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Structural Persistence of Three Dimensional
Autonomous Formations
Changbin Yu1, Julien M. Hendrickx2, Barıs¸ Fidan1, Brian D.O. Anderson1
1National ICT Australia Ltd. and
Research School of Information Sciences & Engineering, the Australian National University,
216 Northbourne Ave., Canberra ACT 2601 Australia
<brian.anderson,baris.fidan,brad.yu>@nicta.com.au? ? ?
2Department of Mathematical Engineering, Universit´
e Catholique de Louvain,
Avenue Georges Lemaitre 4, B-1348 Louvain-la-Neuve, Belgium
Hendrickx@inma.ucl.ac.be†
Abstract. Built upon a recently developed theoretical framework, we consider
some practical issues raised in multi-agent formation control in three dimensional
space. We introduce the partial equilibrium problem, which is associated with
unsafe control of a formation in practical 3-dimensional applications. We define
structurally persistent graphs, a class of persistent graphs free of any partial equi-
librium problem. In real deployment of control of multi-agent systems, forma-
tions with underlying structurally persistent graphs are of interest. We study the
connections between the allocation of degrees of freedom (DOFs) across agents
and the characteristics of persistence and/or structural persistence of a directed
graph. We also show how to transfer degrees of freedom between agents, when
the formation changes with new agent(s) added, to preserve persistence and/or
structural persistence.
1 INTRODUCTION
In [1], we have generalized the definition of persistence to <dfor d≥3, seeking
to provide a theoretical framework for real world applications, which often are in 3-
dimensional space as opposed to the plane. We also have derived some new properties of
persistent graphs and given an operational criterion to determine if a graph is persistent.
In this paper, we demonstrate that a persistent formation, as defined in [1], may also
??? The work of Brian D.O. Anderson,Barıs¸ Fidan and Changbin Yu 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.
†The work of Julien M. Hendrickx is supported by the Belgian Programme 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.
2
suffer from a practical problem where each agent can move to a position which satisfies
the constraints on it once all the other agents are fixed but it is not possible to satisfy all
the constraints on all the agents at the same time.
In Section 2, we formally characterize the above problem, which we call the partial
equilibrium problem, and which is closely associated with unsafe control of a forma-
tion in practical 3-dimensional applications. We then introduce the definition of a struc-
turally persistent graph, a class of persistent graphs free of any partial equilibrium prob-
lem. In real deployment of control of multi-agent system, formations with underlying
structurally persistent graphs are of interest. It is established in Secction 2 incidentally
that in two dimensions, structural persistence and persistence are equivalent.
In Section 3, we focus on the connections between allocation of degrees of freedom
(DOFs) across agents and the characteristics of persistence and/or structural persistence
of a directed graph. We also show how to transfer degrees of freedom between agents,
when the formation changes with new agent(s) added, to preserve persistence and/or
structural persistence. We study cycle-free graphs in <3and show some more powerful
results that exist in this special case, such as the existence of a quadratic time criterion to
verify the cycle-free property and to decide persistence, which automatically guarantees
structural persistence.
We end the paper with concluding remarks in Section 4. Note that all the proofs are
omitted due to space limitations. However, a full version of this work together with the
campanion paper [1] is available in preprint from the authors.
2 PARTIAL EQUILIBRIUM PROBLEM AND
STRUCTURALLY PERSISTENT GRAPHS
Consider a persistent graph G= (V, E )in <d(d∈ {2,3, . . .}). The partial equilibrium
problem we want to avoid is the following: There is a subset ˜
V⊂Vof vertices such
that all the vertices in ˜
Vare at fitting positions whatever the positions of the vertices
in V\˜
Vare, but there exists no position assignment for the vertices in V\˜
Vsuch that
the whole representation is fitting. For example, consider the 3-dimensional persistent
graph ¯
Gshown in Figure 1, an associated set ¯
dof desired lengths dij >0for all the
edges −−→
(i, j), and a realization ¯pof ¯
din agreement with Figure 1. Identify ˜
Vwith {1,2}.
Since the vertices 1 and 2 have zero out-degrees, they are at fitting positions for any
representation of the graph, whatever the positions of 3, 4, 5 are. However, there are
representations of ¯
Garbitrarily close to ¯pwhere the vertices 3, 4, and 5 cannot be at
fitting positions at the same time. From the perspective of formations, in the formation
represented by ¯
G, there exist two leaders, 1 and 2, which are allowed to move freely in
<3without any constraint. This freedom, however, makes it impossible in some cases
for the agents 3, 4, and 5 to meet all the distance constraints on them, although ¯
Gis
persistent, according to the definition given in Section 3 of [1]. In such a case, we will
say that ¯
Gis in partial equilibrium.
The existence of such partial equilibrium problems in three and higher dimensional
spaces makes it necessary to analyse persistent graphs further and introduce new con-
cepts such as structural persistence that will be defined in this section. In <2, how-
ever, there is no persistent graph suffering from partial equilibrium problems, as ex-
3
1
2
3
4
5
Fig.1. A persistent graph in <3which is not structurally persistent.
plained later in Theorem 1. Let us consider a persistent graph G= (V, E )in <d
(d∈ {2,3, . . .}) with a representation p. Let ¯
dbe the set of distances corresponding
to p.Gis in partial equilibrium, and thus has the partial equilibrium problem if there
exists a non-empty vertex subset ˜
V⊂V, a constant ε > 0and a mapping p¯εindexed
by ¯ε,np¯ε:˜
V→ <d|0<¯ε≤εosuch that for any ¯ε≤εthe following hold:
1. d(p(i), p¯ε(i)) ≤¯ε,∀i∈˜
V.
2. For all i∈˜
V,p¯ε(i)is a fitting position with respect to ¯
d, irrespective of the posi-
tions of the vertices in V\˜
V.
3. There exist no fitting representation p0:V→ <din B(p, ¯ε)∆
=©¯p:V→ <d|d(p, ¯p)≤¯εª,
with respect to ¯
d, such that p0(i) = p¯ε(i),∀i∈˜
V.
If a persistent graph is not in partial equilibrium, it is called a structurally persistent
graph. To analyse this concept further, it is defined that for a given directed graph G=
(V, E ), a subgraph G0= (V0, E0)is a practically closed subgraph of Gif for any vertex
i∈V0,d+
G0(i)≥min{d, d+
G(i)}, where d+
G0(i)denotes the number of outgoing edges
incident to the vertex iof a graph G0. We remark that a closed subgraph is always a
practically closed subgraph, since each vertex of it satisfies the criterion defined above.
In the 2-dimensional example shown in Figure 2, where V0={1,2,3},G0is a
practically closed subgraph of Gbut not a closed subgraph of G. All the outgoing edges
of 1 and 2 in Gremain in the subgraph G0. Vertex 3, on the other hand, has two outgoing
edges in G0(making G0a practically closed subgraph) and another one not in G0. From
a perspective of formations where vertices denote agents and edges denote awareness,
in G, although 3 is aware of 4, it may not be able to react to correctly maintain its
distance from 4 because its position is locked by the constraints with respect to the
vertices 1 and 2.
The relation between partial equilibrium problems and practically closed subgraphs
is examined in the following propositions.
Proposition 1 Consider a persistent graph G= (V, E)in <d(d∈ {2,3, . . .}) with a
representation pand a set ¯
dof distances corresponding to p. Let G0= (V0, E0)be a
subgraph of Gwhere V0is a non-empty vertex subset of V. Then, G0is a practically
closed subgraph of Gif and only if there exist a constant ε > 0and a mapping p¯ε
indexed by ¯ε,©p¯ε:V0→ <d|0<¯ε≤ε , p¯εis the restriction of p →V0ªsuch that
for any ¯ε≤εthe following hold:
4
1
2
3
4
1
2
3
4
G G’
Fig.2. A practically closed subgraph in <2.
1. d(p(i),p¯ε(i)) ≤¯ε,∀i∈V0, for all p¯εin the mapping set.
2. For all i∈V0,p¯
ε(i)is a fitting position with respect to ¯
d, irrespective of the
positions of the vertices in V\V0.
Proposition 2 Consider a persistent graph G= (V, E)in <d(d∈ {2,3, . . .}) with a
representation pand a set ¯
dof distances corresponding to p.Gis structurally persistent
if and only if every non-empty practically closed subgraph of Gis persistent.
The following theorem, which uses Proposition 2, states that there is no partial
equilibrium problem in <2, as mentioned in the beginning of the section.
Theorem 1 Any persistent graph G∈ <2is structurally persistent and has all its
practically closed subgraphs persistent.
Proposition 2 states that a graph is not structurally persistent if and only if it contains
a practically closed subgraph that is not persistent. Development of this notion leads to
the following proposition, which gives another necessary and sufficient condition for a
persistent graph to be structurally persistent.
Proposition 3 Let G= (V , E)be a persistent graph in <d.Gis structurally persistent
if and only if there is no non-persistent closed subgraph of Gwith less than dvertices.
The following corollary, which is a major result of the section, and which immedi-
ately follows from Proposition 3, gives a more explicit necessary and sufficient condi-
tion for 3-dimensional persistent graphs to be structurally persistent. It also gives more
insight for the problem encountered in the example in Figure 1.
Corollary 1 A persistent graph G= (V, E)in <3is structurally persistent if and only
if there is at most one leader3in G.
Remark 1 For some dimensions d > 3, one can have a non-persistent closed subgraph
with less than dvertices in a graph that has only one leader. An example in <6is shown
in Figure 3. In general, in a given d-dimensional persistent graph, the presence of a
non-persistent closed subgraph can be checked by looking only at the vertices with out-
degree less than d−1, which are finite in number because of the bound on the total
DOF count.
5
6-DOF
5-DOF
5-DOF
5-DOF
Fig.3. A non-persistent closed subgraph in <6that does not have two leaders. Note that in <6,
the total DOF count of a persistent graph can be up to 21.
Remark 2 In contrast to the case d= 3, for d≥4, a non-persistent closed subgraph of
a persistent graph can be connected. Consider, for example, the 4-dimensional directed
graph G= (V, E )shown in Figure 4. Gis constraint consistent because no vertex has
an out-degree larger than 4. Moreover, it is minimally rigid and it can be obtained by
removing an edge from the complete graph K6, which is trivially rigid. On the other
hand, the closed subgraph G2of Gis non-rigid and hence non-persistent. Therefore, G
is not structurally persistent. Note that G2is connected although it is non-persistent.
all possible
edges from
G1 to G2
G1
G2
G
Fig.4. A 4-dimensional persistent graph with a non-persistent and connected closed subgraph.
3 ALLOCATION OF DOFs IN <3AND TRILATERATION
In Section 2, we have seen that for a directed graph, persistence is not enough to avoid
partial equilibrium problems. In the following subsections, we study how the way de-
grees of freedom happen to be allocated to the vertices of a directed graph is a key de-
terminant of the structural persistence or otherwise of that graph. In an application sce-
nario, this corresponds to giving/restricting the autonomy of certain agents (abstracted
as degrees of freedom of vertices) of the formation [2].
3An agent is a leader if it has no constraints on its movement, e.g. in Figure 1, both agents
represented by vertices 1 and 2 are leaders. In associated graphs, corresponding vertices have
no outgoing edges.
6
3.1 DOF Allocation and Transfer via Directed Trilateration in <3
In this section, we study the properties of the directed version of Henneberg-like vertex
addition in 3 and higher dimensions, which is an abstraction of the event that new
agents join a formation, one at a time. We give examples of applying such operations to
manipulate DOF allocation of persistent graphs, in particular, in <3.
Let us consider a persistent graph G= (V, E )in <d(d∈ {2,3, . . .})where |V| ≥
d. A directed d-vertex addition, DVA (d, n)where n∈ {0, . . ., d}, transforms Gto
another persistent graph G0= (V0, E0)where V0=V∪ { i},E0=E∪ {−−−→
(i, k) :
∀k∈V1} ∪ −−→
(j, i) : ∀j∈V2},V1, V2⊆V , V1∩V2=∅,|V1|=d−n , |V2|=n
, and DOF (j)≥1,∀j∈V2,4provided that the vertices of V1∪V2do not lie in any
q-dimensional hyperplane where q < d.
We note that from Lemma 2 of [1], constraint consistency is preserved with the
directed d-vertex addition defined above. Moreover, from the following lemma which
is drawn from [3, 4], we see that the rigidity is also preserved.
Lemma 1 [3, 4] A graph obtained by adding one vertex to a graph G = (V, E) in <d
and d edges from this vertex to other vertices of G is rigid if and only if G is rigid.
Hence by Theorem 2 of [1], the graph obtained after applying a directed d-vertex
addition on a persistent graph in <dis persistent, i.e., the d-directed vertex addition
defined above preserves the persistence of the graphs.
Remark 3 Consider a persistent graph G = (V, E) in <d. Let G’ = (V’, E’) be the graph
obtained by applying the operation DVA(d,n) to G, where V’=V ∪ {i}. Then we have:
DOFG0(i) = n;DOFG0(j)≤DOFG(j),∀j∈V. and DOFG(j)−DOFG0(j)∈
{0,1},∀j∈V.
In the remaining part of this section, we only consider <3, although results can
be easily expanded to higher dimensions. As a more convenient nomenclature in <3,
we use the term directed trilateration operation, abbreviated DT(·), DT(n) in place of
directed 3-vertex addition or DVA(3, n).
An undirected graph formed by a sequence of trilateration operations starting with
an initial undirected triangle, often called a trilateration graph, is guaranteed to be
generically rigid in <3and globally rigid in <2. A trilateration graph can always be con-
structed/deconstructed using a polynomial time algorithm, where a reverse trilateration
can be performed by removing a vertex with degree 3 at each step. Note that a seed with
3 vertices is needed to initiate a trilateration sequence. However, two different directed
triangular seeds can start a directed trilateration operation in <3as defined in Figure
5(a) and Figure 5(b) are called the leader-first follower-second follower (L−F F −S F )
and the balanced triangle (B1B2B3) seeds, respectively.
Remark 4 The leader-first follower-second follower seed is analogous to the leader-
follower structure defined for a 2-dimensional cycle-free graph [5]. The set of DOF
counts of the seed vertices is {3 ,2 ,1}. The balanced triangle is nothing more than a
directed triangle (cycle) in a cyclic graph and the corresponding DOF count set is {2
,2 ,2}.
4Non-existence of V2means the corresponding DVA(n) cannot be performed for the graph.
7
L FF
SF
B1 B2
B3
(a) (b)
Fig.5. The two directed triangular seeds.
Specifically in the application to 3 dimensional agent formations, note the meanings
of the DT(i) operation for different ican be interpreted as follows,
–DT(3) means election of a new leader.
–DT(2) may result in either breaking/restoring the balanced control structure, or
election of a new first-follower.
–DT(1) may also result in either breaking/restoring the balanced control structure at
more a detailed level, or creation/change of second follower.
–DT(0) preserves the control structure and no decision has to be made by pre-
existing agents.
Noting that in a 3-dimensional persistent graph, there are at most 6 DOFs (as op-
posite to 3 DOFs in the <2case) to be allocated among the vertices, we can list the
following six types of DOF allocation (abbreviated DOF allocation state S1to S6with
DOF counts of vertices):
–S1={3, 2, 1, 0, 0, . . .},S2={2, 2, 2, 0, 0,. . .}
–S3={3, 1, 1, 1, 0, 0, . . .},S4={2, 2, 1, 1, 0, 0,. . .}
–S5={2, 1, 1, 1, 1, 0, 0,. . .},S6={1, 1, 1, 1, 1, 1, 0, 0,. . .}
Further, we define a transient type of DOF assignment S0={3, 3, 0, 0, . . .}, which
can (only) be obtained by applying a DT(3) operation to S3.S0is named “transient”
because it apparently allows two leaders simultaneously in control of a formation, and
hence this creates instability and we want the DOF assignment to avoid this state. Recall
that the underlying directed graph of such a formation is a persistent graph with a partial
equilibrium problem, i.e. it is persistent but NOT structurally persistent (An example of
a graph that is in transient state S0can be seen from Figure 1).
We study the transformational relationship between the possible distribution of
DOFs by applying the appropriate DT(·) operation using the “state transition diagram”
shown in Figure 6. We have the following observations:
–Starting from any one of the two directed triangular seeds, we can build any graph
with any of S0-S6by adding at most three vertices using directed trilateration.
–Any desired DOF reallocation pattern(with no allocation to a specific vertex) can
be achieved by at most four directed trilaterations starting with any of the six types
of DOF allocation.
–Any desired stable DOF reallocation pattern can be achieved by at most three di-
rected trilaterations starting with any stable DOF allocation.
8
3, 2, 1, 0, 0… 2, 2, 2 , 0, 0…
3, 1, 1, 1, 0, 0 … 2, 2 , 1 ,1, 0 , 0…
2, 1, 1, 1,1, 0, 0…
1,1, 1 ,1 ,1,1, 0, 0…
3, 3, 0, 0 …
Legend:
DT(1)
DT(2)
DT(3)
Fig.6. The state transition diagram for directed trilaterations.
The observed results above gives an upper bound on the number of agents required
in order to perform a system reconfiguration operation, such as replacement or elimina-
tion of leaders/first-follower/second-follower, or change to balanced cooperative control
of 3 leaders. And it also gives the possible consequences in a closing ranks problem5,
where the lost agent has a certain positive number of DOFs .
3.2 Cycle-Free Graphs
Persistence of cycle-free graphs in <2was studied in [5]. In this section, we derive a
simple criterion to decide the persistence and structural persistence of cycle-free graphs
in <d. We also show an explicit way to build all the persistent cycle-free graphs.
Proposition 4 A graph obtained by adding one vertex to a graph G = (V, E) in <dand
at least d edges leaving this vertex is persistent if and only if G is persistent.
We thus know that a cycle-free graph obtained by successively adding vertices all
with out-degree d, i.e. DVA(d,0), to an initial seed of a cycle-free persistent graph con-
taining only dvertices is persistent.
Next,we focus on cycle-free persistent graphs in <3, which have important applica-
tion in safe control of multi-agent formations.
5The closing ranks problem for a given rigid formation which has lost a single agent, is to
find new links between some agent pairs which, if maintained cause the resulting formation to
again be rigid.
9
Theorem 2 A cycle-free graph in <3having more than 2 vertices is persistent if and
only if it has a closed subgraph which is the leader-first follower-second follower trian-
gle, and every other vertex has an out-degree larger than or equal to 3.
Moreover, every cycle-free persistent graph in <3can be obtained from an original
seed composed by the leader-first follower-second follower by adding vertices one by
one in the way described in Proposition 4, i.e., each vertex is added with every incident
edge outwardly directed.
We can also progressively deconstruct the cycle-free (persistent) graph by recur-
sively removing one vertex at a time, where that vertex has at least 3 outgoing edges
until we obtain the leader-first follower-second follower triangle. We conclude from
these observations that the computational complexity of verifying both persistence and
cycle-free properties of 3-dimensional graphs is quadratic in the number of vertices. In
other word, if the deconstruction process cannot proceed for the graph, then the graph
we are dealing with is not a persistent cycle-free graph. Note the cycle-free property
allows only one leader in the graph, thus following Corollary 1, we have
Proposition 5 All cycle-free persistent graph in <3are also structurally persistent.
4 CONCLUSION AND FURTHER WORKS
In this paper, we considered some practical issues raised in multi-agent formation con-
trol in three dimensional space, building upon a recently developed theoretical frame-
work. We introduced the partial equilibrium problem. We defined structurally persistent
graphs, a class of persistent graphs free of any partial equilibrium problem, noting that
for d= 2, structural persistence is no different to persistence. We studied the connec-
tions between the allocation of degrees of freedom (DOFs) across agents and the char-
acteristics of persistence and/or structural persistence of a directed graph. We proposed
directed d-vertex addition operations for <d. We also showed how to reallocate degrees
of freedom between agents, when the formation changes with new agent(s) added, to
preserve persistence and/or structural persistence. Finally, we gave some powerful re-
sults about cycle-free persistent graphs in <3.
References
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tence of three and higher dimensional directed formations. To appear in the 1st International
Workshop on Multi-Agent Robotic Systems–MARS05’ as a companion of this paper.
[2] C. Yu, B. Fidan, and B.D.O. Anderson. Persistence acquisition and maintenance for au-
tonomous formations. Submitted to the 2nd International Conference on Intelligent Sensors,
Sensor Networks and Information Processing, 2005.
[3] W. Whiteley. Matroid Theory, volume 197 of Contemporary Mathematics, pages 171–311.
American Mathemtical Society, 1996.
[4] W. Whiteley. Handbook of Discrete and Computational Geometry, chapter Rigidity and
Scene Analysis, pages 893–916. CRC Press, 1997.
[5] J.M. Hendrickx, B.D.O. Anderson, V.D. Blondel, and J.-C. Delvenne. Directed graphs for
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