TY - GEN
T1 - Undirected rigid formations are problematic
AU - Mou, S.
AU - Morse, A. S.
AU - Belabbas, M. A.
AU - Anderson, B. D.O.
N1 - Publisher Copyright:
© 2014 IEEE.
PY - 2014
Y1 - 2014
N2 - By an undirected rigid formation of mobile autonomous agents is meant a formation based on graph rigidity in which each pair of 'neighboring' agents is responsible for maintaining a prescribed target distance between them. In a recent paper, a systematic method was proposed for devising gradient control laws for asymptotically stabilizing a large class of rigid, undirected formations in two-dimensional space assuming all agents are described by kinematic point models. The aim of this paper is to explain what happens to such formations if neighboring agents have slightly different understandings of what the desired distance between them is supposed to be, or equivalently, if neighboring agents have differing estimates of what the actual distance between them is. In either case, what one would expect would be a gradual distortion of the formation from its target shape as discrepancies in desired or sensed distances increase. While this is observed for the gradient laws in question, something else quite unexpected happens at the same time. It is shown that for any rigidity-based, undirected formation which is comprised of three or more agents, that if some neighboring agents have slightly different understandings of what the desired distances between them are suppose to be, then almost for certain, the trajectory of the resulting distorted but rigid formation will converge exponentially fast to a closed circular orbit in two-dimensional space which is traversed periodically at a constant angular speed.
AB - By an undirected rigid formation of mobile autonomous agents is meant a formation based on graph rigidity in which each pair of 'neighboring' agents is responsible for maintaining a prescribed target distance between them. In a recent paper, a systematic method was proposed for devising gradient control laws for asymptotically stabilizing a large class of rigid, undirected formations in two-dimensional space assuming all agents are described by kinematic point models. The aim of this paper is to explain what happens to such formations if neighboring agents have slightly different understandings of what the desired distance between them is supposed to be, or equivalently, if neighboring agents have differing estimates of what the actual distance between them is. In either case, what one would expect would be a gradual distortion of the formation from its target shape as discrepancies in desired or sensed distances increase. While this is observed for the gradient laws in question, something else quite unexpected happens at the same time. It is shown that for any rigidity-based, undirected formation which is comprised of three or more agents, that if some neighboring agents have slightly different understandings of what the desired distances between them are suppose to be, then almost for certain, the trajectory of the resulting distorted but rigid formation will converge exponentially fast to a closed circular orbit in two-dimensional space which is traversed periodically at a constant angular speed.
UR - http://www.scopus.com/inward/record.url?scp=84988221454&partnerID=8YFLogxK
U2 - 10.1109/CDC.2014.7039453
DO - 10.1109/CDC.2014.7039453
M3 - Conference contribution
AN - SCOPUS:84988221454
T3 - Proceedings of the IEEE Conference on Decision and Control
SP - 637
EP - 642
BT - 53rd IEEE Conference on Decision and Control,CDC 2014
PB - Institute of Electrical and Electronics Engineers Inc.
T2 - 2014 53rd IEEE Annual Conference on Decision and Control, CDC 2014
Y2 - 15 December 2014 through 17 December 2014
ER -