@inproceedings{3fcc5d6722b549e8a5115038fe84ed46,
title = "Rigid formation shape control in general dimensions: An invariance principle and open problems",
abstract = "The control systems resulting from seeking to control the shape of a rigid formation are frequently described by gradient flows which are designed to minimize some predefined and relevant potential functions linked to the desired formation shape. This paper discusses and establishes a rank-preserving property of such formation shape stabilization systems. We further show some properties of the degenerate critical formations that live in a lower dimensional space, and prove that they are unstable. The implication of these results is that if all the agents start with generic initial positions, then their trajectories will be strictly bounded away from the set of degenerate formations at any finite or infinite time. By establishing this invariance principle, some previous results on equilibrium analysis for rigid formation systems can be greatly simplified. The results also have applications in other fields such as the multidimensional scaling study. The results are obtained from a joint analysis of rank-preserving flow theory, graph rigidity theory and invariant manifold theory.",
keywords = "Control systems, Convergence, Jacobian matrices, Shape, Stability analysis, Sun, Symmetric matrices",
author = "Zhiyong Sun and Uwe Helmke and Anderson, {Brian D.O.}",
note = "Publisher Copyright: {\textcopyright} 2015 IEEE.; 54th IEEE Conference on Decision and Control, CDC 2015 ; Conference date: 15-12-2015 Through 18-12-2015",
year = "2015",
month = feb,
day = "8",
doi = "10.1109/CDC.2015.7403178",
language = "English",
series = "Proceedings of the IEEE Conference on Decision and Control",
publisher = "Institute of Electrical and Electronics Engineers Inc.",
pages = "6095--6100",
booktitle = "54rd IEEE Conference on Decision and Control,CDC 2015",
address = "United States",
}