Research on formation control and cooperative localization for multirobot systems has been an active field over the last years. A powerful theoretical framework for addressing formation control and localization, especially when exploiting onboard sensing, is that of formation rigidity (mainly studied for the cases of distance and bearing measurements). Rigidity of a formation depends not only on the topology of the sensing/communication graph but also on the spatial arrangement of the robots, since special configurations (“singularities” of the rigidity matrix), which are hard to detect in general, can cause a rigidity loss and prevent convergence of formation control/localization algorithms based on formation rigidity. The aim of this paper is to gain additional insights into the internal structure of bearing rigid formations by considering an alternative characterization of formation rigidity using tools borrowed from the mechanical engineering community. In particular, we show that bearing rigid graphs can be given a physical interpretation related to virtual mechanisms, whose mobility and singularities can be analyzed and detected in an analytical way by using tools from the mechanical engineering community (screw theory, Grassmann geometry, and Grassmann-Cayley algebra). These tools offer a powerful alternative to the evaluation of the mobility and singularities typically obtained by numerically determining the spectral properties of the bearing rigidity matrix (which typically prevents drawing general conclusions). We apply the proposed machinery to several case formations with different degrees of actuation and discuss known (and unknown) singularity cases for representative formations. The impact on the localization problem is also discussed.

References

References
1.
Fox
,
D.
,
Ko
,
J.
,
Konolige
,
K.
,
Limketkai
,
B.
,
Schulz
,
D.
, and
Stewart
,
B.
,
2006
, “
Distributed Multirobot Exploration and Mapping
,”
Proc. IEEE
,
94
(
7
), pp.
1325
1339
.
2.
Michael
,
N.
,
Fink
,
J.
, and
Kumar
,
V.
,
2009
, “
Cooperative Manipulation and Transportation With Aerial Robots
,”
2009 Robotics: Science and Systems
.
3.
Robuffo Giordano
,
P.
,
Franchi
,
A.
,
Secchi
,
C.
, and
Bülthoff
,
H. H.
,
2013
, “
A Passivity-Based Decentralized Strategy for Generalized Connectivity Maintenance
,”
Int. J. Robot. Res.
,
32
(
3
), pp.
299
323
.
4.
Hausman
,
K.
,
Müller
,
J.
,
Hariharan
,
A.
,
Ayanian
,
N.
, and
Sukhatme
,
G. S.
,
2015
, “
Cooperative Multi-Robot Control for Target Tracking With Onboard Sensing
,”
IEEE Trans. Robot.
,
34
(
13
), pp.
1660
1677
.
5.
Montijano
,
E.
,
Cristofalo
,
E.
,
Zhou
,
D.
,
Schwager
,
M.
, and
Sagues
,
C.
,
2016
, “
Vision-Based Distributed Formation Control Without an External Positioning System
,”
IEEE Trans. Robot.
,
32
(
2
), pp.
339
351
.
6.
Michieletto
,
G.
,
Cenedese
,
A.
, and
Franchi
,
A.
,
2016
, “
Bearing Rigidity Theory in SE(3)
,”
Proceedings of the 55th IEEE Conference on Decision and Control (CDC 2016)
.
7.
Palacios-Gasós
,
J. M.
,
Montijano
,
E.
,
Sagues
,
C.
, and
Llorente
,
S.
,
2016
, “
Distributed Coverage Estimation and Control for Multi-Robot Persistent Tasks
,”
IEEE Trans. Robot.
,
32
(
6
), pp.
1444
1460
.
8.
Anderson
,
B. D. O.
,
Yu
,
C.
,
Fidan
,
B.
, and
Hendrickx
,
J. M.
,
2008
, “
Rigid Graph Control Architectures for Autonomous Formations
,”
IEEE Control Syst. Mag.
,
28
(
6
), pp.
48
63
.
9.
Spica
,
R.
,
Robuffo Giordano
,
P.
, and
Daejeon
,
K.
,
2016
, “
Active Decentralized Scale Estimation for Bearing Based Localization
,”
2016 IEEE IROS
,
Oct
.
10.
Dimarogonas
,
D. V.
, and
Johansson
,
K. H.
,
2008
, “
On the Stability of Distance-Based Formation Control
,”
47th Conference on Decision and Control
, pp.
1200
1205
.
11.
Dimarogonas
,
D.
, and
Johansson
,
K.
,
2009
, “
Further Results on the Stability of Distance-Based Multi-Robot Formations
,”
American Control Conference
,
2009
(ACC ’09), pp.
2972
2977
.
12.
Krick
,
L.
,
Broucke
,
M. E.
, and
Francis
,
B. A.
,
2009
, “
Stabilisation of Infinitesimally Rigid Formations of Multi-Robot Networks
,”
Int. J. Control
,
82
(
3
), pp.
423
439
.
13.
Oh
,
K.-K.
, and
Ahn
,
H.-S.
,
2011
, “
Formation Control of Mobile Agents Based on Inter-Agent Distance Dynamics
,”
Automatica
,
47
(
10
), pp.
2306
2312
.
14.
Garcia de Marina
,
H.
,
Cao
,
M.
, and
Jayawardhana
,
B.
,
2015
, “
Controlling Rigid Formations of Mobile Agents Under Inconsistent Measurements
,”
IEEE Trans. Robot.
,
31
(
1
), pp.
31
39
.
15.
Cornejo
,
A.
,
Lynch
,
A. J.
,
Fudge
,
E.
,
Bilstein
,
S.
,
Khabbazian
,
M.
, and
McLurkin
,
J.
,
2013
, “
Scale-Free Coordinates for Multi-Robot Systems With Bearing-Only Sensors
,”
Int. J. Robot. Res.
,
12
(
32
), pp.
1459
1474
.
16.
Shames
,
I.
,
Bishop
,
A. N.
, and
Anderson
,
B. D. O.
,
2013
, “
Analysis of Noisy Bearing-Only Network Localization
,”
IEEE Trans. Autom. Control
,
1
(
58
), pp.
247
252
.
17.
Zhao
,
S.
, and
Zelazo
,
D.
,
2016
, “
Bearing Rigidity and Almost Global Bearing-Only Formation Stabilization
,”
IEEE Trans. Autom. Control
,
61
(
5
), pp.
1255
1268
.
18.
Zelazo
,
D.
,
Robuffo Giordano
,
P.
, and
Franchi
,
A.
,
2015
, “
Bearing-Only Formation Control Using an SE(2) Rigidity Theory
,”
2015 IEEE CDC
,
Dec.
, pp.
6121
6126
.
19.
Schiano
,
F.
,
Franchi
,
A.
,
Zelazo
,
D.
, and
Robuffo Giordano
,
P.
,
2016
, “
A Rigidity-Based Decentralized Bearing Formation Controller for Groups of Quadrotor UAVs
,”
2016 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2016)
.
20.
Zhao
,
S.
, and
Zelazo
,
D.
,
2017
, “
Translational and Scaling Formation Maneuver Control Via a Bearing-Based Approach
,”
IEEE Trans. Control Netw. Syst.
,
4
, pp.
429
438
.
21.
Zhao
,
S.
, and
Zelazo
,
D.
,
2017
, “
Bearing Rigidity Theory and Its Applications for Control and Estimation of Network Systems: Life Beyond Distance Rigidity
,”
IEEE Control Syst. Mag
.
22.
Schiano
,
F.
, and
Robuffo Giordano
,
P.
,
2017
, “
Bearing Rigidity Maintenance for Formations of Quadrotor UAVs
,”
Proceedings of the 2017 IEEE International Conference on Robotics and Automation (ICRA 2007)
.
23.
Trinh
,
M. H.
,
Mukherjee
,
D.
,
Zelazo
,
D.
, and
Ahn
,
H.-S.
,
2018
, “
Formations on Directed Cycles With Bearing-Only Measurements
,”
Int. J. Robust Nonlinear Control
,
28
(
3
), pp.
1074
1096
.
24.
Briot
,
S.
, and
Martinet
,
P.
,
2013
, “
Minimal Representation for the Control of Gough-Stewart Platforms Via Leg Observation Considering a Hidden Robot Model
,”
Proceedings of the 2013 IEEE International Conference on Robotics and Automation (ICRA 2013)
.
25.
Briot
,
S.
,
Martinet
,
P.
, and
Rosenzveig
,
V.
,
2015
, “
The Hidden Robot: An Efficient Concept Contributing to the Analysis of the Controllability of Parallel Robots in Advanced Visual Servoing Techniques
,”
IEEE Trans. Robot.
,
31
(
6
), pp.
1337
1352
.
26.
Andreff
,
N.
,
Marchadier
,
A.
, and
Martinet
,
P.
,
2005
, “
Vision-Based Control of a Gough-Stewart Parallel Mechanism Using Legs Observation
,”
Proceedings of the IEEE International Conference on Robotics and Automation (ICRA’05)
, pp.
2546
2551
.
27.
Briot
,
S.
,
Chaumette
,
F.
, and
Martinet
,
P.
,
2017
, “
Revisiting the Determination of the Singularity Cases in the Visual Servoing of Image Points Through the Concept of “Hidden Robot”
,”
IEEE Trans. Robot.
,
33
(
3
), pp.
536
546
.
28.
Briot
,
S.
,
Martinet
,
P.
, and
Chaumette
,
F.
,
2017
, “
Singularity Cases in the Visual Servoing of Three Image Lines
,”
IEEE Robot. Autom. Lett.
,
2
(
2
), pp.
412
419
.
29.
Ben-Horin
,
P.
, and
Shoham
,
M.
,
2006
, “
Singularity Analysis of a Class of Parallel Robots Based on Grassmann-Cayley Algebra
,”
Mech. Mach. Theory
,
41
(
8
), pp.
958
970
.
30.
Merlet
,
J.-P.
,
2006
,
Parallel Robots
,
2nd ed.
,
Springer
,
New York
.
31.
Schiano
,
F.
,
Franchi
,
A.
,
Zelazo
,
D.
, and
Robuffo Giordano
,
P.
,
2016
, “
A Rigidity-Based Decentralized Bearing Formation Controller for Groups of Quadrotor UAVs
,”
Proceedings of the 2016 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2016)
.
32.
Godsil
,
C.
, and
Royle
,
G.
,
2001
,
Algebraic Graph Theory
,
Springer
,
New York
.
33.
Eren
,
T.
,
2012
, “
Formation Shape Control Based on Bearing Rigidity
,”
Int. J. Control
,
9
, pp.
1361
1379
.
34.
Kong
,
X.
, and
Gosselin
,
C.
,
2007
,
Type Synthesis of Parallel Mechanisms
,
Springer
,
New York
.
35.
Zelazo
,
D.
,
Robuffo Giordano
,
P.
, and
Franchi
,
A.
,
2015
, “
Bearing-Only Formation Control Using an SE(2) Rigidity Theory
,”
Proceedings of the 54th IEEE Conference on Decision and Control (CDC 2015)
.
36.
Waldron
,
K.
,
1966
, “
The Constraint Analysis of Mechanisms
,”
J. Mech.
,
1
, pp.
101
114
.
37.
Gogu
,
G.
,
2008
,
Structural Synthesis of Parallel Robots
,
Springer
,
New York
.
38.
Huo
,
X.
,
Sun
,
T.
, and
Song
,
Y.
,
2017
, “
A Geometric Algebra Approach to Determine Motion/Constraint, Mobility and Singularity of Parallel Mechanism
,”
Mech. Mach. Theory
,
116
, pp.
273
293
.
39.
Briot
,
S.
, and
Khalil
,
W.
,
2015
,
Dynamics of Parallel Robots—From Rigid Links to Flexible Elements
,
Springer
,
New York
. ISBN:978-3-319-19787-6.
40.
Gosselin
,
C.
, and
Angeles
,
J.
,
1990
, “
Singularity Analysis of Closed-Loop Kinematic Chains
,”
IEEE Trans. Robot. Autom.
,
6
(
3
), pp.
281
290
.
41.
Bonev
,
I.
,
2002
, “
Geometric Analysis of Parallel Mechanisms
,” Ph.D. thesis,
Université Laval
,
Qubec City, QC, Canada
.
42.
Franchi
,
A.
,
Masone
,
C.
,
Grabe
,
V.
,
Ryll
,
M.
, and
Bülthoff
,
H. H.
,
2012
, “
Modeling and Control of UAV Bearing-Formations With Bilateral High-Level Steering
,”
Int. J. Robot. Res.
,
31
, pp.
1504
1525
.
43.
Tischler
,
C.
,
Hunt
,
K.
, and
Samuel
,
A.
,
1998
, “
A Spatial Extension of Cardanic Movement: Its Geometry and Some Derived Mechanisms
,”
Mech. Mach. Theory
,
33
, pp.
1249
1276
.
44.
Chablat
,
D.
,
Wenger
,
P.
, and
Bonev
,
I.
,
2006
, “
Self Motions of a Special 3-RPR Planar Parallel Robot
,”
Proceedings of the 10th International Symposium on Advances in Robot Kinematics (ARK 2006)
, pp.
221
228
.
45.
Karger
,
A.
, and
Husty
,
M.
,
1998
, “
Classification of All Self-Motions of the Original Stewart-Gough Platform
,”
Comput. Aided Des.
,
30
(
3
), pp.
205
215
.
46.
Michel
,
H.
, and
Rives
,
P.
,
1993
, “
Singularities in the Determination of the Situation of a Robot Effector From the Perspective View of 3 Points
,” Technical Report,
INRIA
.
47.
Hamel
,
T.
, and
Samson
,
C.
,
2017
, “
Riccati Observers for the Non-Stationary PnP Problem
,”
IEEE Trans. Autom. Control
,
63
(
3
), pp.
726
741
.
48.
Fischler
,
M.
, and
Bolles
,
R.
,
1981
, “
Random Sample Consensus: A Paradigm for Model Fitting With Applications to Image Analysis and Automated Cartography
,”
Commun. ACM: Graph. Image Process.
,
24
(
6
), pp.
381
395
.
49.
Kneip
,
L.
,
Scaramuzza
,
D.
, and
Siegwart
,
R.
,
2011
, “
A Novel Parametrization of the Perspective-Three-Point Problem for a Direct Computation of Absolute Camera Position and Orientation
,”
Proceedings of the 2011 IEEE Conference on Computer Vision and Pattern Recognition (CVPR 2011)
,
June
.
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