Rigidity and Persistence of Three and Higher

Dimensional Formations

Julien M. Hendrickx

, Barıs¸ Fidan

, Changbin Yu

, Brian D. O. Anderson

and

Vincent D. Blondel

Department of Mathematical Engineering, Universit

e Catholique de Louvain,

Avenue Georges Lemaitre 4, B-1348 Louvain-la-Neuve, Belgium

National ICT Australia Ltd. and

Research School of Information Sciences & Engineering., the Australian National University,

216 Northbourne Ave., Canberra ACT 2601 Australia

Abstract. In this paper, we generalize the notion of persistence, which has been

originally introduced for two-dimensional formations, to ℜ

for d ≥ 3, seeking

to provide a theoretical framework for real world applications, which often are

in three-dimensional space as opposed to the plane. We verify that many of the

properties of rigid and/or persistent formations established in ℜ

are also valid for

higher dimensions. Analysing the closed subgraphs and directed paths in persis-

tent graphs, we derive some further properties of persistent formations. We also

provide an easily checkable necessary condition for persistence.

1 Introduction

Multi-agent systems have attracted considerable attention recently as witnessed by ex-

plosion of papers in the area (see for example [1–4]). Agents, abstracted as vertices

of graphs in this paper (following [5,6]), can be thought as any autonomous agents

including combat robots, underwater vehicles, unmanned aerial vehicles, and ground

vehicles.

Many control tasks for point-agent systems involve maintaining of the distance be-

tween nominated pairs of agents. For such tasks, a graph can naturally be used to depict

the control architecture as follows: To each agent corresponds a vertex, and for each

⋆ ⋆ ⋆

The work of Julien M. Hendrickx and Vincent D. Blondel is supported by the Belgian Pro-

gramme on Interuniversity Attraction Poles initiated by the Belgian Federal Science Policy

Ofﬁce, and by the Concerted Research Action (ARC) “Large Graphs and Networks” of the

French Community of Belgium. The scientiﬁc responsibility rests with its authors. Julien M.

Hendrickx is a FNRS (Belgian Fund for Scientiﬁc Research) fellow.

†

The work of Barıs¸ Fidan, Changbin Yu, and Brian D.O. Anderson is supported by National ICT

Australia, which is funded by the Australian Government’s Department of Communications,

Information Technology and the Arts and the Australian Research Council through the Back-

ing Australia’s Ability Initiative. Changbin Yu is an Australia-Asia Scholar supported by the

Australian Government’s Department of Education, Science and Training through Endeavours

Programs.

M. Hendrickx J., Fidan B., Yu C., D. O. Anderson B. and D. Blondel V. (2005).

Rigidity and Persistence of Three and Higher Dimensional Formations.

In Proceedings of the 1st International Workshop on Multi-Agent Robotic Systems, pages 39-46

DOI: 10.5220/0001193600390046

 SciTePress

agent (vertex) pair i, j there is a directed edge

−−→

(i, j) from i to j if i has a constraint on

the distance it must actively maintain from j.

In the recent control literature, the characterization of a system of the above type has

started to be attempted using the notion of rigidity of a directed graph [1, 5], which is

called persistence of a directed graph [6] as well. In this paper, we prefer to use the term

persistence in order to distinguish it from the undirected notion of rigidity. In Section

2, formal deﬁnition of persistence given in [6] is generalized to ℜ

for d ≥ 3, seeking

to provide a theoretical framework for real world applications, which often are in 3-

dimensional space as opposed to the plane. This deﬁnition has the following intuitive

meaning: a graph is persistent if, provided that all the agents are trying to satisfy their

distance constraints, the global structure of the formation is preserved, i.e. when the

formation moves, it necessarily moves as a cohesive whole. We will see that rigidity

of the underlying undirected graph is a necessary but not sufﬁcient condition. This

will lead us to the notion of constraint consistence of graph, which is the additional

condition for a rigid graph to be persistent. Intuitively, a graph is constraint consistent

if every agent is able to satisfy all its distance constraints provided that all the others

are trying to do so. We will then show that a graph is persistent if and only if it is rigid

and constraint consistent.

In Section 3, we generalize some of the main properties of persistent graphs to 3-

or high dimensional graphs, drawing on established results in ℜ

In Section 4, we reason about the persistence of closed subgraphs of persistent

graphs and use this reasoning to analyze the directed paths in persistent graphs. As

results of this analysis, we present some further properties of persistent graphs and an

easily checkable necessary condition for persistence.

The paper is ended with the concluding remarks in Section 5. Note that all the

proofs in the paper are omitted because of space limitations. However, a full version

of this work together with the companion paper [7] is available in preprint from the

authors.

2 Rigidity and Persistence

In [6], rigidity, persistence, and some other related notions have been deﬁned for di-

rected graphs in ℜ

. In this section, we generalize these deﬁnitions to be applicable

for any space ℜ

, d ∈ {1, 2, 3, . . .}. Some of the terms we use such as “rigidity” are

undirected notions, i.e., notions that are deﬁned for undirected graphs. These notions,

however, apply to directed graphs as well, e.g., we call a directed graph rigid iff its un-

derlying undirected graph is rigid. Note that, in directed graphs, rigidity and the other

undirected notions are not affected by modiﬁcation of the edge directions.

In ℜ

(d ∈ {1, 2, 3, . . .}), a representation of an undirected graph G = (V, E) with

vector set V and edge set E is a function p : V → ℜ

. We say that p(i) ∈ ℜ

is the

position of the vertex i, and deﬁne the distance between two representations p

and p

of the same graph by

d(p

, p

) = max

i∈V

||p

(i) − p

(i)|| .

A distance set

d for G is a set of distances d

> 0, deﬁned for all edges (i, j) ∈ E.

A distance set is realizable if there exists a representation p of the graph for which

||p(i) − p(j)|| = d

for all (i, j) ∈ E. Such a representation is then called a realiza-

tion. Note that each representation p of a graph induces a realizable distance set (deﬁned

by d

= ||p(i) − p(j)|| for all (i, j) ∈ E), of which it is a realization.

A representation p is rigid if there exists ǫ > 0 such that for all realizations p

′

the distance set induced by p and satisfying d(p, p

′

) < ǫ, there holds ||p

′

(i) − p

′

(j)|| =

||p(i) − p(j)|| for all i, j ∈ V . (We say in this case that p and p

′

are congruent). A

graph is said to be generically rigid if almost all its representations are rigid. Note that

the reasons for which we only require almost all representations to be rigid instead of

all of them will be detailed in Remark 1.

As mentioned above, rigidity is an undirected notion, and is therefore insufﬁcient to

characterize persistence. As noted in [6], rigidity of a representation only means that if

an external observer (or some physical properties) makes sure that the distance between

the positions of any pair of vertices connected by an edge remains constant, then all the

sufﬁciently close realizations of the induced distance set are congruent to each other.

But, in a typical system of autonomous agents, there is no such external observer. Each

agent is only aware of its own distance constraints, and can “move freely” as long

as these particular constraints are satisﬁed. For example, agents that have only one

constraint can move along a hyper-sphere centered on the position of the only other

agent of which they are aware. So, it could happen that because one agent i is moving

on such a hyper-sphere, it becomes impossible for another agent j to satisfy all its

constraints, especially if j has d + 1 or more constraints. So, in order to have a more

formal deﬁnition of persistence, we ﬁrst need to characterize the fact that each agent is

trying to keep the distances from its neighbors constant.

Let us thus ﬁx a directed graph G = (V, E) depicting a point-agent system archi-

tecture, where each agent corresponds to a vertex in V , and for each agent (vertex) pair

i, j there is a directed edge

−−→

(i, j) ∈ E from i to j if i has a constraint on the distance

it must actively maintain from j. Let us also ﬁx desired distances d

> 0, ∀

−−→

(i, j) ∈ E

and a representation p. We say that the edge

−−→

(i, j) ∈ E is active if ||p(i) − p(j)|| = d

We also say that the position of the vertex i ∈ V is ﬁtting for the distance set

d if it is

not possible to increase the set of active edges leaving i by modifying the position of

i while keeping the positions of the other vertices unchanged. More formally, given a

representation p, the position of vertex i is ﬁtting if there is no p

∗

∈ ℜ

for which the

following strict inclusion holds:

{

−−→

(i, j) ∈ E : ||p(i) − p(j)|| = d

} ⊂ {

−−→

(i, j) ∈ E : ||p

∗

− p(j)|| = d

} (1)

This condition intuitively means that the agent i cannot satisfy additional distance con-

straints without breaking some that it already satisﬁes, as shown in the two-dimensional

example in Figure 2, which is drawn from [6]. A representation of a graph is a ﬁtting

representation for a certain distance set

d if all the vertices are at ﬁtting positions for

Note that any realization is a ﬁtting representation for its distance set.

We can now give a formal deﬁnition of persistence: A representation p is persistent

if there exists ǫ > 0 such that every representation p

′

ﬁtting for the distance set induced

(a) (b)

Fig.1. Suppose that d

= d

= c. The position of 4 in (a) is not ﬁtting because it only

makes

−−−→

(4, 1) active while there exists a position that would make both

−−−→

(4, 1) and

−−−→

(4, 3) active. On

the other hand, its position in (b) is ﬁtting because no point can be at the same time at a distance

c from 1, 2 and 3.

by p and satisfying d(p, p

′

) < ǫ is congruent to p. A graph is then generically persistent

if almost all its representations are persistent.

One can show that a generically persistent graph is always generically rigid. A suf-

ﬁcient condition for a generically rigid graph to be generically persistence is the generic

constraint consistence. A representation p is constraint consistent if there exists ǫ > 0

such that any representation p

′

ﬁtting for the distance set

d induced by p and satisfying

d(p, p

′

) < ǫ is a realization of

d. Intuitively, the constraint consistence of a representa-

tion means that if each agent tries to satisfy its distance constraints (i.e., is at a ﬁtting

position), then all the distance constraints will be satisﬁed, or equivalently, no agent

will be in a situation where it cannot satisfy some constraint. The illustration of such a

situation in ℜ

can be found in [6]. Again, we say that a graph is generically constraint

consistent if almost all its representations are constraint consistent.

We have the following useful equivalences for directed graphs in any d-dimensional

space ℜ

(d ∈ {1, 2, 3, . . .}), which have already been established for ℜ

in [6].

Theorem 1 A representation in ℜ

(d ∈ {1, 2, 3, . . .}) is persistent iff it is rigid and

constraint consistent. A graph in ℜ

(d ∈ {1, 2, 3, . . .}) is generically persistent iff it is

generically rigid and generically constraint consistent.

Remark 1 In our deﬁnitions of generic rigidity, persistence and constraint consistence,

a graph has a generic property if almost all its representations have the property. Some

discussions on using “generic” and “almost all” can be found in [6,8]. One reason

for using these terms, in ℜ

(d ∈ {1, 2, 3, . . .}), is to avoid the problems arising from

having d+1 or more vertices lying on a d

-dimensional hyper-surface where d

≤ d−1.

In the sequel, we avoid the use of “generic”.

3 Characterization of Persistent Graphs

In this section, we examine the properties of persistent graphs in three and higher di-

mensions, and present the fundamental results related to persistence. These results are

comparable to the properties of two-dimensional persistent graphs presented in [6], and

hence the corresponding propositions and lemmas are given as generic ones that are

applicable to ℜ

as well.

We begin the characterization of persistent graphs by giving a lower bound on the

number of active edges, and a sufﬁcient condition for a graph to be constraint consistent.

In the sequel, d

−

(i) and d

(i) designate respectively the in- and out-degree of the

vertex i.

Lemma 1 Let i be a vertex of a graph G = (V, E). For almost all representations

p of G, there exists ǫ > 0 such that in every representation p

′

∈ B(p, ǫ) (i.e., such

that d(p, p

′

) < ǫ) ﬁtting for the distance set induced by p, the number of active edges

leaving i is at least min (d, d

(i)). Consequently, a graph in which all the vertices have

an out-degree smaller than or equal to d is always constraint consistent.

The next proposition which is the generalization of Propositions 1 and 2 in [6] for

any arbitrary dimension d ∈ {1, 2, 3, . . .}, allows us to delete edges in a persistent

(constraint consistent) graph and maintain persistence (constraint consistence).

Proposition 1 A persistent graph in ℜ

(d ∈ {1, 2, 3, . . .}) remains persistent after

deletion of any edge

−−→

(i, j) for which d

(i) ≥ d + 1. Similarly, a constraint consistent

graph in ℜ

(d ∈ {1, 2, 3, . . .}) remains constraint consistent after deletion of any edge

−−→

(i, j) for which d

(i) ≥ d + 1.

An interesting corollary of Proposition 1 concerns the total number of degrees of

freedom. The number of degrees of freedom (DOF count) of a vertex is the maximal

dimension, over all representations of the graph, of the set of possible ﬁtting positions

for this vertex. For example, in ℜ

, the DOF counts of the vertices with zero, one, and

two out-degrees are respectively 3, 2, and 1; all the other vertices have zero DOF. The

following result provides a natural bound on the total DOF count, i.e., the sum of the

vertex DOF counts of a persistent graph.

Corollary 1 The total DOF count of a persistent graph in ℜ

(d ∈ {1, 2, 3, . . .}) can

at most be d(d + 1)/2.

Remark 2 There are d-dimensional persistent graphs having a total DOF count less

than d(d + 1)/2. Figure 2 shows a three-dimensional persistent graph each vertex of

which has 1-DOF. The total DOF count for this example is 5, i.e., less than 3(3 + 1)/2.

As stated in Proposition 1, a persistent graph remains persistent after deletion of any

edge

−−→

(i, j) for which d

(i) ≥ d + 1. After successive deletions, we can thus reach in

this way a persistent graph whose vertices all have an outgoing degree that is smaller

than or equal to d. In the next theorem, which is analogous to Theorem 3 in [6] stated

for ℜ

, we see that a graph is persistent if and only if all the graphs obtained in this way

are rigid. This criterion allows us to note that a graph obtained by adding an edge to a

persistent graph is not necessarily persistent, as shown on the example in Figure 3.

Theorem 2 A d-dimensional graph is persistent if and only if all those subgraphs are

rigid which are obtained by successively removing outgoing edges from vertices with

out-degree larger than d until all such vertices have an out-degree equal to d.

Fig.2. A persistent graph in ℜ

with all the vertices having out-degree 2 and hence 1-DOF.

Theorem 2 provides a non-polynomial time algorithm to check the persistence of

any d-dimensional graph for d ∈ {1, 2, 3, . . .}: It is sufﬁcient to check the rigidity of all

subgraphs obtained by deleting the edges leaving vertices with out-degree larger than or

equal to d + 1 until all the vertices have an out-degree less or equal to d. An algorithm

with a smaller complexity would be useful in the case of large graphs, especially if there

is a high number of vertices with high out-degrees, but no such algorithm is currently

available. More discussions on determining the persistence of two-dimensional directed

graphs in polynomial time can be found in [6]. Moreover, [7] presents results leading

to a quadratic time algorithm for the case d = 3 for cycle-free graphs, which can be

generalized easily to any d ∈ {1, 2, 3, . . .}.

(a) (b) (c)

Fig.3. The two-dimensional graph represented in (a) can be obtained by adding an edge to the

persistent graph (b). However, by Theorem 2, it is not persistent because the subgraph represented

in (c) is not rigid. In the corresponding multi-agent system, this could arise from a combination

of unfortunate information architecture selections for the three agents of the cycle.

4 Closed Subgraphs and Directed Paths

In this section, we focus on the directed paths in persistent graphs and analyze some

related properties. As a part of this analysis, we reason about the persistence of closed

subgraphs of persistent graphs. An important outcome of our analysis will be an easily

checkable necessary condition (Proposition 3) for persistence. Note that the proofs of

many results concerning closed subgraphs rely on the properties of minimally persistent

graphs, i.e., persistent graphs for which no single edge can be removed without losing

persistence. These properties were omitted here due to space limitation, but a extended

studies of minimal persistence and its connections with the analogous notion of mini-

mal rigidity can be found in [6] or in the full version of this work, available in preprint

from the authors.

Consider a directed graph G = (V, E) in ℜ

(d ∈ {1, 2, 3, . . .}) and a subgraph

′

= (V

′

, E

′

) of G. G

′

is called a closed subgraph of G if for any vertex i ∈ V

′

−−→

(i, j) ∈ E implies j ∈ V

′

and

−−→

(i, j) ∈ E

′

. From the perspective of autonomous agent

formations, the agents corresponding to the vertices of V

′

are unaware of the existence

of those of V \ V

′

. We call then total DOF count of V

′

with respect to G the sum of

the DOF counts of all the vertices of V

′

. Using these notions, we reach the following

proposition.

Proposition 2 Let G = (V, E) be a persistent graph in ℜ

(d ∈ {1, 2, 3, . . .}) having

at least d vertices. If a vertex v ∈ V belongs to a closed subgraph of G containing

less than d vertices, the out-degree of v has to be smaller than or equal to d − 2. For

the set V

of all such vertices v, we have |V

| ≤ d − 1. On the other hand, any vertex

′

∈ V that does not belong to any closed subgraph of G having less than d vertices

is connected by a directed path (leaving v

′

) to all the vertices of G with positive DOF

count.

Remark 3 Given a directed graph G = (V, E) in ℜ

(d ∈ {1, 2, 3, . . . }), for any

vertex i ∈ V , there exists a certain closed subgraph of G containing i, which we call

the reachability subgraph of G for i. For a given i ∈ V , the reachability subgraph of

G for i can be formally deﬁned as the subgraph G

′

= (V

′

, E

′

) of G where V

′

is the

set of all the vertices that can be reached from i (including the vertex i) by a directed

path in G and E

′

⊆ E is the set of all the edges e such that e joins a pair of vertices in

′

. It is easy to see that the reachability subgraph of G for the vertex i is the smallest

closed subgraph of G containing i, and equivalently, the intersection of all the closed

subgraphs of G containing i.

The following corollary, which immediately follows from Proposition 2, gives a

more explicit criterion to check the existence of a directed path between given two

vertices of a persistent graph in ℜ

, one of which has a positive DOF count.

Corollary 2 Let G = (V, E) be a persistent graph in ℜ

having at least 3 vertices. Any

vertex i is connected by a directed path (leaving i) to all the vertices of G with positive

DOF count unless one of the following holds:

1. i is a ﬁrst leader, i.e., d

(i) = 0.

2. i is a ﬁrst follower, i.e., d

(i) = 1 and there exists a j ∈ V such that

−−→

(i, j) ∈ E

and d

(j) = 0.

Using the above propositions about closed subgraphs and directed paths in persis-

tent graphs, we reach the following easily checkable necessary condition for persis-

tence. The detailed derivation of the criterion can be found in the full version of this

work, which is available in preprint from the authors.

Proposition 3 Let G = (V, E) be a persistent graph in ℜ

(d ∈ {1, 2, 3, . . .}) with at

least d vertices. Then all the closed subgraphs of G having more than d− 1 vertices are

persistent.

In ℜ

, Proposition 3 leads to the following corollary.

Corollary 3 Let G be a persistent graph in ℜ

with at least three vertices. Then any

closed subgraph G

′

of G is persistent unless G

′

consists of two disconnected vertices.

In other words, G has a non-persistent closed subgraph if and only if it contains two

vertices each having 3 DOFs.

5 Concluding Remarks

In this paper, we have generalized the notion of persistence given in [6] for 2-dimensional

directed graphs to dimensions higher than two, seeking to provide a theoretical frame-

work for real world applications, which often are in 3-dimensional space. We have veri-

ﬁed that many of the properties already established for persistent graphs in ℜ

are valid

for higher dimensions as well. We have also analyzed the directed paths in persistent

graphs, exposed some further properties of such graphs in three or higher dimensions,

and given an easily checkable necessary condition (Proposition 3) for persistence. In

the companion paper [7], we analyze the “partial equilibrium problem”, a problem ob-

served in some persistent formations associated with feasibility of satisfying all the

constraints on all the agents simultaneously. We provide some criteria to check whether

a given persistent graph suffers from the partial equilibrium problem.

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