Stiffness matrix and quantitative measure of formation rigidity

Rigidity of formation is an important concept in multi-agent localization and control problems. There are well-developed existing methods to test rigidity of a given graph. However, little work is done on quantitative measurement of formation rigidity. In this paper, the concept of stiffness matrix of a formation is proposed to represent both its rigidity and structural information, from which we then derive the worst-case rigidity index that can be used as an applicable quantitative measure of formation rigidity. Its validity is shown through the establishment of its related properties that conform to intuitive assumptions and practical applications as well as some examples.