# An analysis of rigidity indices of formation in multi-agent systems

#### Abstract

The formation control of multi-agent systems has been studied by many researchers in various contexts, such as robotics, unmanned air vehicle, underwater vehicles [1] and so on *Rigidity* is a critical concept in the study of formation control as it characterizes the ability of a multi-agent system to persist a desired formation despite the presence of (and possibly significant) sensing errors, communication delays, and environmental perturbations. An early paper on applying the notion of rigidity to multi-agent systems can be found in [2], which employs results in graph theory such as the Laman's theorem [3] and the rigidity matrix theorem [4, 5] to determine *qualitatively * whether a formation is rigid or non-rigid. Dynamics of formation such as splitting, merging [6], and closing ranks [7] have also been explored. ^ In [8], *quantitative measures* of formation rigidity are proposed based on the notion of a stiffness matrix by analogizing a formation to a mass-spring system: each node (agent) corresponds to a mass and each edge (communication link) connecting a pair of nodes corresponds to a spring with a given elastic constant. The rigidity indices are then derived from the eigenvalues of the stiffness matrix associated with the overall elastic structure. With these quantitative measures, one can now formally compare the degree of rigidity among different formations previously only known to be rigid. ^ In this thesis, we focus on two quantitative measures of formation rigidity: the worst rigidity index (WRI) and a novel measure called the mean rigidity index (MRI).We observe and verify the properties of the quantitative measurement through analysis in Henneberg construction using rigidity indices. Furthermore, we try to find the most rigid formations as measured by these indices through an iterative algorithm that optimizes both the node locations and the formation topology simultaneously. Such optimized formations typically result in better robustness of the multi-agent systems on formation control or localization tasks.^

#### Degree

M.S.E.C.E.

#### Advisors

JIANGHAI HU, Purdue University.

#### Subject Area

Engineering, Automotive|Engineering, Electronics and Electrical

Off-Campus Purdue Users:

To access this dissertation, please **log in to our
proxy server**.