This paper deals with a refined analysis and modification of existing results on the flocking algorithms proposed for the second order dynamic agents. In the present work, the limiting condition of ever connectivity is removed and it is proved that the flocking can be reached if only the union of the network proximity graphs during nonoverlapping time intervals becomes connected frequently enough. Also, it is proved that including a static virtual leader cannot model the group objective of achieving the desired velocity and it will stop eventually at a predefined point in the space. The convergence rate to this fixed point is determined too. The last contribution of this paper is definition of group configuration when only a fraction of agents are informed about the virtual leader.

References

References
1.
Couzin
,
I. D.
,
Krause
,
J.
,
James
,
R.
,
Ruxton
,
G. D.
, and
Franks
,
N. R.
,
2002
, “
Collective Memory and Spatial Sorting in Animal Groups
,”
Theor. Biol.
,
218
(
1
), pp.
1
11
.10.1006/jtbi.2002.3065
2.
Parrish
,
J. K.
,
Viscido
,
S. V.
, and
Grunbaum
,
D.
,
2002
, “
Self-Organized Fish Schools: An Examination of Emergent Properties
,”
Biol. Bull.
,
202
(
3
), pp.
296
305
.10.2307/1543482
3.
O'Loan
,
O. J.
, and
Evans
,
M. R.
,
1999
, “
Alternating Steady State in One Dimensional Flocking
,”
J. Phys. A: Math. General
,
32
(
8
), pp.
99
105
.10.1088/0305-4470/32/8/002
4.
Mogilner
,
A.
, and
Edelstein-Keshet
,
L.
,
1999
, “
A Non-Local Model for a Swarm
,”
J. Math. Biol.
,
38
, pp.
534
570
.10.1007/s002850050158
5.
Reynolds
,
C. W.
,
1987
, “
Flocks, Herds, and Schools: A Distributed Behavioral Model
,”
Comput. Graph.
,
21
(
4
), pp.
25
34
.10.1145/37402.37406
6.
Cortes
,
J.
,
Martinez
,
S.
, and
Bullo
,
F.
,
2004
, “
Robust Rendezvous for Mobile Autonomous Agents via Proximity Graphs in Arbitrary Dimensions
,”
IEEE Trans. Autom. Control
,
51
(
8
), pp.
1289
1298
.10.1109/TAC.2006.878713
7.
Su
,
H.
,
Chen
,
G.
,
Wang
,
X.
, and
Lin
,
Z.
,
2011
, “
Adaptive Second-Order Consensus of Networked Mobile Agents With Nonlinear Dynamics
,”
Automatica
,
47
(
2
), pp.
368
375
.10.1016/j.automatica.2010.10.050
8.
Tian
,
Y. P.
, and
Zhang
,
Y.
,
2012
, “
High-Order Consensus of Heterogeneous Multi-Agent Systems With Unknown Communication Delays
,”
Automatica
,
48
(
6
), pp.
1205
1212
.10.1016/j.automatica.2012.03.017
9.
Notarstefano
,
G.
,
Egerstedt
,
M.
, and
Haque
,
M.
,
2011
, “
Containment in Leader-Follower Networks With Switching Communication Topologies
,”
Automatica
,
47
(
5
), pp.
1035
1040
.10.1016/j.automatica.2011.01.077
10.
Moreau
,
L.
,
2005
, “
Stability of Multi-Agent Systems With Time-Dependent Communication Links
,”
IEEE Trans. Autom. Control
,
50
(
2
), pp.
169
182
.10.1109/TAC.2004.841888
11.
Tanner
,
H. G.
,
Pappas
,
G. J.
, and
Kumar
,
V.
,
2004
, “
Leader to Formation Stability
,”
IEEE Trans. Rob. Autom.
,
20
(
3
), pp.
443
455
.10.1109/TRA.2004.825275
12.
Lin
,
Z.
,
Francis
,
B.
, and
Maggiore
,
M.
,
2005
, “
Necessary and Sufficient Graphical Conditions for Formation Control of Unicycles
,”
IEEE Trans. Autom. Control
,
50
(
1
), pp.
121
127
.10.1109/TAC.2004.841121
13.
Su
,
H.
,
Wang
,
X.
, and
Chen
,
G.
,
2009
, “
A Connectivity-Preserving Flocking Algorithm for Multi-Agent Systems Based Only on Position Measurements
,”
Int. J. Control
,
82
(
7
), pp.
1334
1343
.10.1080/00207170802549578
14.
Dimarogonas
,
D. V.
, and
Kyriakopoulos
,
K. J.
,
2008
, “
Connectedness Preserving Distributed Swarm Aggregation for Multiple Kinematic Robots
,”
IEEE Trans. Rob.
,
24
(
5
), pp.
1213
1223
.10.1109/TRO.2008.2002313
15.
Su
,
H.
,
Wang
,
X.
, and
Chen
,
G.
,
2010
, “
Rendezvous of Multiple Mobile Agents With Preserved Network Connectivity
,”
Syst. Control Lett.
,
59
(
5
), pp.
313
322
.10.1016/j.sysconle.2010.03.006
16.
Ogren
,
P.
,
Egerstedt
,
M.
, and
Hu
,
X.
,
2002
, “
A Control Lyapunov Function Approach to Multi-Agent Coordination
,”
IEEE Trans. Rob. Autom.
,
18
(
5
), pp.
847
851
.10.1109/TRA.2002.804500
17.
Leonard
,
N.
, and
Friorelli
,
E.
,
2001
, “
Virtual Leaders, Artificial Potentials and Coordinated Control of Groups
,”
Proceedings of 40th IEEE Conference Decision Control
, Orlando, FL, Vol. 3, pp.
2968
2973
.
18.
Fax
,
J. A.
, and
Murray
,
R. M.
,
2002
, “
Graph Laplacians and Stabilization of Vehicle Formations
,”
Presented in the 15th IFAC Congress
, Barcelona, Spain.
19.
Arcak
,
M.
,
2007
, “
Passivity as a Design Tool for Group Coordination
,”
IEEE Trans. Automatic Control
,
52
(
8
), pp.
1380
1390
.10.1109/TAC.2007.902733
20.
Qu
,
Z.
,
Wang
,
J.
, and
Hull
,
R. A.
,
2008
, “
Cooperative Control of Dynamical Systems With Application to Autonomous Vehicles
,”
IEEE Trans. Autom. Control
,
53
(
4
), pp.
894
911
.10.1109/TAC.2008.920232
21.
Khatib
,
O.
,
1986
, “
Real-Time Obstacle Avoidance for Manipulators and Mobile Robots
,”
Int. J. Rob. Res.
,
5
(
1
), pp.
90
98
.10.1177/027836498600500106
22.
Rimon
,
E.
, and
Koditschek
,
D. E.
,
1992
, “
Exact Robot Navigation Using Artificial Potential Functions
,”
IEEE Trans. Rob. Autom.
,
8
(
5
), pp.
501
518
.10.1109/70.163777
23.
Tanner
,
H. G.
,
Jadbabaie
,
A.
, and
Pappas
,
G. J.
,
2007
, “
Flocking in Fixed and Switching Networks
,”
IEEE Trans. Autom. Control
,
52
(
5
), pp.
863
868
.10.1109/TAC.2007.895948
24.
Olfati Saber
,
R.
,
2006
, “
Flocking for Multi-Agent Dynamic Systems: Algorithms and Theory
,”
IEEE Trans. Autom. Control
,
51
(
3
), pp.
401
420
.10.1109/TAC.2005.864190
25.
Cucker
,
F.
, and
Smale
,
S.
,
2007
, “
Emergent Behavior in Flocks
,”
IEEE Trans. Autom. Control
,
52
(
5
), pp.
852
862
.10.1109/TAC.2007.895842
26.
Su
,
H.
,
Wang
,
X.
, and
Chen
,
G.
,
2009
, “
A Connectivity-Preserving Flocking Algorithm for Multi-Agent Systems Based Only on Position Measurements
,”
Int. J. Control
,
82
(
7
), pp.
1334
1343
.10.1080/00207170802549578
27.
Zavlanos
,
M. M.
,
Tanner
,
H. G.
,
Jadbabaie
,
A.
, and
Pappas
,
G. J.
,
2009
, “
Hybrid Control for Connectivity Preserving Flocking
,”
IEEE Trans. Autom. Control
,
54
(
12
), pp.
2869
2875
.10.1109/TAC.2009.2033750
28.
Su
,
H.
,
Chen
,
G.
,
Wang
,
X.
, and
Lin
,
Z.
,
2011
, “
Adaptive Second-Order Consensus of Networked Mobile Agents With Nonlinear Dynamics
,”
Automatica
,
47
(
2
), pp.
368
375
.10.1016/j.automatica.2010.10.050
29.
Guo
,
W.
,
Lu
,
J.
,
Chen
,
S.
, and
Yu
,
X.
,
2011
, “
Second-Order Tracking Control for Leader-Follower Multi-Agent Flocking in Directed Graphs With Switching Topology
,”
Syst. Control Lett.
,
60
(
12
), pp.
1051
1058
.10.1016/j.sysconle.2011.09.020
30.
Su
,
H.
,
Wang
,
X.
and
Lin
,
Z.
,
2009
, “
Flocking of Multi-Agents With a Virtual Leader
,”
IEEE Trans. Autom. Control
,
54
(
2
), pp.
293
307
.10.1109/TAC.2008.2010897
31.
Luo
,
X.
,
Li
,
S.
, and
Guan
,
X.
,
2010
, “
Flocking Algorithm With Multi-Target Tracking for Multi-Agent Systems
,”
Pattern Recogn. Lett.
,
31
(
9
), pp.
800
805
.10.1016/j.patrec.2010.01.014
32.
Wen
,
G.
,
Duan
,
Z.
,
Li
,
Z.
, and
Chen
,
G.
,
2012
, “
Flocking of Multi-Agent Dynamical Systems With Intermittent Nonlinear Velocity Measurements
,”
Int. J. Rob. Nonlinear Control
,
22
(
16
), pp.
1790
1805
.10.1002/rnc.1784
33.
Godsil
,
C.
, and
Royle
,
G.
,
2001
,
Algebraic Graph Theory
,
Springer-Verlag
,
New York
.
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