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Persistent multi-robot formations with redundancy

TL;DR: This paper presents an approach based on rigidity theory for constructing persistent leader-follower formations with redundancy; specified robots may experience sensor link failure without losing the persistence of the formation.
Abstract: A multi-robot formation composed of autonomous agents may need to maintain an overall rigid shape for tasks such as collective transport of an object. To distribute control, we construct leader-follow formations in the plane that are persistent: designated “leader” robots control the movement of the entire formation, while the remaining “follower” robots maintain directed local links sensing data to other robots in such a way that the entire formation retains its overall shape. In this paper, we present an approach based on rigidity theory for constructing persistent leader-follower formations with redundancy; specified robots may experience sensor link failure without losing the persistence of the formation. Within this model, we consider the impact of special positions due to certain geometric conditions and provide simulation results confirming the expected behavior.

Summary (2 min read)

1 Introduction

  • For applications such as collective transport, multi-robot formations need to maintain a global shape.
  • Since this may increase sensing and communication costs, a “persistence theory” for directed distance constraints between points was proposed by Hendrickx et al. [5] (see also [4]), effectively cutting costs in half by assigning one of the two agents to be responsible for sensing and maintaining a distance.
  • Redundancy is wellunderstood in rigidity theory, and the associated objects form the foundation for the main contribution of this paper: an approach for constructing persistent leader-follower formations with redundancy.
  • Restricting to acyclic formations implies that such special conditions do not impact their construction, and the authors present simulation results (Section 5) verifying their approach.

2 Preliminaries

  • For a given graph, almost all associated embeddings, called generic embeddings, share the same rigidity properties (see, e.g., [19]).
  • One can interpret each of the vertices as representing an autonomous agent with out-going edges specifying distance constraints to neighbors that it is responsible for satisfying.
  • While the formations in Figures 2(a) and 2(b) have the same underlying undirected graph, only one is persistent.
  • The authors consider a specific type of persistent formations called leader-follower formations.

3 Redundancy for persistence theory

  • Communication links and sensors can fail, motivating the need for redundancy in a multi-robot system.
  • Refer back to the formation in Figure 2(d).
  • While its underlying undirected graph is redundantly rigid, the formation is not redundantly persistent; without the edge −→ 32, the resulting formation ) is no longer persistent.
  • Furthermore, the authors can use “pebble collection” moves in the pebble game to find an orientation H where exactly one vertex vL has out-degree 0, another vertex vC incident to vL, has out-degree 1 and all other vertices have out-degree 2.
  • Proposition 2 gives a recursive approach for constructing persistent leader-follower formations with any desired number of vertices whose out-edge sets each contain redundancy.

4 Special geometric conditions

  • In Section 3, the authors presented approaches for constructing generically persistent graphs, applying to situations where agents are positioned with a generic embedding in the plane.
  • As noted in Section 2, for particular geometric configurations, a generically rigid graph may become flexible.
  • The authors illustrate these two types of special positions with some simple examples.
  • Observe that the persistent formations of Fig. 5 and 6 are acyclic, while the non-persistent formations are not.

5 Simulation

  • Each vertex in the graph is implemented by a robot that is equipped with a generic emitter, and each directed edge by equipping the source robot with a receiver that listens to the target robot’s emitter.
  • Every follower computes and moves to a goal position based on its assigned distance constraints: with two constraints, the closest point of the 2 intersection points of 2 circles, and, with three constraints, the average of 3 points computed for each pair of constraints.
  • The expected position of the follower robots is computed using the position of the leader and co-leader positions; since the co-leader is on the same robot as the leader, its distance constraint is always satisfied.
  • Data for each simulation was then computed as the mean of the mt , µt and Mt across all simulation steps.
  • The results confirm the expected behavior of the formations; the simulations testing redundancy performed comparably to the control with all edges present.

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Persistent Multi-Robot Formations with
Redundancy
Alyxander Burns, Bernd Schulze, and Audrey St. John
Abstract A multi-robot formation composed of autonomous agents may need to
maintain an overall rigid shape for tasks such as collective transport of an object.
To distribute control, we construct leader-follow formations in the plane that are
persistent: designated “leader” robots control the movement of the entire formation,
while the remaining “follower” robots maintain directed local links sensing data to
other robots in such a way that the entire formation retains its overall shape. In this
paper, we present an approach based on rigidity theory for constructing persistent
leader-follower formations with redundancy; specified robots may experience sen-
sor link failure without losing the persistence of the formation. Within this model,
we consider the impact of special positions due to certain geometric conditions and
provide simulation results confirming the expected behavior.
1 Introduction
For applications such as collective transport, multi-robot formations need to main-
tain a global shape. To do so in a distributed fashion, we focus on formations com-
posed of autonomous agents that use local sensing to maintain a global rigid struc-
ture. In particular, we consider persistent leader-follower formations where des-
ignated leader robots control the trajectory of the entire formation; the remaining
robots autonomously sense and adjust their positions locally to follow specified
robots in a way that maintains the global structure.
Alyxander Burns, Audrey St. John (e-mail: {burns22l, astjohn}@mtholyoke.edu)
Computer Science Department, Mount Holyoke College, South Hadley, MA
Partially supported by NSF IIS-1253146 and the Clare Boothe Luce Foundation
Bernd Schulze (e-mail: b.schulze@lancaster.ac.uk)
Department of Mathematics and Statistics, Fylde College Lancaster University, Lancaster, UK
Supported by EPSRC grant EP/M013642/1
1

2 Alyxander Burns, Bernd Schulze, and Audrey St. John
In our model, each autonomous robot is represented as a point in the plane,
and we work with range-only measurements, represented as distance constraints
between pairs of points. This model is known in an area called “rigidity theory”
as the 2D bar-and-joint framework (see, e.g., [19]) and is well-understood, with a
quadratic algorithm for determining the bar-and-joint rigidity properties [8]. While
rigidity theory has been applied to the construction and analysis of formations of
autonomous agents [11, 3], the approach assumes undirected constraints, leading to
a model where both agents would be responsible for the constraint. Since this may
increase sensing and communication costs, a “persistence theory” for directed dis-
tance constraints between points was proposed by Hendrickx et al. [5] (see also [4]),
effectively cutting costs in half by assigning one of the two agents to be responsible
for sensing and maintaining a distance. Unlike decentralized approaches for col-
lective transport where robots maintain constraints to the transported object (e.g.,
[7, 16]), a persistent formation could be used to carry delicate items, such as a par-
tially constructed vehicle. In particular, we use a leader-follower architecture [2]; as
described in [15], local sensing and communication can achieve specific geometric
formations, allowing dynamic adaptation based on the surrounding environment.
Contributions. In this paper, we focus on accommodating sensing and communi-
cation failures by incorporating redundancy into our model. Redundancy is well-
understood in (undirected) rigidity theory, and the associated objects form the foun-
dation for the main contribution of this paper: an approach for constructing (di-
rected) persistent leader-follower formations with redundancy.
We work within the basic model of persistence, following the definitions from
[5], and present a class of directed graphs where any edge from a vertex with out-
degree 3 is redundant; after removal of such an edge, the resulting formation re-
mains persistent. Algorithms for constructing these graphs, as well as simulation
results confirming the expected behavior of acyclic formations, are presented. We
also include a discussion of the impact of special geometric conditions that can
affect the “generic” behavior of the combinatorial model. To the best of our knowl-
edge, these graphs are the first to incorporate redundancy into persistent formations.
While the redundancy is restricted to specified sets of edges, it is a first step towards
the stronger notion of redundantly persistent formations, defined in Section 3, where
any edge in the formation could be removed.
Structure. In Section 2, we provide an overview of the relevant definitions and re-
sults from rigidity and persistence theory. We present an approach for constructing
persistent leader-follower formations with redundancy in Section 3 before consider-
ing special geometric conditions that may affect persistence in Section 4. However,
restricting to acyclic formations implies that such special conditions do not impact
our construction, and we present simulation results (Section 5) verifying our ap-
proach. We conclude with future directions in Section 6.

Persistent Multi-Robot Formations with Redundancy 3
2
1
4
3
4
3
(a) A flexible framework with another (gray) incon-
gruent embedding satisfying the distance function.
2
1
4
3
(b) Adding the edge 14 results in a mini-
mally rigid framework.
Fig. 1 Flexible and rigid frameworks in the plane.
2 Preliminaries
Let G = (V, E) be an undirected graph with vertex set V = [1..n] and edge set E
of unordered pairs of vertices. An embedding of G in the Euclidean plane is an
assignment p (R
2
)
n
of the vertices to points in the plane; the pair (G, p) is called
a framework. Another embedding q is congruent to p if ||p
i
p
j
|| = ||q
i
q
j
||
for every pair of vertices i and j. Given a framework, we can extract a distance
function d : E R, where d(i j) = ||p
i
p
j
||. If all q in the neighborhood of p
satisfying the distance function d are congruent to p, the framework is rigid and
flexible otherwise; refer to Figure 1. The rigid framework of Figure 1(b) is minimally
rigid as the removal of any edge results in a flexible framework.
For a given graph, almost all associated embeddings, called generic embeddings,
share the same rigidity properties (see, e.g., [19]). Therefore, we may call a graph
generically rigid or flexible, referring to the behavior of generic embeddings. The
formal definition of genericity is captured by a polynomial whose vanishing indi-
cates a non-generic embedding, or special position, and is outside the scope of this
paper. The impact of special positions is discussed in Section 4.
For multi-robot formations, where minimizing the cost of communication and
sensing is desirable, we work with a notion closely related to rigidity called persis-
tence. We build upon the foundations of [5] and include here only the relevant defi-
nitions and results. Persistence is framed in terms of a directed graph and intuitively
defines the directed analog of rigidity. One can interpret each of the vertices as rep-
resenting an autonomous agent with out-going edges specifying distance constraints
to neighbors that it is responsible for satisfying. This eliminates the cost for sensing
and communication costs on one endpoint of a constraint edge, which would be re-
quired if working with undirected graphs and rigidity. If (1) every agent can find a
position to satisfy its distance constraints, and (2) the corresponding framework is
rigid, then the formation is considered persistent.
Let H = (V, E) be a directed graph with vertex set V = [1..n] and edge set E of
ordered pairs of vertices. For clarity, we denote a directed edge from the source i to
j with
i j to contrast with an undirected edge i j. For a graph H and an embedding p,
the pair (H, p) is called a formation. Given a formation, we can extract the distance
function d as before.

4 Alyxander Burns, Bernd Schulze, and Audrey St. John
We can now state the technical definitions from [5] for persistence. Given d, let
q (R
2
)
n
be an embedding of the vertices of H. If
i j E and ||q
i
q
j
|| = d(
i j ),
then the edge
i j is active in q. The set A
q
(i) denotes the set of active edges in q
whose source is i. The position q
i
is fitting for vertex i if there does not exist another
embedding q
such that (1) q
i
6= q
i
, (2) q
j
= q
j
for all j 6= i, and (3) A
q
(i) ( A
q
(i).
Intuitively, a position is fitting for a vertex if it cannot be moved to satisfy additional
constraints. If the positions for all vertices are fitting, then q is a fitting embedding.
With this notion of fitting embeddings formalized, we can define persistence.
Definition 1. Let (H, p) be a formation. If all fitting embeddings q in the neighbor-
hood of p are congruent to p, then the formation is persistent.
See Figure 2 for examples of persistent and non-persistent formations. While the
formations in Figures 2(a) and 2(b) have the same underlying (rigid) undirected
graph, only one is persistent. The embedding shown in Figure 2(c) is fitting for the
formation in Figure 2(b) as each vertex is in a position that maximizes the number of
its out-going constraints; vertex 4 cannot satisfy the dashed constraint to 3 without
violating at least one of its other constraints to 1 or 2. Since it is not congruent, this
certifies that the formation in Figure 2(b) is not persistent.
As with rigidity, almost all embeddings of a directed graph exhibit the same
persistence properties, so we may refer to a directed graph as generically persistent.
We rely on the following result of [5]:
Theorem 1 (Theorem 3 of [5]). A graph is generically persistent if and only if
the underlying undirected graph of every subgraph obtained by removing out-edges
from vertices with out-degree > 2 until all vertices have out-degree 2 is generi-
cally rigid.
Leader-follower formations. In this paper, we consider a specific type of persis-
tent formations called leader-follower formations. In the persistent formations of
Figures 2(a) and 2(d): only vertex 1 has out-degree 0 and is called the leader, only
vertex 2 incident to it has out-degree 1 and is called the co-leader, and all other ver-
tices (3 and 4) have out-degree 2 and are called followers. Since a point in 2D has
two degrees of freedom (translation along the x- and y-axes), but an entire formation
has three (translation along the x- and y-axes along with rotation about the origin),
the simplest leader-follower formation must have both a leader and a co-leader; the
entire formation cannot be controlled by a single lead point agent.
3 Redundancy for persistence theory
Communication links and sensors can fail, motivating the need for redundancy in a
multi-robot system. In this section, we present an approach for constructing persis-
tent leader-follower formations with redundancy.
We begin by reviewing redundancy in rigidity; a graph is generically redundantly
rigid if removing any edge results in a rigid graph. Minimality is defined as follows:

Persistent Multi-Robot Formations with Redundancy 5
2
1
4
3
(a) A persistent for-
mation with leader 1,
co-leader 2 and fol-
lowers 3 and 4.
2
1
4
3
(b) Reversing the
edge from 3 to 4
results in a non-
persistent formation.
2
1
4
3
(c) A fitting embed-
ding for the forma-
tion in (b) that is not
congruent.
2
1
4
3
(d) Adding an edge
from 3 to 2 results
in a persistent forma-
tion.
Fig. 2 Persistent and non-persistent frameworks in the plane.
an undirected graph is a generic rigidity circuit if removing any edge results in a
generically minimally rigid graph. Circuits are standard in matroid theory (see, e.g.,
[12]) and the 2D bar-and-joint rigidity matroid captures the behavior of the distance
constraints described in Section 2 [18]. In this section, we only consider the generic
behavior of graphs; for brevity, we omit the word “generically” for the remainder.
The smallest example of a rigidity circuit can be seen in Figure 2(d); removing any
edge from the (undirected) K
4
graph gives a minimally rigid graph.
We analogously define a directed graph to be redundantly persistent if the re-
moval of any edge results in a persistent graph. However, the behavior of redundant
rigidity does not easily extend to the persistence model. Refer back to the formation
in Figure 2(d). While its underlying undirected graph is redundantly rigid, the for-
mation is not redundantly persistent; without the edge
32, the resulting formation
(of Figure 2(b)) is no longer persistent. We leave the question of redundantly per-
sistent graphs open; it is challenging to even come up with a simple formation that
satisfies the definition.
In the remainder of this section, we present a class of graphs with a more re-
stricted notion of redundancy. These arise from considering leader-follower forma-
tions whose underlying undirected graphs are rigidity circuits, beginning with the
following result.
Proposition 1. Let G be a rigidity circuit. Then there exists a leader-follower orien-
tation of G that is persistent.
Proof. Let G = (V, E) be a rigidity circuit and let e = i j E be any edge. Then
G
0
= (V, E \ {e}) is a minimally rigid graph. By using an algorithm called the (2, 3)-
pebble game [8, 10], there exists an orientation of G
0
where every vertex has out-
degree at most 2. Furthermore, we can use “pebble collection” moves in the pebble
game to find an orientation H where exactly one vertex v
L
has out-degree 0, another
vertex v
C
incident to v
L
, has out-degree 1 and all other vertices have out-degree 2.
By Theorem 1, this formation is persistent; there are no vertices with out-degree >
2, so we only need to consider the underlying undirected graph G
0
of H, which is
(minimally) rigid.
We now add the edge i j back to the formation, orienting it as
i j if i 6= v
C
, v
L
and
ji otherwise. By this construction, the source of the edge e now has out-degree 3.

Citations
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Journal ArticleDOI
01 May 2020
TL;DR: This problem combines k‐coverage and the cooperative multirobot observation of multiple moving targets problem, and thereby captures key features of rapidly deployed camera networks, including redundancy and team‐based tracking of evasive or unpredictable targets.
Abstract: In this article, we explore the online multiobject k‐coverage problem in visual sensor networks. This problem combines k‐coverage and the cooperative multirobot observation of multiple moving targets problem, and thereby captures key features of rapidly deployed camera networks, including redundancy and team‐based tracking of evasive or unpredictable targets. The benefits of using mobile cameras are demonstrated and we explore the balance of autonomy between cameras generating new subgoals, and those responders able to fulfill them. We show that higher performance against global goals is achieved when decisions are delegated to potential responders who treat subgoals as optional, rather than as obligations that override existing goals without question. This is because responders have up‐to‐date knowledge of their own state and progress toward goals where they are situated, which is typically old or incomplete at locations remote from them. Examining the extent to which approaches overprovision or underprovision coverage, we find that being well suited for achieving 1‐coverage does not imply good performance at k‐coverage. Depending on the structure of the environment, the problems of 1‐coverage and k‐coverage are not necessarily aligned and that there is often a trade‐off to be made between standard coverage maximization and achieving k‐coverage.

11 citations

Journal ArticleDOI
01 Jun 2021
TL;DR: In this article, the authors describe the formation control problem in multi-robot systems and survey recent advances focusing on aspects of maintaining a formation by a group of robots distinguished by the means of analysis.
Abstract: Formation control is a canonical problem in multi-robot systems, which focuses on the ability of a group of robots to travel in coordination through an area, while maintaining a certain shape or a particular behavior. The robot groups vary in their communication, computation, and sensing capabilities. Moreover, the formation control task itself may have various objectives. These divergences force the use of different models for controlling the formation and for analyzing the task performance. In this paper, we describe the formation control problem and survey recent advances focusing on aspects of maintaining a formation by a group of robots distinguished by the means of analysis. Various approaches may be applied for the sake of formation maintenance, whereas each approach possesses a different perspective in regard with formation control. Recent research focuses on combining those approaches, due to their applicability regarding certain scenarios. For instance, consensus-based control and collision avoidance are usually intertwined together for the sake of reaching a consensus in a manner which is collision-free. Furthermore, machine learning (ML)–based methods for navigating a robot team through unknown complex environments can be incorporated, where the robot team aims to reach a goal position while avoiding collisions and maintaining connectivity. Moreover, recent approaches focus on developing new mechanisms or adapt existing ones for formation control for tolerating limitations in sensing, communication, and coordination, preferably distributively while providing performance guarantees. Such combined approaches yield that the means of analysis, which can be applied to each one separately, can also be utilized in an intertwined manner, and thus provide us with novel methods for preserving formation. Whereas some approaches were vastly investigated (e.g., consensus-based formation control) and need to be adapted to distributed imperfect settings, others still require further insight for unveiling brand new architectures and tools (e.g., ML-based formation control).

7 citations

Proceedings Article
01 Jan 2019
TL;DR: Redundant persistence to multi-robot systems as a mechanism for incorporating robustness to sensing failure and an inductive construction for generating redundantly persistent graphs are applied.
Abstract: We present theoretical and experimental results on the application of acyclic persistent leader-follower formations with redundancy to a distributed multi-agent system. A leader-follower formation is defined on a set of point agents constrained by fixed distance assignments for following other agents; if satisfying the constraints results in the distances between all pairs of agents being maintained, the formation is persistent. The (generic) persistence of a leader-follower formation in 2D is combinatorially characterized by a directed graph with one “leader” vertex having no out-edges, one “co-leader” vertex having exactly one out-edge (to the leader), all other “follower” vertices having out-degree at least 2, and an underlying minimally rigid (undirected) graph. We provide theoretical results for three types of persistent formations with redundancy, including an inductive construction for generating redundantly persistent graphs (the strongest notion of redundancy). We apply redundant persistence to multi-robot systems as a mechanism for incorporating robustness to sensing failure. In particular, we implement the approach on a vision-based distributed multi-robot platform. Using acyclic orientations permits a simple, “wave”-based control that converges reliably, and redundant edges allow the control to recover e↵ectively from sensing limitations (e.g., a camera’s limited field of view or obstruction by another robot).

1 citations

Proceedings ArticleDOI
01 Oct 2018
TL;DR: This paper presents an algorithm that allows each individual robot to estimate the overall formation accuracy of the other robots in their field of view via a tree reconstruction algorithm, used to select the most accurate local leader, or to generate virtual local leader via a weighted average of all visible robots.
Abstract: Leader-Follower is a hierarchical form of multi-robot formation control, where each robot aims to maintain specific predefined angle and distance from one or more robots in the team (referred to as its local leaders), while a single robot is selected to lead the entire formation to a desired destination. When the robots are given a specific formation to maintain, their goal is usually to minimize the deviation from this desired formation (maximizing the accuracy) during their journey. Previous work has considered optimality in an uncertain environment only in centralized setting (or using perfect, or almost perfect communication). In this paper we examine the problem of optimal multi-robot formation control in a distributed setting, while accounting for two challenges: sensory uncertainty and absence of communication. Specifically, we present an algorithm that allows each individual robot to estimate the overall formation accuracy of the other robots in their field of view via a tree reconstruction algorithm. The algorithm is used to select the most accurate local leader, or to generate virtual local leader via a weighted average of all visible robots. We provide both theoretical analysis and an extensive empirical evaluation (in ROS/Gazebo simulated environment) showing the effectiveness of the two approaches.

1 citations


Additional excerpts

  • ...distributed systems [2], [7], [11], [9], [10], [3], [18], [8]....

    [...]

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Journal ArticleDOI
10 Dec 2002
TL;DR: In this article, the authors describe a framework for cooperative control of a group of nonholonomic mobile robots that allows them to build complex systems from simple controllers and estimators, and guarantee stability and convergence in a wide range of tasks.
Abstract: We describe a framework for cooperative control of a group of nonholonomic mobile robots that allows us to build complex systems from simple controllers and estimators. The resultant modular approach is attractive because of the potential for reusability. Our approach to composition also guarantees stability and convergence in a wide range of tasks. There are two key features in our approach: 1) a paradigm for switching between simple decentralized controllers that allows for changes in formation; 2) the use of information from a single type of sensor, an omnidirectional camera, for all our controllers. We describe estimators that abstract the sensory information at different levels, enabling both decentralized and centralized cooperative control. Our results include numerical simulations and experiments using a testbed consisting of three nonholonomic robots.

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TL;DR: In this article, a graph theoretical framework is proposed to formally define formations of multiple vehicles and the issues arising in uniqueness of graph realizations and its connection to stability of formations, as well as formal representation of split, rejoin, and reconfiguration maneuvers for multi-vehicle formations.
Abstract: We provide a graph theoretical framework that allows us to formally define formations of multiple vehicles and the issues arising in uniqueness of graph realizations and its connection to stability of formations. The notion of graph rigidity is crucial in identifying the shape variables of a formation and an appropriate potential function associated with the formation. This allows formulation of meaningful optimization or nonlinear control problems for formation stabilization/tacking, in addition to formal representation of split, rejoin, and reconfiguration maneuvers for multi-vehicle formations. We introduce an algebra that consists of performing some basic operations on graphs which allow creation of larger rigid-by-construction graphs by combining smaller rigid subgraphs. This is particularly useful in performing and representing rejoin/split maneuvers of multiple formations in a distributed fashion.

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"Persistent multi-robot formations w..." refers background in this paper

  • ...While rigidity theory has been applied to the construction and analysis of formations of autonomous agents [11, 3], the approach assumes undirected constraints, leading to a model where both agents would be responsible for the constraint....

    [...]

Journal ArticleDOI
TL;DR: In the context of generic rigidity percolation, it is shown how to calculate the number of internal degrees of freedom, identify all rigid clusters, and locate the overconstrained regions.

379 citations

Journal ArticleDOI
TL;DR: In this article, the authors consider formations of autonomous agents moving in a two-dimensional space, each agent tries to maintain its distances toward a pre-specified group of other agents constant and the problem is to determine if one can guarantee that the distance between every pair of agents (even those not explicitly maintained) remains constant, resulting in the persistence of the formation shape.
Abstract: We consider in this paper formations of autonomous agents moving in a two-dimensional space Each agent tries to maintain its distances toward a pre-specified group of other agents constant and the problem is to determine if one can guarantee that the distance between every pair of agents (even those not explicitly maintained) remains constant, resulting in the persistence of the formation shape We provide here a theoretical framework for studying this problem We describe the constraints on the distance between agents by a directed graph and define persistent graphs A graph is persistent if the shapes of almost all corresponding agent formations persist Although persistence is related to the classical notion of rigidity, these are two distinct notions We derive various properties of persistent graphs, and give a combinatorial criterion to decide persistence We also define minimal persistence (persistence with the least possible number of edges), and we apply our results to the interesting special case of cycle-free graphs Copyright (C) 2006 John Wiley & Sons, Ltd

213 citations


"Persistent multi-robot formations w..." refers background or methods or result in this paper

  • ...We can now state the technical definitions from [5] for persistence....

    [...]

  • ...[5] (see also [4]), effectively cutting costs in half by assigning one of the two agents to be responsible for sensing and maintaining a distance....

    [...]

  • ...Then, by Propositions 1 and 3 in [5], we obtain an acyclic generically minimally persistent graph G....

    [...]

  • ...We rely on the following result of [5]: Theorem 1 (Theorem 3 of [5])....

    [...]

  • ...However, if a persistent graph is acyclic, then there exists an ordering of the vertices such that (1) the first vertex has out-degree 0, (2) the second out-degree 1, and (3) every other vertex has ≥ 2 out-edges to vertices earlier in the ordering [5]....

    [...]

Book ChapterDOI
01 Jan 1997

209 citations


"Persistent multi-robot formations w..." refers background or methods in this paper

  • ...Graphs of this type are a generalization of Henneberg I graphs, given the use of the “vertex addition” step for the followers that was first described by Henneberg for minimally rigid graphs [18, 19]....

    [...]

  • ..., [19]) and is well-understood, with a quadratic algorithm for determining the bar-and-joint rigidity properties [8]....

    [...]

Frequently Asked Questions (2)
Q1. What future works have the authors mentioned in the paper "Persistent multi-robot formations with redundancy" ?

The authors recognize that the redundancy property exhibited by the class of graphs presented here is quite restrictive and wish to address the strong notion of redundantly persistent formations defined in Section 2. The authors hope to further develop and evaluate a theoretical framework that would be robust to noise and disturbances. 

To distribute control, the authors construct leader-follow formations in the plane that are persistent: designated “ leader ” robots control the movement of the entire formation, while the remaining “ follower ” robots maintain directed local links sensing data to other robots in such a way that the entire formation retains its overall shape. In this paper, the authors present an approach based on rigidity theory for constructing persistent leader-follower formations with redundancy ; specified robots may experience sensor link failure without losing the persistence of the formation. Within this model, the authors consider the impact of special positions due to certain geometric conditions and provide simulation results confirming the expected behavior.