Plane Formation by Synchronous Mobile Robots without Chirality

Plane Formation by Synchronous Mobile Robots without Chirality

Yusaku Tomita Graduate School of Information Science and Electrical Engineering,
Kyushu University, Japan
11email: tomita@tcslab.csce.kyushu-u.ac.jp,
{yamauchi, kijima, mak}@inf.kyushu-u.ac.jp
   Yukiko Yamauchi Graduate School of Information Science and Electrical Engineering,
Kyushu University, Japan
11email: tomita@tcslab.csce.kyushu-u.ac.jp,
{yamauchi, kijima, mak}@inf.kyushu-u.ac.jp
   Shuji Kijima Graduate School of Information Science and Electrical Engineering,
Kyushu University, Japan
11email: tomita@tcslab.csce.kyushu-u.ac.jp,
{yamauchi, kijima, mak}@inf.kyushu-u.ac.jp
   Masafumi Yamashita Graduate School of Information Science and Electrical Engineering,
Kyushu University, Japan
11email: tomita@tcslab.csce.kyushu-u.ac.jp,
{yamauchi, kijima, mak}@inf.kyushu-u.ac.jp
Abstract

We consider a distributed system consisting of autonomous mobile computing entities, called robots, moving in a specified space. The robots are anonymous, oblivious, and have neither any access to the global coordinate system nor any explicit communication medium. Each robot observes the positions of other robots and moves in terms of its local coordinate system. To investigate the self-organization power of robot systems, formation problems in the two dimensional space (2D-space) have been extensively studied. Yamauchi et al. (DISC 2015) introduced robot systems in the three dimensional space (3D-space). While existing results for 3D-space assume that the robots agree on the handedness of their local coordinate systems, we remove the assumption and consider the robots without chirality. One of the most fundamental agreement problems in 3D-space is the plane formation problem that requires the robots to land on a common plane, that is not predefined. It has been shown that the solvability of the plane formation problem by robots with chirality is determined by the rotation symmetry of their initial local coordinate systems because the robots cannot break it. We show that when the robots lack chirality, the combination of rotation symmetry and reflection symmetry determines the solvability of the plane formation problem because a set of symmetric local coordinate systems without chirality is obtained by rotations and reflections. This richer symmetry results in the increase of unsolvable instances compared with robots with chirality and a flaw of existing plane formation algorithm. In this paper, we give a characterization of initial configurations from which the robots without chirality can form a plane and a new plane formation algorithm for solvable instances.

1 Introduction

Distributed coordination of mobile computing entities has been gaining increasing attention from many areas such as robotics, transportation, construction, material engineering, DNA computing, and so on. Though these wide areas of applications require complicated operations, they can be classified into fundamental tasks, for example, gathering, formation, exploration, surveillance, flocking, and partitioning. The underlying goals of these distributed coordination tasks are agreement and self-organization. We focus on a theoretical aspect of one of such mobile computing entity models, called autonomous mobile robots [2, 4, 7, 9, 11, 12, 13, 14, 15, 16, 17]. A mobile robot system consists of a set of robots each of which autonomously moves in a specified space. Each robot is an anonymous (indistinguishable) point, and it executes a common distributed algorithm. Each robot repeats a Look-Compute-Move cycle, where it takes a snapshot of the positions of other robots in a Look phase, computes its next position in the Compute phase, and moves to the next position in the Move phase. A configuration of such a system is the set of positions of the robots observed in the global coordinate system, in other words, a set of points. The robots have neither any access to the global coordinate system nor any explicit communication medium. Each robot observes and moves in terms of its local coordinate system. Though observation is the only way for the robots to cooperate with each other, they have to tolerate inconsistency among observations. A robot is oblivious if in a Compute phase, it does not remember the past observations and the past computations, and can use the observation obtained in the Look phase of the current cycle. Otherwise, a robot is non-oblivious, which means it is equipped with local memory. Existing literature introduces the following three asynchrony models: In the fully-synchronous (FSYNC) model, the robots execute the th Look-Compute-Move cycle at the same time. Thus the robots execute a cycle at each time step . In the semi-synchronous (SSYNC) model, the robots follow discrete time steps, but some robots may skip cycles. In the asynchronous (ASYNC) model, no assumption is made except that the length of each cycle is finite.

The self-organization power of mobile robot systems has been studied for robots in a discrete space (e.g., graphs) [5, 6], in the two-dimensional space (2D-space or plane) [2, 4, 7, 9, 11, 12, 13, 15, 11, 13, 15], and in the three-dimensional space (3D-space) [14, 16, 17]. The formation problem requires the robots to form a specified pattern from a given initial configuration. The set of formable patterns indicates the self-organization power of a robot system. Depending on the specified pattern, the formation problem is classified into the following problems; the point formation problem, which is the simplest form of the agreement problem among the robots [2, 8], the circle formation problem [7, 12], and the pattern formation problem for arbitrary target pattern [9, 11, 13, 15]. Since real systems work in 3D-space and applications such as drones become widely available, robot systems in 3D-space form an important and promising field. Yamauchi et al. proposed the plane formation problem that requires the robots to land on a common plane without making any multiplicity.111As the plane formation problem does not allow multiplicity, point formation is not a solution. The plane formation problem is one of the simplest agreement problems in 3D-space and it bridges between the robots in 3D-space and the robots in 2D-space, so that existing techniques in 2D-space can be used in 3D-space.

In this paper, we consider the plane formation problem by mobile robots that lack chirality. A robot system does not have chirality when the robots may not agree on the handedness (right-handed or left-handed) of their local coordinate systems. On the other hand, a robot system has chirality when the handedness of all local coordinate systems are identical. Lack of chirality introduces heterogeneity among the robots, and the model is expected to reveal the self-organization power of the weakest robot model. For example, Flocchini et al. and Mamino et al. showed that more than four oblivious ASYNC robots can form a circle without chirality [7, 12].

Figure 1: Symmetric initial positions and local coordinate systems.

Existing studies show that the set of formable patterns in 2D-space is determined by the initial symmetry among the robots. Consider an initial configuration of the four robots in 2D-space, where they form a square and their local coordinate systems are symmetric regarding the center of the square (Fig. 1). Since the robots execute a common algorithm, from this initial configuration, they keep square positions forever if they execute cycles synchronously. Yamashita et al. introduced the notion of symmetricity that gives formal explanation for such situation [13, 15]. We consider the decomposition of a set of points into regular -gons centered at one point. We consider that one point is a regular -gon with an arbitrary center and two points form a regular -gon with the center being the midpoint. Then the maximum value of such is the symmetricity of in 2D-space. When is greater than one, the common center is the center of the smallest enclosing circle of , denoted by , and is generally the order of the cyclic group that acts on . However, when , this definition allows , which means the symmetry of can be broken. This is achieved by the robot on leaving its current position. It has been shown that irrespective of obliviousness and asynchrony, the robots with chirality in 2D-space can form a target pattern from an initial configuration if and only if divides except the case where is a point with multiplicity two [11, 13, 15]. The exception is called the rendezvous problem, which is trivially solvable by FSYNC robots while not solvable by SSYNC (thus ASYNC) robots.

The notion of symmetricity is later extended to the robots in 3D-space [17]. In 3D-space, a set of symmetric local coordinate systems with chirality is obtained by rotations on the global coordinate systems, and there are five types of rotation symmetry; the cyclic groups, the dihedral groups, the tetrahedral group, the octahedral group, and the icosahedral group. Each rotation symmetry forms a group that can be recognized as the set of symmetric rotation operations on a prism, a pyramid, a regular tetrahedron, a regular octahedron, and a regular icosahedron, respectively. In other words, each rotation group is determined by the arrangement of rotation axes and their foldings. A rotation axis is a -fold axis if it admits rotations by for . Yamauchi et al. introduced rotation group and symmetricity for a set of points in 3D-space. The rotation group of a set of points is the rotation group that acts on and none of its supergroup in the set of rotation groups acts on . The symmetricity of is the set of rotation groups such that the group action of on divides into -sets where is the order of . In the same way as 2D-space, the definition of symmetricity implies symmetry breaking by movement of the robots because when some robots are on the rotation axes of , the robots do not allow the specified decomposition regarding the rotation axis. In other words, consists of the rotation groups formed by “unoccupied” rotation axes of . Actually, the robots on rotation axes can remove the rotation axes by leaving their current positions. Yamauchi et al. showed that irrespective of obliviousness, the FSYNC robots with chirality can form a target pattern from an initial configuration if and only if is a subset of  [17].

However, all these results assume chirality among the robots. After Yamashita et al. present pattern formation algorithms for the oblivious SSYNC robots with chirality in 2D-space [13, 15], Fujinaga et al. investigate the embedded pattern formation problem, where a target pattern is given as a set of landmarks on the plane [10]. They showed that oblivious ASYNC robots can form any embedded target pattern by presenting an algorithm that is based on the “clockwise” minimum-weight perfect matching between the robots and the landmarks. Based on this clockwise matching algorithm, Fujinaga et al. presented a pattern formation algorithm for oblivious ASYNC robots with chirality [11]. Later Cicerone et al. pointed out that the clockwise matching algorithm does not work when the robots lack chirality, and showed a new embedded target pattern formation algorithm [1]. They also pointed out that robots without chirality may forever move symmetrically regarding an axis of symmetry.

Our contribution.  The goal of our study is to formalize the degree of symmetry among the robots without chirality in 3D-space and investigate their formation power. The contribution of this paper is twofold. First, we give a definition of symmetricity among the robots without chirality in 3D-space. We consider both rotation symmetry and reflection symmetry because when the robots lack chirality, a local coordinate system is obtained by a uniform scaling, a translation, a rotation, a reflection by a mirror plane, or a combination of them on the global coordinate system.222 When the robots have chirality, reflection is not necessary since reflection changes the handedness of local coordinate system. The combination of rotation symmetry and reflection symmetry introduces seventeen types of symmetry groups, which is well studied in group theory and crystal symmetry [3]. We extend the notion of symmetricity in [17] to these seventeen symmetry groups. We validate the definition by showing that the robots cannot resolve their symmetricity forever. Then, we give a necessary and sufficient condition for FSYNC robots without chirality to solve the plane formation problem. To show the sufficiency, we present a new plane formation algorithm since the existing plane formation algorithms for robots with chirality [14, 16] do not work in our model.

We focus on the FSYNC robots with rigid movement, that is, all robots synchronously execute a cycle and reach their next positions in each cycle. If a robot stops en route, its movement is non-rigid. While most existing results assume non-rigid movement, the worst case is when the robots cannot resolve their symmetry. Thus the worst case is determined by synchrony and rigid movement. Formally, any execution of the FSYNC robots with rigid movement appears in the SSYNC (thus ASYNC) model with non-rigid movement.

In [16], the cyclic groups and the dihedral groups are called 2D rotation groups because one rotation axis is recognized, and when the rotation group of the current configuration is a 2D rotation group, the robots with chirality can easily land on a “horizontal” plane perpendicular to this rotation axis (Fig. (a)a and Fig. (b)b). On the other hand, the remaining three rotation groups do not act on a set of points on a plane, and they are called 3D rotation groups. The necessary and sufficient condition in [16] is rephrased as follows: The FSYNC robots with chirality can form a plane from an initial configuration if and only if consists of 2D rotation groups. This characterization implies that the FSYNC robots with chirality can form a plane from an initial configuration where they form a regular polyhedron (except a regular icosahedron) or an icosidodecahedron, while they cannot form a plane from the remaining (convex) uniform polyhedra. Clearly, the necessity of this result holds for the robots without chirality.

(a)
(b)
(c)
(d)
Figure 2: Examples of plane formation procedure. (a) Robots directly land on a plane. (b) Robots with chirality follow a “right-screw rule”. (c) Robots without chirality cannot avoid multiplicity. (d) Robots on the horizontal mirror plane can break the reflection symmetry.

When the robots lack chirality, even when is a 2D rotation group, the “horizontal” plane can be a mirror plane and the robots cannot resolve the symmetry regarding this mirror plane (Fig. (c)c). Actually, the only plane that the robots can agree is this mirror plane, but they cannot avoid multiplicity on it. As a result, a cube is removed from the set of solvable instances, when compared with the robots with chirality. However, we will show that when there is at least one robot on the horizontal mirror plane, the robots can remove the mirror plane and can form a plane. Intuitively, our necessary and sufficient condition for the FSYNC robots without chirality requires that an initial symmetricity contains neither any 3D rotation group nor any combination of a 2D rotation group and an “empty” horizontal mirror plane. Our current results are preliminary in the sense of symmetry by rotation axes and mirror planes in 3D-space. For example, we defined the symmetricity among the robots, and prove that the robots cannot resolve it, but the symmetry breaking is not fully explored as we will address in the conclusion section.

Organization.  In Section 2, we define our robot model and introduce rotation symmetry and reflection symmetry in 3D-space. We present a necessary and sufficient condition for plane formation by FSYNC robots without chirality. We show the necessity of the condition in Section 3, and we prove the sufficiency by presenting a plane formation algorithm in Section 4. We conclude this paper with Section 5.

2 Preliminary

2.1 Robot Model

Let be a set of anonymous robots, each of which is a point in 3D-space. We use just for description. We consider discrete time and let be the position of at time in the global -- coordinate system , where is the set of real numbers. The configuration of at time is . We denote the set of all possible configurations of by . We assume that the initial positions of robots are distinct, i.e., for and .333When more than one robots are at one point, it is impossible to separate them by a deterministic algorithm. We also assume that since any three robots are on one plane.

Each robot has no access to the global coordinate system, and it uses its local -- coordinate system . The origin of is the current position of while the unit distance, the directions, and the orientations of the , , and axes of are arbitrary and never change. Hence, it is appropriate to denote , but we use a shorter description. Each is either right-handed or left-handed. Thus the robots do not have chirality. We denote the coordinates of a point in by .

We consider the fully-synchronous (FSYNC) model, where the robots start the th Look-Compute-Move cycle at the beginning of time and finishes it before time (). Each of the Look phase, the Compute phase, and the Move phase of a cycle is completely synchronized at each time step. At time , each robot obtains a set in the Look phase. Then computes its next position by using a common algorithm in the Compute phase. A robot is oblivious if it does not remember the past observations and the past computations, thus the input to is . Otherwise, it is non-oblivious and the input to contains the past observations and the past computations. Finally, moves to the next point in the Move phase. We assume that each robot always reaches its next position in a move phase and we do not care for the route to reach there. Thus we consider rigid movement.

An execution of an algorithm from an initial configuration is a sequence of configurations . When the initial local coordinate systems of , the algorithm , and initial local memory content (if any) are fixed, the FSYNC execution is uniquely determined.

The plane formation problem requires that the robots land on a plane, which is not predefined, without making any multiplicity. Hence point formation is not a solution for the plane formation problem. We say that an algorithm forms a plane from an initial configuration , if, regardless of the choice of initial local coordinate systems for each , any execution there exists a finite such that (i) is contained in a plane, (ii) , i.e., all robots occupy distinct positions, and (iii) once the system reaches , the robots do not move anymore.

For a set of points , we denote the smallest enclosing ball (SEB) of by and its center by . A point on the sphere of a ball is said to be on the ball, and we assume that the interior or the exterior of a ball does not include its sphere. The innermost empty ball is the ball whose center is , that contains no point of in its interior and contains at least one point of on its sphere. When all points of are on , we say is spherical.

2.2 Symmetry by Rotations and Reflections

We consider symmetry among the robots which is caused by not only symmetric positions of the robots but also symmetric local coordinate systems of them. Since any local coordinate system is obtained by a uniform scaling, a translation, a rotation, a reflection by a mirror plane, or a combination of them on the global coordinate system, we focus on symmetry operations by rotation axes and mirror planes.

A -fold axis admits rotations by (). These operations form the cyclic group of order . When there are more than one rotation axes, they also form a group, and there are five kinds of rotation groups in 3D-space, each of which is determined by the types of rotation axes and the arrangement of them [3]. Clearly, these multiple rotation axes intersect at one point. The dihedral group consists of a single -fold axis called the principal axis and -fold axes perpendicular to the principal axis, and its order is .444The rotation group consists of three -fold rotation axes, and it has been shown that the principal axis of can be also recognized [16]. This is because we do not consider only, but a set of points and a rotation (or symmetry) group that acts on the points. We can recognize by the rotations on a prism with regular -gon bases. We abuse the term “principal axis” for the single rotation axis of a cyclic group.

The remaining three rotation groups are the tetrahedral group, the octahedral group, and the icosahedral group, and we can recognize them by the rotations on the corresponding regular polyhedra. The tetrahedral group consists of three -fold axes and four -fold axes, and its order is . The octahedral group consists of six -fold axes, four -fold axes, and three -fold axes, and its order is . The icosahedral group consists of fifteen -fold axes, ten -fold axes, and six -fold axes, and its order is . We call the cyclic groups and the dihedral groups 2D rotation groups, and we call the remaining three rotation groups , , and 3D rotation groups because a 3D rotation group does not act on a point on a plane.

A mirror plane changes the handedness and a mirror image of an object has a different handedness from the original object. This is the reason why we need to consider reflection symmetry when we consider the robots without chirality. The bilateral symmetry consists of one mirror plane and its order is . When there are more than one mirror planes, an intersection of mirror planes introduces a rotation axis. We consider the compositions of rotation symmetry and mirror planes. Each symmetry type also forms a group. Clearly, the rotation axes and mirror planes of the symmetry type intersect at one point. The composition of () and a mirror plane perpendicular to the principal axis is denoted by , where represents the “horizontal” mirror plane. The order of is . The composition of () and mirror planes containing the principal axis is denoted by , where represents the “vertical” mirror planes. The order of is . The composition of () and a horizontal mirror plane regarding the principal axis is denoted by . However, this horizontal mirror plane together with rotation axes of forces vertical mirror planes each of which contains two -fold axes and the principal axis. The order of is . The composition of () and vertical mirror planes is denoted by . The vertical mirror planes do not contain any -fold axes, otherwise the rotation axes of forces a horizontal mirror plane. The order of is .

The composition of and three mutually perpendicular mirror planes, each of which contains two -fold axes is denoted by . The order of is . The composition of and six mirror planes, each of which contains two -fold axes is denoted by . The order of is . The composition of and nine mirror planes is denoted by . Three of the mirror planes are mutually perpendicular and each of them contains two -fold axes. Each of the remaining six mirror planes contains two -fold axes. The order of is . The composition of and fifteen mirror planes, each of which contains two -fold axes is denoted by . The order of is .

Another type of composite symmetry is rotation reflection, where a rotation regarding a single rotation axis and taking a mirror image regarding a mirror plane perpendicular to the rotation axis are alternated. This type of symmetry is denoted by . Because of the alternation, the folding of the rotation axis is even. corresponds to the central inversion, which is denoted by . See Appendix A for more detail.

Let where consists of only the identity element. We call the elements of symmetry groups. These seventeen types of symmetry groups describe all symmetry in 3D-space [3]. In this paper we consider rotation symmetry separately. We call the elements of the rotation groups.

We denote the order of with . When is a subgroup of (), we denote it by . If is a proper subgroup of (i.e., ), we denote it by . For example, we have , , but . If has a -fold axis, if divides . For symmetry groups containing mirror planes, but . For , we have .

2.3 Rotation Group, Symmetry Group, and Symmetricity

Let be a set of points. The rotation group of is the rotation group that acts on and none of its proper supergroup in acts on . The symmetry group of is the symmetry group that acts on and none of its proper supergroup in acts on . Clearly, is a subgroup of (), and they are uniquely determined.555See for example [3], that shows an algorithm to uniquely determine the symmetry group of a polyhedra. The algorithm checks rotation axes, mirror planes, and a point of inversion. Since we consider a set of points and their convex-hulls, we can use the same algorithm. By the definition, when is either , because such configuration does not have any rotation axis. Table 1 shows the rotation group of a set of vertices of each regular polyhedron.

Polyhedron Rotation group Symmetry group Symmetricity
Regular tetrahedron
Regular octahedron
Cube
Regular dodecahedron
Regular icosahedron
Table 1: Rotation group, symmetry group, and symmetricity of regular polyhedra

The group action of decomposes into disjoint subsets. Let be the orbit of where denotes the action of on , and the orbit space is called the -decomposition of . Each element is transitive because it is one orbit regarding .

Yamauchi et al. showed that in configuration without any multiplicity, the robots with chirality can agree on the -decomposition of and a total ordering among the elements so that (i) is on , (ii) is on , and (iii) is not in the interior of the ball centered at and containing on its sphere [16]. Though their technique relies on chirality, we can extend it to robots without chirality. In [16], each robot translates its local observations to a “celestial map” by considering as the earth and its current position is on the half line from containing the north pole. Then, the robot selects an appropriate robot to define the prime meridian and translates the position of each robot to a triple consisting of its altitude, latitude, and longitude. The ordered sequence of these triples is the local observation of the robots. However, the lack of chirality does not allow the robots to agree on the direction of longitude. Then we make a robot consider both directions and select the direction that produces the smallest sequence. In the same way as [16], we have the following property.

Lemma 1

Let and be a configuration of robots represented as a set of points and its -decomposition, respectively. Then we have the following two properties:

  1. For each (), all robots in have the same local view.

  2. Any two robots, one in and the other in , have different local views, for all .

Proof

The first property is obvious by the definitions of -decomposition and local view, since for any there is an element such that .

As for the second property, to derive a contradiction, suppose that there are distinct integers and , such that robots and have the same local view. Let and be the local view of and . Thus, we have . Let us consider a function that maps the th element of to that of . More formally, letting the th element of (resp., ) be (resp., ), maps to . Then is a transformation that keeps unchanged by the definition of local view, i.e., is a rotation or an reflection in , which contradicts to the definition of -decomposition. ∎

By Lemma 1, the robots can agree on a total ordering of the elements of the -decomposition of . In the following, we assume that is sorted by this ordering.

We denote the set of local coordinate systems for configuration with where is the position of (i.e., the origin of ) and , , and are the , , and of observed in the global coordinate system . We use to explicitly show the set of local coordinate systems for though contains as . We define the symmetry group of as the symmetry group of that acts on and none of its proper supergroup in acts on it. Clearly, we have . We define the -decomposition of in the same way as the -decomposition of . We note that the robots of cannot obtain nor because they can observe only the positions of themselves.

Given a set of points, determines the arrangement of its rotation axes and mirror planes in . We thus use and the arrangement of its rotation axes and mirror planes in interchangeably. For two groups , an embedding of to is an embedding of each rotation axis and each mirror plane of to one of the rotation axes and one of the mirror planes of with keeping their arrangement in . Any -fold axis of is embedded so that it overlaps a -fold axis of , where divides , and any mirror plane of is embedded to a mirror plane of . However we need careful treatment for . When , its mirror plane is embedded to a mirror plane of , and when , its mirror plane cannot grant any mirror plane of . For example, we can embed to . There are three embeddings of to depending on the choice of the -fold axis. We can embed to , and to . We can embed to but cannot to . Observe that we can embed to if and only if .

We can also consider a -decomposition of a set of points for some for an embedding of in . However, for such -decomposition, the robots cannot agree the ordering among the elements since Lemma 1 does not hold.

We now define symmetricity of a set of points in 3D-space. Intuitively, symmetricity shows all possible symmetry groups to which the robots may forever subject. As the symmetry groups are partially ordered, we consider a set of such rotation groups.

Definition 1

Let be a set of points. The symmetricity of is the set of symmetry groups that acts on (thus ) and there exists an embedding of to such that each element of the -decomposition of is a -set.

We define as a set because the “maximal” symmetry group that satisfies the definition is not uniquely determined. Maximality means that there is no proper supergroup in that satisfies the condition of Definition 1. When it is clear from the context, we denote by the set of such maximal elements. For example, if forms a cube,

and we denote it by . The set does not contain itself since -decomposition of consists of one -set, while . From the definition, contains every element of that is a subgroup of if . See Table 1 as an example.

We can rephrase the definition of symmetricity of a set of points as a set of symmetry groups formed by rotation axes and mirror planes of that do not contain any point of . This is because a point on a rotation axis (a mirror plane, respectively) does not allow a decomposition into -sets for any containing the rotation axis. 666 We assume that a set of points does not contain any multiplicity. In other words, we consider an initial configuration .

We conclude this section with the following two lemmas, that validate the definition of symmetricity. Lemma 2 shows that there exists an arrangement of local coordinate systems for any initial configuration and such that . Then, Lemma 3 shows that the robots are caught in this initial symmetry.

In the proofs, we take a new view of the positions of the robots. In the definition of symmetricity for a set of points , we consider an arrangement of a symmetry group in . On the other hand, to show that the initial symmetry cannot be broken, we consider the cases where the positions of robots are caught in an arrangement of .

Lemma 2

For an arbitrary initial configuration and any , there exists a set of local coordinate systems such that .

Proof

We show a construction of for and . Let be the -decomposition of for some embedding of to . Clearly, such embedding exists since . From the definition, for . For each , we arbitrary fix a local coordinate system of one robot . Then for each , there exists a unique element of such that and if for any . Then we fix the local coordinate system of by applying to the local coordinate system of . The local coordinate systems obtained by this procedure satisfies the property. ∎

Lemma 3

Irrespective of obliviousness, for an arbitrary initial configuration , any , and any algorithm , there exists an execution such that .

Proof

Let be initial local coordinate systems for such that for arbitrary . By Lemma 2, such always exists. Let be the -decomposition of .

From this arrangement of initial local coordinate systems, the robots forming keeps their symmetry group forever for any algorithm . We first show an induction for the oblivious FSYNC robots. For any , when holds, we have and . Let be the positions of robots of in . Thus and . By an easy induction for , we have the property for any .

Non-obliviousness does not improve the situation. When the initial memory contents are identical (for example, empty), the above discussion holds for the transition from to . During this transition, the robots in the same element of the -decomposition of obtain the same local observation, performs the same computation, and exhibits the same movement. Thus, their local memory content are still the same in , and they continue symmetric movement during the transition from to . ∎

3 Impossibility of Plane Formation

The following theorem shows a necessary condition for the FSYNC robots without chirality to form a plane, that will be shown to be a sufficient condition in Section 4. The condition means that to solve the plane formation from an initial configuration , should not contain any of the following symmetry groups: , , , , , , , (), and ().

Theorem 3.1

Irrespective of obliviousness, the FSYNC robots without chirality can form a plane from an initial configuration only if consists of , , , , , , , , and .

Proof

Let be an arbitrary plane formation algorithm for an initial configuration such that contains a 3D rotation group, (), or (). We have the following three cases.

Case A:  contains for some .

Let be a set of initial local coordinate systems for such that (). From Lemma 3, irrespective of obliviousness, for any algorithm , there exists an execution such that for any . Assume that be a terminal configuration. Then is a supergroup of , and has the mirror plane of . The robots are on the mirror plane, otherwise the robots are not on one plane because of their symmetry. Let be the -decomposition of . For each () and , there exists such that and are at symmetric positions regarding the mirror plane of .

By Lemma 3, the robots of move with keeping the rotation axis and the mirror plane of the embedding of in . Thus and occupy the same point on the mirror plane of in . Hence the robots cannot avoid multiplicity and is not a terminal configuration of the plane formation problem.

Case B:  contains for some .

Let be a set of initial local coordinate systems for such that (). By Lemma 3, irrespective of obliviousness, for any algorithm , there exists an execution such that for any . We have the same discussion as Case A. If there exists a terminal configuration, the robots are on the initial horizontal mirror plane of . Hence, the robots cannot avoid multiplicity and is not a terminal configuration of the plane formation problem.

Case C:  contains a 3D-rotation group.

The impossibility for this case has been shown for robots with chirality in [16] and the result holds for our robots because our model allows the robots with chirality. We note that when contains or , then it contains the corresponding rotation group because it is a subgroup without any mirror plane. ∎

4 Plane Formation Algorithm

In this section, we show a plane formation algorithm for oblivious FSYNC robots without chirality and prove our main theorem.

Theorem 4.1

Irrespective of obliviousness, the FSYNC robots without chirality can form a plane from an initial configuration if and only if consists of , , , , , , , , and .

The necessity is clear from Theorem 3.1. We prove the sufficiency by presenting a plane formation algorithm for solvable instances (i.e., initial configurations). Because of the condition of the theorem, solvable instances are classified into the following three types.

Type 1: 

Initial configurations with 3D rotation groups. From the condition of Theorem 4.1, any initial configuration of this type contains one of the following polyhedra as an element of its -decomposition because some robots are on some rotation axes: a regular tetrahedron, a regular octahedron, a regular dodecahedron, and an icosidodecahedron.777 Points on rotation axes of a 3D rotation group also forms a cube, a cuboctahedron, and a regular icosahedron. However, these polyhedra allow () to join their symmetricity.

Type 2: 

Initial configurations with 2D rotation groups with at least one rotation axis. From the condition of Theorem 4.1, any initial configuration of this type satisfies that either does not have the horizontal mirror plane or there are some robots on a horizontal mirror plane.

Type 3: 

Initial configurations without any rotation axis. This case is further divided into asymmetric initial configurations, a symmetric initial configurations with a single mirror plane, and a symmetric initial configurations with point of inversion.

The proposed algorithm handles these three types separately. The robots can agree on the type of the current configuration and they execute the corresponding algorithm. For the first case, the robots first break their symmetry and translates an initial Type 1 configuration to another Type 2 or Type 3 configuration. For the second case, the robots agree on a plane perpendicular to the principal axis and containing the center of their smallest enclosing ball, and land on the plane. For the third case, if the initial configuration is asymmetric, the robots agree on a plane by using the total ordering among themselves. Otherwise, the robots agree on a plane other than the mirror plane by using two elements of their decomposition and lands on it.

Before we go into the detailed description of the proposed algorithm, we show preparation steps for an initial configuration . These steps can be realized very easily in the FSYNC model because the set of robots to move is easy to recognize, and the movement neither makes collisions nor changes the symmetry group and the symmetricity of the robots. In the following, we use a point and a robot at the point interchangeably. For example, the position of robot means , and means the set of robots at positions of .

First, when , the preparation phase sends the robot on to an arbitrary point in the interior of , so that a resulting configuration will be asymmetric. The next position of the robot at is, for example, a point neither on any rotation axis nor on any mirror plane of .

Second, for Type 1 cases, the preparation phase moves an element of the -decomposition of forming one of the specified polyhedra to the interior of .

Finally, for Type 2 cases, the preparation phase moves an element of the -decomposition of on the horizontal mirror plane of to the interior of . At the same time, if there exists another element of the -decomposition of that is on neither the horizontal mirror plane nor the principal rotation axis, the preparation phase makes the element slightly “shrink” so that is kept by . For example, if with consists of a cube and a square on the horizontal mirror plane, the robots forming a cube shrink to translate the cube to a long square prism.

For the second and the third cases, we select the minimum index among the elements satisfying the condition and move each (, respectively) along the line . Since we select the minimum index, this movement introduces no collision. Additionally, in the third case, the robots of move on the horizontal mirror plane toward .

4.1 Symmetry Breaking

We consider a Type 1 configuration . Let the the -decomposition of . The preparation step guarantees that forms one of the following four polyhedra; a regular tetrahedron, a regular octahedron, a regular dodecahedron, and an icosidodecahedron. Then the proposed plane formation algorithm first makes the robots execute the go-to-center algorithm (Algorithm 1) proposed in [16]. Each robot of selects an adjacent face of the polyhedron and moves to the center of the selected face. But it stops before the center to avoid collisions.

Algorithm 1 does not depend on chirality among the robots and it has been shown that the rotation group of a resulting configuration is always a 2D rotation group [16]. Since our robots lack chirality, we should consider the combination of such 2D rotation groups and mirror planes. The following lemma guarantees that any resulting configuration does not have any horizontal mirror plane (except ). We note that we do not have to care for rotation reflections in this phase because such configurations are Type 3 configurations.

Notation
: Current configuration observed in .
: The position of (i.e., the origin of ).
: -decomposition of , where forms one of the four
polyhedra.
, where is the length of an edge of the polyhedron that forms.
Algorithm
If then
If is an icosidodecahedron then
Select an adjacent regular pentagon face of .
Destination is the point before the center of the face on the line
from to the center.
Else
// is a tetrahedron, a octahedron, or a dodecahedron.
Select an adjacent face of .
Destination is the point before the center of the face on the line
from to the center.
Endif
Move to .
Endif
Algorithm 1 Symmetry breaking algorithm for robot  [16]
Lemma 4

Let be a configuration such that is a 3D rotation group and contains neither any 3D rotation group nor any symmetry group with a horizontal mirror plane (except ). Let be the -decomposition of . Then there exists one element of () that forms one of the following polyhedra; a regular tetrahedron, a regular octahedron, a regular dodecahedron, or an icosidodecahedron. We further assume that forms one of the above polyhedra. Then one step execution of Algorithm 1 translates into another configuration that satisfies (i) is a 2D rotation group, and (ii) if , does not have any horizontal mirror plane.

Proof

To show the first property of the lemma, we introduce another technique to check transitive set of points regarding a symmetry group.888 This is an extension of the same technique for rotation groups shown in [16]. Given an arrangement of and a seed point in an arrangement of , by applying all elements of to , we obtain an orbit of . Clearly, is transitive regarding and the location of determines the size of . For example, when is on a -fold rotation axis of , , when is on a mirror plane of but not on a rotation axis of , , and when is neither on any mirror plane nor on any rotation axis, .

Since is a 3D rotation group and does not contain any 3D rotation group, there are some robots on the rotation axes of . A seed point on a rotation axes of a 3D rotation group produces a regular tetrahedron, a regular octahedron, a cube, a cuboctahedron, a regular dodecahedron, a regular icosahedron, or an icosidodecahedron. However, since and , a cuboctahedron and an icosahedron allow to remain in symmetricity. Additionally, a cube allows to remain in symmetricity, and the plane formation is not possible from a cubic initial configuration. We have the first property of the lemma.

Assume that the robots of occupy the points . It suffices to show that does not have any horizontal mirror plane, since is a subgroup of .

We first check the rotation group of any resulting configuration of Algorithm 1 and then we proceed to the combinations of rotation axes and mirror planes. In [16], it has been shown that after one step execution of Algorithm 1, the rotation group of any resulting configuration is one of the rotation groups shown in Table 2. Hence is a composition of these rotation symmetry and reflection symmetry.

Polyhedron of Candidates of
Regular tetrahedron 4
Regular octahedron 6
Regular dodecahedron 20
Icosidodecahedron 30
Table 2: Symmetricity and rotation group of the four polyhedra and rotation groups after the execution of Algorithm 1
(a)
(b)
(c)
(d)
Figure 3: The set of candidate destinations. (a) an -expanded tetrahedron, (b) an -expanded cube, (c) an -expanded icosahedron, and (d) an -truncated icosahedron.

The set of candidate destinations of Algorithm 1 forms the polyhedra shown in Fig. 3. For example, when forms a regular octahedron, possible next positions of the six robots are around the centers of the faces, thus, around the vertices of the dual cube. Since the robots do not move to the center, the candidate destinations form an -expanded cube, which is obtained by expanding the faces of a cube. The rotation group of an -expanded cube is the same as its original polyhedra, i.e., a cube, and it is . Additionally, its vertices form a transitive set regarding . The six robots select a subset of the vertices of this -expanded cube in Algorithm 1. In the same way, when forms a regular tetrahedron, the candidate destinations form an -expanded tetrahedron, when forms a regular dodecahedron, the candidate destinations form an -expanded icosahedron, and when forms an icosidodecahedron, the candidate destinations form an -truncated icosahedron. Each of these polyhedra is also transitive (hence spherical) regarding the rotation group of its original polyhedron.

We check the symmetry group of and depending on , we have the following four cases.

Case A: When forms a regular tetrahedron.  The set of candidate destinations form an -expanded tetrahedron and . By Table 2, we check whether is .

Assume that . When the points of are on the principal axis (secondary axes, respectively), is on one plane. When the points of are on mirror planes but not on any rotation axes, still is on one plane. Otherwise, we have , and we do not have this case.

However, since the four robots select one face of a regular tetrahedron, is not on a plane. There are following four cases: (i) three robots select the same face, (ii) two robots select the same face and the remaining two robots select another face, (iii) robots are divided into a -set and two -sets and the three groups select different faces, and (iv) each robot selects different face. In any of the four cases, the four robots are not on one plane. Hence, we do not have the case where .

Case B: When forms a regular octahedron.  The set of candidate destinations form an -expanded cube and . By Table 2, we check whether is or .

We first show that is not on a plane. The candidate destinations of one forms a square face of an -expanded cube. If is on one plane, say , contains at least one vertex of each square face of an -expanded cube. Clearly, such does not exist.

Assume that . Thus the points of are on some mirror planes, otherwise we have . Since is not on one plane, forms a triangular prism. As Fig. 4 shows, any regular triangle in an -expanded cube is centered at a point on a -fold rotation axis. Additionally, no combination of these triangles form a triangular prism. Hence, we have .

Figure 4: Triangles in an -expanded cube

Assume that . Since is not on one plane, is not on the horizontal mirror plane of and it forms a triangular prism. In the same way as the above discussion, we do not have this case.

Case C: When forms a regular dodecahedron.  The set of candidate destinations form an -expanded icosahedron and . By Table 2, we check whether is , , or .

We first show that is not on a plane. The candidate destinations of one forms a regular triangle face of an -expanded icosahedron. If is on one plane, say , contains at least one vertex of each regular triangle face of an -expanded icosahedron. Clearly, such does not exist.

Assume that . Since is not on one plane, we have the following two cases: (a) contains a pentagonal prism (size ), and (b) contains a transitive -set regarding , (c) contains a set of points on the principal rotation axis of . As Fig. 5 shows, any regular pentagon in an -expanded icosahedron is centered at a point on a -fold rotation axis. Additionally, no combination of these pentagons form a pentagonal prism. Hence, we do not have case (a).

Figure 5: Pentagons in an -expanded icosahedron

Any transitive -set regarding consists of two pentagonal prisms. From the above discussion, we do not have case (b).

As sown in Fig. 5, any pentagon in an -expanded icosahedron has no point above its rotation axis because there is no point of an -expanded icosahedron on its -fold rotation axis. Thus we do not have case (c) and we have .

Assume that . Since is not on one plane, we have the following two cases: (d) contains a pentagonal prism (size ), and (e) contains a set of points on the principal axis. In the same way as above discussion, we have .

Assume that . The size of a transitive set of points regarding is either , (on a mirror plane), or (on a rotation axis). Since is not on one plane, does not consist of transitive -sets. When contains a transitive -set, the number of transitive -sets is greater than one because . However, since an -expanded icosahedron is spherical, at most two points of it are on a line. Thus should contain a transitive -set because of its size.

Fig. 6 shows all possible cuboids in an -expanded icosahedron. Then their mirror planes contain a -fold axis and two -fold axes, in other words, four vertices of the original icosahedron. Since a vertex of a regular icosahedron is broken into five points in an -expanded icosahedron, there is no rectangle containing the mirror plane of any of such cuboids. Hence, we have .

Figure 6: Cuboids in an -expanded icosahedron. The figures show two parallel faces of each possible cuboid.

Case D: When forms a regular icosidodecahedron.  The set of candidate destinations form an -truncated icosahedron and . By Table 2, we check whether is or .

We first show that is not on a plane. The candidate destinations of one forms an edge of an -truncated icosahedron. If is on one plane, say , contains at least one endpoint of each edge of an -expanded icosahedron. Clearly, such does not exist.

Assume that . Since is not on one plane, we have one of the following two cases: (a) contains a pentagonal prism, or (b) contains a set of points on the rotation axis of . As Fig. 7 shows, any regular pentagon in an -truncated icosahedron is centered at a point on a -fold rotation axis. Additionally, no combination of these pentagons form a pentagonal prism. Hence, we do not have case (a).

Figure 7: Pentagons in an -truncated icosahedron

As sown in Fig. 7, any pentagon in an -truncated icosahedron has no point above its rotation axis because there is no point of an -truncated icosahedron on its -fold rotation axis. Thus we do not have case (b) and we have .

Assume that . Since is not on one plane, we have one of the following two cases: (c) contains a triangular prism, or (d) contains a set of points on the rotation axis of . As Fig. 8 shows, any regular triangle in an -truncated icosahedron is centered at a point on a -fold rotation axis. Additionally, no combination of these triangles form a triangular prism. Hence, we do not have case (c).

Figure 8: Triangles in an -truncated icosahedron

As sown in Fig. 7, any regular triangle in an -truncated icosahedron has no point above its rotation axis because there is no point of an -truncated icosahedron on its -fold rotation axis. Thus we do not have case (d) and we have .

From the above four cases, we conclude that is a 2D rotation group and if have at least one rotation axis, it does not have a horizontal mirror plane (except ). Since is a subgroup of , we have the lemma. ∎

We finally note that the robots cannot remove vertical mirror planes of with the go-to-center algorithm. For example, consider an initial configuration where the robots form a regular octahedron. Thus . Consider an embedding of in . The -fold rotation axis of overlaps a -fold rotation axis of and each vertical mirror plane of contains two trajectories of the go-to-center algorithm. Actually, the six robots can take these trajectories and the resulting configuration forms a triangular anti-prism (thus ).

4.2 Landing Algorithm

In this section, we show a plane formation algorithm for Type 2 and Type 3 initial configurations. When of a current configuration is a cyclic group or a dihedral group, our basic strategy is to make the robots agree on the plane perpendicular to the principal axis and containing and then we send the robots to the plane. Each robot moves along a perpendicular to the plane. To avoid multiplicities, we need some tricks for the following two cases: First, when has a horizontal mirror plane, the condition of Theorem 4.1 guarantees that there is at least one element of the -decomposition of on it. To remove this mirror plane, we first make the robots of such an element leave their current positions (Fig. (a)a). The other case is when is a dihedral group and some element, say , of the -decomposition of has a horizontal mirror plane. Since the target plane is this mirror plane, the final destination of any symmetric two robots are the same. However, Theorem 4.1 guarantees that there exists at least one element, say , of the -decomposition of that does not have a horizontal mirror plane. The robots of use to break their symmetric landing points (Fig. (b)b).

(a)
(b)
Figure 9: Removing horizontal mirror plane and resolving collisions. (a) The robots on the horizontal mirror plane move vertically, and removes the plane. (b) The robots forming a prism rotates by using an anti-prism.

The proposed algorithm consists of five phases. The first three phases break the mirror plane of and resolves the collisions on the target plane. The fourth phase makes the robots agree on the target plane and in the fifth phase each robot computes the destinations of all robots to avoid any collision. The fourth and the fifth phases are done in local computation at each robot. Finally, the robots move to their final destinations in the same cycle. In any configuration , the robots execute the algorithm for the smallest phase number. The robots can easily agree on which phase to execute because the condition for each of the five phases divide the set of all configurations with 2D rotation groups into disjoint subsets. Depending on the execution, some phases may be skipped. Since the proposed algorithm is designed for the oblivious robots, it is always described for a current configuration.

4.2.1 First Phase: Removing the Mirror Plane

By the condition of Theorem 4.1, when is a cyclic group or a dihedral group and has a horizontal mirror plane, the mirror plane contains some robots. The preparation step guarantees that this element is of the -decomposition of . Let be the folding of the principal axis of . Intuitively, the first phase makes the robots select the upward direction or the downward direction regarding this mirror plane and the robots move to the selected directions. Any resulting configuration does not have the horizontal mirror plane any more because for each new positions of the robots, there is no corresponding point regarding the horizontal mirror plane. However, for the simplicity of the correctness proof, these next positions are selected more carefully.

The robots consider a fictitious prism with a regular -gon base inscribed in , that share the horizontal mirror plane (Fig. 10). However, the size is selected so that the length of the edge of the regular -gon base is one tenth of the length of its side edge, and its arrangement is determined so that the plane formed by and a side edge contains a point of . Then, each moves toward one of the nearest vertex of this fictitious prism.

Figure 10: Fictitious prism for the first phase.
Lemma 5

Let be a configuration such that is a 2D rotation group () and some robots are one the horizontal mirror plane of . In this configuration, the robots execute the first phase, and a new configuration yields. Then satisfies one of the following conditions: (a) , (b) , or (c) does not have any horizontal mirror plane.

Proof

Let and be the -decomposition of and the folding of the principal axis of . We denote the vertices of the fictitious prism by . Clearly, . During the transition from to , the robots of select a subset and moves to the selected vertices. Other robots of do not move. Thus clearly the mirror plane of is not a mirror plane of (even ) because each does not have the corresponding point regarding this initial mirror plane.

We separately consider the following two cases. First, when , and is a subgroup of . In other words, the symmetry group of is kept by the robots of because these robots do not move during the transition from to , the symmetry group that acts on them does not change. Since is a subgroup of and , we check .

We first consider rotation axes of . We have the following four cases:

Case A:  If the principal axis of remains as some rotation axis of , its folding is a divisor of because of the movement of the robots of . Clearly, has no horizontal mirror plane from the above discussion.

Case B:  If a -fold axis of remains as some ration axis of , its folding remains two because any subset of that is on a plane perpendicular to this -fold axis forms a line or a rectangle because the prism is long. Actually, we do not have the case of a square because is not symmetric regarding the horizontal mirror plane of .

Case C:  If a new rotation axis appears and it has an intersection with the top or the base of , it is a -fold axis because any subset of that is on a plane perpendicular to this axis forms a line.

Case D:  If a new rotation axis appears and it has an intersection with the side face of , it is a -fold axis because any subset of that is on a plane perpendicular to this axis forms a line.

From the above four cases, is neither nor for any because the possible principal axis do not have any mirror plane by Case A. The remaining case is . The rotation axes of Case A and Case B do not form because there is no horizontal mirror plane. Even when the rotation axes of Case C and Case D form , this does not act on since . Hence, is or does not have any horizontal mirror planes.

Second, we consider the case where . From the preparation phase, all points of are on the horizontal mirror plane (thus the plane formation is finished), or all points of are on the principal axis. In the first case, the assumption is used in the combination of the rotation axes of Case C and Case D. Thus what we should check is this case. When the rotation axes of Case C and Case D form , it should act on the points on the original principal axis of