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Title: Weak rigidity theory and its application to formation stabilization
Authors: Jing, GS 
Zhang, GF 
Lee, HWJ 
Wang, L
Issue Date: 2018
Source: SIAM journal on control and optimization, 2018, v. 56, no. 3, p. 2248-2273
Abstract: This paper introduces the notion of weak rigidity to characterize a framework by pairwise inner products of interagent displacements. Compared to distance-based rigidity, weak rigidity requires fewer constrained edges in the graph to determine a geometric shape in an arbitrarily dimensional space. A necessary and sufficient graphical condition for infinitesimal weak rigidity of planar frameworks is derived. As an application of the proposed weak rigidity theory, a gradient-based control law and a nongradient-based control law are designed for a group of single-integrator modeled agents to stabilize a desired formation shape, respectively. Using the gradient control law, we prove that an infinitesimally weakly rigid formation is locally exponentially stable. In particular, if the number of agents is one greater than the dimension of the space, a minimally infinitesimally weakly rigid formation is almost globally asymptotically stable. In the literature of rigid formation, the sensing graph is always required to be rigid. Using the nongradient control law based on weak rigidity theory, it is not necessary for the sensing graph to be rigid for local exponential stability of the formation. A numerical simulation is performed for illustrating the effectiveness of our main results.
Keywords: Graph rigidity
Rigid formation
Multiagent systems
Matrix completion
Publisher: Society for Industrial and Applied Mathematics
Journal: SIAM journal on control and optimization 
ISSN: 0363-0129
EISSN: 1095-7138
DOI: 10.1137/17M1122049
Rights: © 2018 Society for Industrial and Applied Mathematics
Posted with permission of the publisher.
The following publication Jing, G., Zhang, G., Lee, H. W. J., & Wang, L. (2018). Weak Rigidity Theory and Its Application to Formation Stabilization. SIAM Journal on Control and Optimization, 56(3), 2248-2273 is available at
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