According to the most general definition, rigidity theory aims at studying the stiffness of a given system, understood as reaction to an induced deformation. This branch of knowledge originally emerges in the mathematics and geometry field, and extends to several research areas, ranging from mechanics to biology, from robotics to chemistry, properly declining itself according to the study context.
I-a Historical Background
The concept of rigidity dates back to 1776 when Euler has conjectured that every polyhedron is rigid, meaning that every motion of a given polyhedron results in a new polyhedron which is congruent to the first one . This fact has been proven in 1813 by Cauchy for convex polytopes in three dimensional space . Nonetheless, the most interesting part of the Cauchy’s theorem consists in its corollary affirming that if one makes a physical model of a convex polyhedron by connecting together rigid plates for each of the polyhedron faces with flexible hinges along the polyhedron edges, then this ensemble of components necessarily forms a rigid structure. Such a corollary represents a starting point for the study of the rigid structures that affects various research fields, involving mechanical and building structures, biological and artificial compounds and industrial materials, to cite a few (see  and the references therein).
An outstanding result in this direction is constituted by the work of Laman who, in 1970, has provided a definition of a family of sparse graphs describing rigid systems of bars and joints in the plane . According to this definition, a plane skeletal structure can be modeled as a graph so that each vertex corresponds to a joint in the structure and each edge represents a bar connecting two elements. This graphical model is then endowed with a map from the vertex set to the two-dimensional Euclidean space associating each joint to its position on the plane. The pair made up of the underlying graph and the corresponding position map is often referred as a plane Euclidean realization of the graph.
In 1978, to model more complex systems composed of different units interconnected by flexible linkages or hinges, Asimow and Roth have introduced the more general notion of framework. This mathematically corresponds to the graph-based representation of the system jointly with a set of elements belonging to , , each of one associates to a vertex of the graph in order to describe the position in of the corresponding unit composing the structure .
The problem of determining whether a given framework is rigid in , namely if there is no transformation of the graph vertices such that the final configuration is not congruent (in the Euclidean sense) to the original one, has been studied by many authors for well over a century [6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. In most of the cases, the rigidity or flexibility of a given system can be established by computing the rank of a suitable defined matrix, generically called rigidity matrix, that accounts for the interconnections among the structure components.
Recently, overcoming the standard bar-and-joints frameworks, the rigidity theory has enlarged its focus towards autonomous multi-agent systems wherein the connections among the system elements are virtual, representing the sensing relations among the devices (see  and the references therein). The concept of framework has thus been redefined by considering also manifolds more complex than the (-dimensional) Euclidean space. In these cases, the rigidity theory is turned out to be an important architectural property of many multi-agent systems where a common inertial reference frame should be unavailable but the agents involved are characterized by sensing, communication and movement capabilities. In particular, the rigidity concepts and results suitably fit for applications connected to the motion control of mobile robots and to the sensors cooperation for localization, exploration, mapping and tracking of a target (see, e.g., ).
I-B Distance vs. Bearing Rigidity
Within the multi-device systems context, rigidity properties for a given framework deals with agents interactions maintenance, according to the available sensing measurements111In accordance with the existing works about rigidity theory, in the rest of the paper we assume to deal with homogeneous multi-agent systems whose elements are characterized by the same sensing capabilities.. From this perspective, the literature differentiates between distance rigidity and parallel/bearing rigidity. When the agents are able to gather only range measurements, distance constraints can be imposed to preserve the formation distance rigidity. On the other hand, the formation parallel/bearing rigidity properties are determined through the fulfillment of direction constraints defined upon bearing measurements which are available whether the vehicles are equipped with bearing sensors and/or calibrated cameras.
The principal notions about distance rigidity are illustrated in [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. These works explain how distance constraints for a framework can be summarized into a properly defined matrix whose rank determines the rigidity properties of the system analogously to the case of frameworks embedded in . In such a context, indeed, it turns out to be useful to consider the given multi-agent system as a bar-and-joint structure where the agents are represented as particle points (joints) in , , and the interacting agent pairs can be thought as being joined by bars whose lengths enforce the inter-agent distance constraints.
Bearing rigidity in (or parallel rigidity) is instead determined by the definition of normal constraints over the directions of interacting agents, namely the edges of the graph associated to the framework, as explained in [29, 30, 31, 32, 33]. These constraints entail the preservation of the angles formed between pairs of interconnected agents and the lines joining them, i.e., the inter-agent bearings. Similar inter-agent direction constraints can be stated to access the rigidity properties of frameworks embedded in with
, where the bearing measurement between two agents coincides with their normalized relative direction vector[34, 35, 36, 37, 38, 39, 17]. In both cases, the agents are modeled as particle points in ,
, and the necessary and sufficient condition to guarantee the rigidity properties of a given framework rests upon the rank and eigenvalues of a matrix which summarizes the involved constraints.
Dealing with a more realistic scenario, in [40, 41, 42, 43, 44, 45] bearings are assumed to be expressed in the local frame of each agent composing the framework. This implies that each device in the group is model as a rigid body having a certain position and orientation w.r.t. a common inertial frame which is supposed to be unavailable to the group. In particular, in [41, 42] the attention is focus on multi-agent systems acting on a plane, in [40, 43, 44] the study is extended to the 3D space although limiting the agents attitude kinematics to rotations along only one axis, while in 
fully-actuated formations are considered by assuming to deal with systems of agents having six controllable degrees of freedom (dofs). Analogously to the former cases, the rigidity properties of the aforementioned multi-agent systems can be established through the definition and the spectral analysis of a matrix accounting for the inter-vehicle sensing interplay.
I-C A Unified Framework for the Bearing Rigidity Theory
Since in the last decades distance rigidity has been deeply investigated from the theoretical perspective and the related multi-agent systems applications are copious, in this work the attention is focused on bearing rigidity theory.
Motivated by the similarities emerging from the existing works that account for the different framework domains due to agent features, we aim at providing a unified framework for the statement of the principal bearing rigidity notions which are thus defined accounting for generic metric spaces. This constitutes the original contribution of this work that also summarizes the state-of-the-art results by properly specifying the given generic notions for the multi-agent scenarios described in Sec. I-B. In all the cases, we state the main definitions jointly with the conditions to determine whether a given system is rigid and we clarify the theoretical results through graphical examples. Tab. II provide a comprehensive overview about the principal features of the bearing rigidity theory for different framework domains.
The rest of the paper is organized as follows. In Sec. II we recall some notions on graph theory and we state the notation used in the rest of the work. In Sec. III we provide the basics of the bearing rigidity theory that is then particularized in Sec. IV-VI for specific domains. Sec. VII is devoted to the discussion about degenerate formation cases. Finally, we summarize the main conclusions in Sec. VIII.
Ii Preliminaries and Notation
is an ordered pairconsisting of a vertex set , and an edge set , having cardinality and , respectively. We distinguish between undirected, directed and oriented graphs. An undirected graph is a graph in which edges have no orientation, thus is identical to . Contrarily, a directed graph is a graph in which edges have orientation so that the edge is directed from (head) to (tail). An oriented graph is an undirected graph jointly with an orientation that is the assignment of a unique direction to each edge, hence only one directed edge ( or ) can exist between two vertices .
For any graph , the corresponding complete graph is the graph characterized by the same vertex set , while the edge set is completed so that each pair of distinct vertices is joined by an edge if is undirected/oriented () and by a pair of edges (one in each direction) if is directed ().
For a directed/oriented graph, the incidence matrix is the -matrix defined as
and, in a similar way, the matrix is given by
We introduce also the matrices and , where indicates the Kronecker product, is the the
-dimensional identity matrix, andrefers to the dimension of the considered space.
In this perspective, the -sphere embedded in is denoted as . Thus, represents the 1-dimensional manifold on the unit circle in , and represents the 2-dimensional manifold on the unit sphere in . The vectors of the canonical basis of are indicated as , and they have a one in the entry and zeros elsewhere.
Given a vector , its Euclidean norm is denoted as . In addition, we define the operator ,
that maps any (non-zero) vector to the orthogonal complement of the vector (orthogonal projection operator). Hence, indicates the projection of onto the orthogonal complement of . Given two vectors , their cross product is referred as , where the map associates each vector
to the corresponding skew-symmetric matrix belonging to the Special Orthogonal algebra.
Given a matrix , its null space and image space are denoted as and , respectively. The dimension of (or equivalently the number of linearly independent columns of ) is indicated as , whereas stands for the nullity of the matrix, namely . The well-known rank-nullity theorem asserts that .
Finally, we use the notation to indicate the block diagonal matrix associated to the set . In the following we will deal with block diagonal matrices related to the edges of a given graph : each block thus depends on the vertices , so that , i.e., and .
Iii Main definitions
In this section we introduce the main concepts related to the bearing rigidity theory. These will be particularized for the specific metric spaces in the rest of the paper.
Iii-a Framework Formation Model
Consider a generic formation of agents, wherein each agent is associated to an element of the metric space describing its state in terms of controllable variables. In addition, each agent is provided with bearing sensing capabilities, i.e., it is able to recover relative direction measurements w.r.t. some neighbors. From a mathematical point of view, such an -agents formation can be modeled as a framework embedded in the metric space .
Definition III.1 (Framework in ).
A framework in is an ordered pair consisting of a connected (directed or undirected) graph with and , and a configuration .
The framework model characterizes a formation in terms of both agents state and interaction capabilities. Indeed, the graph describes the available bearing measurements associating each agent to a vertex. Note that can be directed or undirected since agents interaction can be unidirectional or bidirectional, however, it is assumed to be not time-varying. The configuration copes with the set describing the agents state so that coincides with the -th agent position when this is modeled as a particle point and with the pair of its position and (partial/full) attitude when the rigid body model is assumed. Concerning the agents position, in the rest of the paper the non-degenerate case is usually considered.
Definition III.2 (Non-Degenerate Formation).
A -agent formation () modeled as a framework in is non-degenerate if the agents are not all collinear, namely if the matrix of the coordinates describing their positions is of rank greater than 1.
For a given formation, the bearing rigidity properties are related to the agents sensing capabilities. For this reason, we introduce the bearing measurements domain . According to the framework formation model in Def. III.1, any edge represents the bearing measurement recovered from the -th agent which is able to sense the -th agent, . The set of the available measurements is related to the framework configuration according to the following definition where an arbitrary edges labeling is introduced.
Definition III.3 (Bearing Rigidity Function).
Given a -agent formation () modeled as a framework in , the bearing rigidity function is the map
where the vector stacks all the available bearing measurements.
Hereafter, the framework model is adopted to refer a -agents formation and the two concepts are assumed to be equivalent. Moreover, we always suppose that .
Iii-B Static Rigidity Properties
Def. III.3 allows to introduce the first two notions related to the bearing rigidity theory, namely the equivalence and the congruence of different frameworks.
Definition III.4 (Bearing Equivalence).
Two frameworks and are bearing equivalent (BE) if .
Definition III.5 (Bearing Congruence).
Two frameworks and are bearing congruent (BC) if , where is the complete graph associated to .
Two frameworks defining by the same graph (and different configurations) are BE if they are characterized by the same set of bearing measurements for the interacting agents, i.e., for all . On the contrary, they are BC when the bearing measurements are the same for each pair of agents in the formation, namely for all . Accounting for the preimage222Let be a function. Let , and . Then is called the image of under , and is called preimage of under . under the bearing rigidity function (4), the set includes all the configurations such that is BE to , while the set contains all the configurations such that is BC to . Trivially, it holds that .
The definition of these sets is needed to introduce the (local and global) property of bearing rigidity.
Definition III.6 (Bearing Rigidity in ).
A framework is (locally) bearing rigid (BR) in if there exists a neighborhood of such that
Definition III.7 (Global Bearing Rigidity in ).
A framework is globally bearing rigid (GBR) in if every framework which is BE to is also BC to , or equivalently if .
A framework is thus bearing rigid whether any framework , which is BE to with sufficiently closed to , is also BC to . Fig. 1 provides a graphical interpretation of condition (6) highlighting the relation between the sets and . The requirement of “closeness” in the configurations space is missed in Def. III.7 of global bearing rigidity. As a consequence, this property results to be stronger than the previous one as proved in the next theorem.
A GBR framework is also BR.
For a GBR framework , it holds that . Consequently, condition (6) is valid for demonstrating that the framework is BR. ∎
Iii-C Dynamic Rigidity Properties
All the properties previously defined concern rigidity for static frameworks. Nevertheless, in real-world scenarios agents belonging to a formation are generally required to be able to move in order to accomplish global tasks, such as exploring and mapping of harsh environments and monitoring areas of interest. For this reason, in this section we assume to deal with dynamic agent formations, i.e., frameworks where the configuration can change over time, namely , while the agents interaction is fixed, namely the topology can not vary. Our aim is to identify the constraints under which a given dynamic formation is able to move by maintaining its rigidity, i.e., by preserving the existing bearing measurements among the agents.
We thus introduce the instantaneous variation vector that represents a deformation of the configuration taking place in an infinitesimal time interval. This vector belongs to the instantaneous variations domain whose identity depends on the considered metric space , i.e., on the space of agent controllable variables. The introduction of allows to describe the bearing measurement dynamics in terms of configuration deformations. The relation between and the time derivative of the bearing rigidity function, clarified in the next definition, constitutes the starting point for the analysis of the rigidity properties of a given dynamic formation.
Definition III.8 (Bearing Rigidity Matrix).
For a given (dynamic) framework , the bearing rigidity matrix is the matrix that satisfies the relation
The dimensions of the bearing rigidity matrix depend on the spaces and . Nevertheless, one can observe that the null space of always identifies all the (first-order) deformations of the configuration that keep the bearing measurements unchanged. From a physical perspective, such variations of can be considered as sets of command inputs to provide to the agents to instantaneously drive the formation from the initial state to a final state belonging to .
Definition III.9 (Infinitesimal Variation).
For a given (dynamic) framework , an infinitesimal variation is an instantaneous variation that allows to preserve the relative directions among the interacting agents.
For a given (dynamic) framework , an infinitesimal variation is an instantaneous variation such that .
On the other hand, when the bearing measurements remain unchanged for each pair of agents in the formation () the shape uniqueness is guaranteed.
Definition III.10 (Trivial Variation).
For a given (dynamic) framework , a trivial variation is an instantaneous variation such that shape uniqueness is preserved.
For a given (dynamic) framework , a trivial variation is an instantaneous variation such that , where is the bearing rigidity matrix computed for the complete graph associated to .
Given a (dynamic) framework and denoting as the complete graph associated to , it holds that
Since each edge of the graph belongs to the graph , the equations set constitutes a subset of the equations set . Then implies . ∎
Condition (8) is fundamental for the next definition that constitutes the core of the rigidity theory.
Definition III.11 (Infinitesimal Bearing Rigidity in ).
A (dynamic) framework is infinitesimally bearing rigid (IBR) in if
Otherwise, it is infinitesimally bearing flexible (IBF).
A framework is IBR if all its infinitesimal variations are also trivial. Contrarily, a framework is IBF if there exists at least an infinitesimal variation that warps the configuration in .
It is notable how the trivial variation assumes a specific physical meaning when the non-degenerate formation case is specified according to the domain of interest, as detailed in the following. This leads to a characterization of the dimension of that is exploited to derive a (necessary and sufficient) condition to check whether a given framework is IBR.
In the rest of the paper we investigate the rigidity properties of formation characterized by a specific metric space . In detail, we limit our analysis to the dynamic frameworks case, however the time dependency is dropped out to simplify the notation.
Iv Bearing Rigidity Theory in
In this section we focus on dynamic (non-degenerate) formations of agents controllable in , , and endowed with bearing sensing capabilities. This is, for example, the case of a team of mobile sensors interacting in a certain (two-dimensional or three-dimensional) area of interest. We aim at describing the rigidity properties of these multi-agent structures recasting the results provided in [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 17] within the generic context described in Sec. III.
Iv-a Formation Description
Each element of a formation composed of agents acting in the metric space can be modeled as a particle point whose state is defined by its (controllable) position , with , in the global inertial frame that is assumed to be known by all the agents333Note that two agents can not have the same position, hence the agents non-overlapping condition for stands. Moreover, because of non-degenerate formation assumption, for each -th component of the position vectors () it also holds for any .. We also suppose that neighboring agents are able to reciprocally recover bearing measurements, namely agent interactions are assumed bidirectional, supporting the particle point choice.
Adopting the framework formation model introduced in Sec. III-A, a formation in can thus be described by the pair , where the configuration is associated to the position vector and the graph is undirected.
Let consider an arbitrary orientation for obtaining an oriented graph. The bearing measurement associated to the (directed) edge results to be
where , and . Note that , namely any orientation for entails the same amount of bearing information, and that .
where , introduced in Sec. II, is obtained from the incidence matrix of the (oriented) graph . Note that the non-degenerate formation assumption in this case translates into a condition about the non-collinearity of the bearing measurements, thus ensuring that the bearing rigidity function is such that where the vector identifies a direction in and for .
Iv-B Rigidity Properties
To characterize the rigidity properties of a formation in , it is necessary to derive a suitable expression of its corresponding rigidity matrix (Def. III.8).
To this end, note that each agent in is characterized by translational degrees of freedom (tdofs) as its position can vary over time in a controllable manner. In this perspective, the instantaneous variation vector introduced in Sec. III-C can be selected as
Thus the variation domain coincides with and the selection (12) corresponds to assuming a first-order model for the agents dynamics.
Furthermore, using (10), we observe that the dynamics of the bearing measurements depends on the position variation of the interacting agents. Indeed, it holds that
One can observe that the matrix (14) coincides with the gradient of the bearing rigidity function along the positions vector , i.e., .
According to Lemma III.2, the null space of the bearing rigidity matrix (14) allows to identify the infinitesimal variations of . However, because of Lemma III.3 and Def. III.11, to check the infinitesimal rigidity of the framework is necessary to account also for its trivial variations, namely to study the null space of the bearing rigidity matrix computed by considering the complete graph associated to . Given a (non-degenerate) -agents formation , in the following we prove that its trivial variation set coincides with the -dimensional set
describing the (instantaneous) translation and uniform scaling of the entire configuration .
Lemma IV.1 (Lemma 4 in ).
For a framework in , it holds that
We prove that all the vectors spanning belong to . From (14), for any graph , one has that . Furthermore, since for each pair of non-coincident vertices , , then also for any graph . Choosing proves the thesis. ∎
Lemma IV.1 holds also for degenerate frameworks and implies for both degenerate and non-degenerate cases. This preliminary result is exploited in the proof of the next statement where the attention is restricted to the cases of (non-degenerate) formations with , which are of interest for real world scenarios.
For a non-degenerate framework , with and , it holds that
The bearing rigidity matrix associated to has the form
where is the block corresponding to the edge .
For , this explicitly is
with . For each edge , we consider only one opportunely scaled row of in (18), obtaining the matrix
This has the same rank of but lower dimensions, so hereafter, we consider instead of and we prove thesis by induction on the number of agents in the formation.
We aim at proving that . To do so, we observe that
is full-rank if there not exist such that , i.e., whether the agents are not collinear. Because of non-degenerate formation hypothesis the thesis is thus proved.
Note that, given a set of agents, for each subset containing elements, it is possible to partition so that
where the first block has rows related to the edges incident to the first agents, while the second block has rows related to the edges connecting the -th agent with the first agents.
For inductive hypothesis the thesis holds for , i.e., . Exploiting this fact we aim at showing that for we get .
For the inductive hypothesis, the first block of in (22) contains linearly independent rows. Moreover, there are at least two agents, for instance the -th and -th agent, that are not aligned with the -th agent, hence it does not exist such that and the rows related to the edges , and are linearly independent w.r.t. the rows of the first block. has thus at least linearly independent rows, and, since for Lemma IV.1, then it must be .
For (19) is substituted by
where are the (scalar) components of vector along the -axis, -axis, and -axis of the global inertial frame, respectively. The proof for this case thus follows the same inductive reasoning performed for . ∎
Theorem IV.3 (Condition for IBR, Thm. 4 in ).
A non-degenerate framework in is IBR if and only if
V Bearing Rigidity Theory in
In this section we deal with (non-degenerate) formations whose agents state is defined accounting for the manifold () where is the unit circle. This is, for instance, the case of teams of unicycle-modeled ground robots () or of standard under-actuated quadrotors () whose controllable variables are the position and the steering/yaw angle. The rigidity properties of these systems have been studied in [41, 42, 43, 44, 40] assuming that the agents are equipped with bearing sensing capabilities (e.g., with on-board cameras). In the following we recall the main results about bearing rigidity theory in particularized the concepts introduced in Sec. III.
V-a Formation Description
Let consider a team of -agents () whose state is depicted by an element of the Cartesian product where (real-world scenarios). We assume that each agent is equipped with an on-board camera that allows to recover bearing measurements w.r.t. some neighbors.
Under these premises, each agent can be modeled as a rigid body having a certain position and orientation w.r.t. the global inertial frame . In detail, we assume that each -th agent, , is associated to a local reference frame whose origin coincides with the agent center of mass (com) while the -axis is parallel to the direction of the focal axis of the camera bearing sensor. The introduction of allows to characterize the (controllable) agent state . This coincides with the pair composed of the vector , that indicates the position of in , and the angle , that specifies the orientation of w.r.t. .
It can be proved that is isomorphic to the interval and also to the two-dimensional Special Orthogonal group
, namely to the set of length-preserving linear transformations inwhose matrix representations have unitary determinant. Hence, a rotation on a plane can be equivalently described by a rotation matrix in or by an angle in .
When we consider a formation on a plane, i.e., for , the orientation of each agent is (completely) specified by an angle , , that is univocally associated to a rotation matrix , i.e., . When we account for the 3D case (), instead, the single angle specifies the -th agent orientation only along a single direction in . In this case, the matrix , belonging to the three-dimensional Special Orthogonal group, denotes the rotation of angle around the arbitrary unit vector that identifies the unique controllable direction in .
According to the discussion in Sec. III-A, the described formation can be modeled as a framework embedded in . In this case, it results that and we can distinguish the position vector and the orientation vector . In addition, because of the mutual visibility constraints deriving from the use of cameras as bearing sensors, the sensing capabilities are not necessarily reciprocal between pair of agents. Moreover, we assume that agents do not have access to the global coordinate system, so the gathered measurements are inherently expressed in the local frames. These facts entail that the graph is assumed to be directed.
Given these premises, the directed edge refers to the bearing measurement of the -th agent obtained by the -agent. Although measured in the -th agent local frame, this can be expressed in terms of the relative position and orientation of the agents in the inertial frame, namely
where is the normalized relative position vector introduced in Sec. IV-A and is the rotation matrix that describe the orientation of w.r.t. . Note that as in the previous case.
V-B Rigidity Properties
To evaluate the infinitesimal rigidity properties of a formation modeled by a framework in , one can observe that each agent belonging to this system is characterized by tdofs and only one rotational dof (rdof) that are assumed to be independently controllable. Hence, the instantaneous variation vector belonging to results from the contribution of two components related to variation of the position and of the orientation, namely where
Note that, for , the variation of the angle corresponds to the variation of the -th agent orientation on the plane. For , instead, it identifies a variation of the -th agent orientation only along the direction determined by , according to Rmk. 3.
To determine the rigidity matrix, we focus on the time derivative of the generic bearing measurement in (24). For , this results to be
where with . For , the expression of is more complex involving the skew-symmetric of the unit rotation vector , i.e.,
and , are derived from .
Note that the two matrix blocks in (30) correspond to the gradients of the bearing rigidity function along vectors and , i.e., to and respectively. In addition, evaluating (30), we can state the next theorem about the instantaneous infinitesimal variations of a framework in .
Theorem V.1 (Thm. III.7 in ).
Given a non-degenerate framework in , with and , any infinitesimal variation satisfies the condition
being and related to the graph .
Due to Def. III.9 any infinitesimal variations satisfies the condition . Pre-multiplying this relation for with , for each edge