Rigidity of similarity-based formation and formation shape stabilization

In this paper, a novel formation, i.e., similarity-based formation, is proposed and its rigidity is explored. Based on the concept of geometric similarity, the definition of similarity-based formation is given. As the inherent property of geometric similarity, the proposed formation has the adaptability of similarity transformations, including translation, rotation and scaling, which provides great flexibility of formation control. To describe the expected formation shape mathematically, three different approaches with two different geometric variables are proposed, and the corresponding conditions of uniquely determining the formation shape are derived. The distributed formation control laws with different kinds of similarity constraints are designed to make the formation locally converge to the desired formation shape with desired orientation and scale. Simulations demonstrate the validity of the approaches of defining similarity-based formation and the effectiveness of formation control laws.