Cooperative Coordination and Formation Control for Multi-agent Systems by Zhiyong Sun

Author:Zhiyong Sun
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

6.1.2 Chapter Organization

The remaining parts of this chapter are organized as follows. In Sect. 6.2, we introduce some results relating to the null space of a rigidity matrix as well as the problem formulation. Section 6.3 provides the main result. Typical simulation results are shown in Sect. 6.4. Finally, Sect. 6.5 concludes this chapter. Proofs for some key lemmas are given in the Appendix Sect. 6.6.

6.2 Preliminaries and Problem Setup

6.2.1 Rigidity Matrix and Its Null Space

We refer the readers to Sect. 2.​3 of Chap. 2 for background on graph rigidity theory. From Theorem 2.​1 in Chap. 2, one knows that the dimension of the null space of R(p) for an infinitesimally rigid framework in the d-dimensional space is . The following lemma characterizes the structure of its null space.

Lemma 6.1

(Null space of the rigidity matrix) Suppose the framework is infinitesimally rigid with the associated rigidity matrix denoted as R(p).

The case: The null space of R(p) is of dimension 3 and is described as , where

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