south toward goal ell C. Cell i;j is in the entral setion of olumn i, with an edge direted toward A, by Observation. Likewise, ells A and B are in the entral setion of their respetive olumns. Sine Q is in the north setion, C has no marked goal ell to the northeast, in ell K. C must be in the entral setion of its olumn with an edge direted to SE neighbor D K D

9 SPECIAL ISSUE OF IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION: MULTI-ROBOT SSTEMS, 00 9 beause there are no goal ells south of C. Any ell with an edge direted north must be in the south setion of its olumn, whih is, by definition, south of the entral setion. Therefore, olumns i+, i+, and i+3 ontain no ells with edges direted north. Thus, if remove i;j+ 6= ; when i;j is appended to path, then goal ell i;j annot have an edge direted north, and neither an any goal ells in olumns i+, i+,or i+3. N NE j SE S H O T A V P B C Q Lemma 3: For any north segment formed by goal ells i;j+, i;j,ifremove i;j+ 6= ; when i;j is appended to path, then i;j+ will be a prefix of an admissible path from i;j+ to olumn i+4 after line 9 of Find Path is exeuted. Proof: Sine Find Path adds ells to path from west to east, then no olumn east of i+ will have had a ell unmarked at the time i;j is appended to path in line 7 of Find Path. Therefore, Observations,, and 3 must hold for all olumns east of i+ when i;j is appended to path in line 7 of Find Path. If i;j is appended to path, itmust be marked. Referring to labels used in Figure 4, after i;j is appended to path, remove i;j+ = fp g. Then A, B, C, and Q are goal ells and Q has an edge direted south. If Q has an edge direted south, then C must be marked. Therefore, A and B must be marked, by Observation 3. Sine neither K nor are marked goal ells, C must have an edge direted east to marked ell D in olumn i+4. So there must be a path of marked goal ells from ell i;j+ to ell D in olumn i+4 after line 9 of Find Path is exeuted. Thus, the lemma holds. Consider the last exeution of line 9 of Find Path that removes a marked neighbor of j, either to the north (N), northeast (NE), southeast (SE), or south (S) of j (see Figure 5 ), say at time t. Reall that a ell is added to a remove set in proedure IsAdmissible only in Case, when vertial edges are direted. Without loss of generality, assume the removal was aused by the addition of a north segment (an analogous argument an be made for a south segment). Figure 5 shows a labeling on ells when a north segment O, H is added to path at time t, ausing ell j to lose its last marked neighbor. From IsAdmissible (Figure 9), we an see that only ell P an possibly be unmarked when segment O, H is added to path. In the following ase analysis, all possible positions for ell j in relation to ell P are onsidered when proedure IsAdmissible is alled with d =S.Itisshown that, in eah of these ases, either j will have a marked neighbor after P is unmarked at time t, or j annot possibly have been inluded on path after time t, a ontradition. Case : The last neighbor of goal ell j to be unmarked is N. Then j = B, whih would still have marked neighbor C. Case : The last neighbor of goal ell j to be unmarked is NE. Then j = A, whih would still have marked neighbor B. Cases 3 and 4: The last neighbor of goal ell j to be Fig. 5. Labels used in proof of Theorem (in part, possible goal ells are unshaded). unmarked is SE or S. Then j = T or V. But neither A nor H an have edges direted to the north, by Lemma, and therefore there is no path from H to either T or V. Thus, j 6= T and j 6=V. Thus, in all ases, Find Path will not get stuk". Therefore, if ontains an admissible substrate path p, thereby satisfying Definition 3, then Find Path will return a path that ends in olumn m. Theorem : If algorithm Find Path returns a path ending in olumn m, then is admissible. Proof: Let the returned path be p = ; ;:::; l, where l is in m. Note that it is easy to show that p is a substrate path. The key thing remaining to be shown is that p is admissible, meaning that p satisfies the onditions in Definition 3. It is easy to show that p never goes west, by the way H is onstruted. We will show, by indution, that for» i» l, prefix p i = ;:::; i of p satisfies onditions and 3 for being an admissible path. The basis, i =, is true beause p = = ;0, whih an ause no violation of onditions or 3 in Definition 3, sine p onsists of a single ell. For the indutive hypothesis, assume that p i = ;:::; i is admissible. Now, we will show that p i+ = ;:::; i+ is also admissible. Referenes to ells A, B, C, P, Q,,, and in this proof refer to the labels used in the IsAdmissible proedure (Figures 9 and 0). In the remainder of the figures used in this proof, goal ells have solid borders and unoupied ells have dashed borders. The labels on unoupied ells provide orientation in relation to the labels used in the IsAdmissible proedure. We use a ase analysis of the diretion of two onseutive edges in p. Note that if eah possible diretion of a segment (N, S, NE, or SE) an be followed by eah of four others, that there are 6 possible ases. However, the ombinations of a N segment followed by a S segment and vie versa are not possible due to the assumption that p is omposed of unique ells. A N segment annot be followed immediately by a SE segment (and likewise a S segment annot be followed immediately by a NE segment), nor an a NE segment be following immediately by a S segment (and likewise for a SE segment followed by a N segment), by Claim. This leaves 0 possible ases. We onsider only 5 of these beause the rest are vertial inversions of the ases shown.

10 SPECIAL ISSUE OF IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION: MULTI-ROBOT SSTEMS, 00 0 Case : i, i, i+ form part of a N segment (see Figure 6). There is no violation of ondition beause if there are goal ells in,, or, then the edge from i to i+ would not have been direted, by Case of IsAdmissible, so this segment would not be added to path, a ontradition. Sine the N segment i, i did not violate ondition of Definition 3 for any S segment ending to the west, no violation will be aused by the N segment formed by i, i, and i+. then it would satisfy the onditions for an admissible path by the indutive hypothesis and would not ause the N segment formed by i and i+ to violate these onditions either. i i+ i i 3 i i i+ i i+ i i i+ i i i 4 i 3 B i A i P () i+ C i Q i 4 i 3 i i (d) i+ i i i+ i () i i (d) i+ Fig. 6. Cases through 4 in proof of Theorem. Case : i is the end of a N segment and i+ forms part of a NE segment (see Figure 6). Then the N segment i, i would not have violated ase of IsAdmissible beause,, and are not goal ells. Sine the segment ending in i+ is not a S segment, p i+ will satisfy all onditions of Definition 3. Cases 3 and 4: i is the end of a SE segment and i+ forms part of a NE segment (see Figure 6()) or i is the end of a NE segment and i+ forms part of a NE segment (see Figure 6(d)). Sine neither segment in either ase is vertial, p i+ would satisfy Definition 3. Case 5: i is the end of a NE segment and i+ forms part of a N segment. Any violations of ondition of Definition 3 by N segment i, i+ ourring to the east of i+ are averted by the ation of algorithm Diret Edges. By Case of IsAdmissible, if there are goal ells in positions,, or (see Figure 7), then the edge from i to i+ would not be direted, and therefore i+ ould not be added to p, a ontradition. Condition of Definition 3 ould be violated at a later time, when a S segment is added to p to the east of i+, but any violation will our at the time this S segment is added to p, not when segment i, i+ is added. In order for p i+ to ause a violation of Definition 3in relation to ells to the west, there must be a S segment in the segments formed by ells i 4 and i 3 or i 3 and i. Below, eah of these segments is onsidered as a prefix to the segments formed by i, i, and i, i+. Note that there are 3 ases beause we need onsider only S segments followed by a SE segment in the segments and, by Claim. If either segment or were a N segment, Fig. 7. Case 5 in proof of Theorem.. There is a SE segment ending in i and a S segment ending in i (see Figure 7). By ase ofisadmissible, segment i 3, i ould not have been added to p i beause i+ is in ell.. There is a SE segment ending in i, a SE segment ending in i, and a S segment ending in i 3 (see Figure 7()). Notie that there are goal ells in ells A and C. If there is no goal ell in B, then segment i 4, i 3 ould not have been added to p i beause it violates ase of IsAdmissible. If there is a goal ell in B, then ell i (P) would have been unmarked when segment i 4, i 3 was added to p i,by ase of IsAdmissible. Sine ell P is marked by assumption, it must be that there is no goal ell in B, so segment i 4, i 3 ould not have been added to p i. 3. There is a NE segment ending in i, a SE segment ending in i, and a S segment ending in i 3 (see Figure 7(d)). By ase of IsAdmissible, segment i 4, i 3 ould not have been added to p i beause i+ is in ell. These ases show that it is not possible for p i+ to violate ondition of Definition 3 when i+ is added to p i, sine there an be no onfliting S segment to the west in p i. So p i+ must satisfy all onditions of Definition 3. Sine we assume Find Path returns a path ending in m, the lemma implies that p is an admissible substrate path. Theorems and imply that algorithm Find Path will return only an admissible substrate path and will find an admissible substrate path if one exists in. In other words, the algorithms presented in this setion will orretly identify admissible onfigurations of. VI. DISTRIBUTED RECONFIURATION ALORITHM In this setion, we present the distributed reonfiguration algorithm that performs the reonfiguration of I to after an admissible substrate path is found using the algorithms in the previous setion.

11 SPECIAL ISSUE OF IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION: MULTI-ROBOT SSTEMS, 00 A. Algorithm Assumptions. Eah module knows the total number of modules in the system, n, and the goal onfiguration,.. Initially, one module is in eah ell of I. 3. I is a straight hain. 4. is an admissible onfiguration. 5. I and overlap in goal ell ;, as desribed in Set. V- A. To simplify the presentation of our reonfiguration algorithm, we assume the oordinates of are ordered at eah module as follows: ffl The oordinates of ells on the substrate path are stored in a list, in the order in whih the ells our on the direted path from to m, beginning with the ell on the substrate path whih has a direted edge inoming from ell ;. ffl The oordinates of ells in that are north of the substrate path are stored in a list starting with the ell adjaent to and north of the ell on the substrate path in m to m;, followed by the ell adjaent to and north of the ell on the substrate path in m to m ;, and so on, ending with the northwesternmost ell north of the substrate path in. ffl The oordinates of ells in that are south of the substrate path are stored starting with the ell adjaent to and south of the ell on the substrate path in m to m;j, where j = j m j, followed by the ell adjaent to and south of the ell on the substrate path in m to m ;k, where k = j m j, and so on, ending in the southwesternmost ell south of the substrate path in. B. Overview of Algorithm The algorithm works in synhronous rounds. In eah round, eah module alulates whether it is free (f. Fig. 8). In this figure, the modules labeled trapped are unable to move due to hardware onstraints and those labeled free represent modules that are allowed to move in our algorithm, possibly after some initial delay. The modules in the other ategory are restrited from moving by our algorithm, not by hardware onstraints. its position in I to the length of the arrays of oordinates on, north, and south of the substrate path Fig. 9. Correspondene of initial module positions to final goal positions. Let p be the substrate path, starting with the ell that has an edge inoming from ell ;. Modules in positions»jpjfill in the substrate path first. After p is filled, modules alternate rotation diretions, filling the olumns projeting north and south of p from east, m, to west,. Figure 9 has numbered goal ells showing how initial module positions orrespond to final goal positions. As in our previous paper[], modules use speifi patterns of rotation and delay in our algorithm, as listed below. Note that only patterns and 4 are used in the algorithm presented in this setion.. (0,0)-bidiretional: modules alternate diretion with no delay after free.. (,0)-bidiretional: modules alternate diretion with delay of time unit after free for modules in positions > rotating CW and no delay after free for modules rotating CCW. 3. -unidiretional: modules rotate same diretion with delay of after free for modules in positions >. 4. -unidiretional: modules rotate same diretion with delay of after free for modules in positions >. TRAPPED FREE OTHER Fig. 8. Indiates non ontat edge Indiates ontat edge Contat patterns possible in algorithm. Modules in I initially alulate their position in I, diretion of rotation, possible delay and final oordinates in by determining their lattie distane from ell ;. A module alulates the goal ell it will oupy by omparing The reonfiguration proeeds as follows: ffl For modules in positions through jpj: - Modules use -unidiretional pattern in CW diretion. - When a module is in the goal ell that it should oupy in p, it stops in that ell. ffl For modules in positions > jpj: - Modules use (,0)-bidiretional pattern until all ells on one side of p are filled. After this, modules use -unidiretional pattern, with either CW or CCW diretion, depending on whether there are ells remaining to be filled on the north or south side of p. - When a module is in the goal ell it should oupy, it stops. ffl One a module stops in the goal ell it should oupy for a round it never moves out of that goal ell.