This paper hands in a review of the basic issues about the statics of tensegrity structures. Definitions and notation for the most important concepts, borrowed from the vast existing literature, are summarized. All of these concepts and definitions provide a complete mathematical framework to analyze the rigidity and stability properties of tensegrity structures from three different, but related, points of view: motions, forces and energy approaches. Several rigidity and stability definitions are presented in this paper and hierarchically ordered, from the strongest condition of infinitesimal rigidity to the more wide concept of simple rigidity, so extending some previous classifications already available.Important theorems regarding the relationship between these definitions are also put together to complete the static overview of tensegrity structures. Examples of different tensegrity structures belonging to each of the rigidity and stability categories presented are described and analyzed. Concluding the static analysis of tensegrity structures, a review of existing form-finding methods is presented.

The main objective of this paper is twofold. First, to conclude the overview about tensegrity frameworks, started by the same authors in a previous work, covering the most important dynamic aspects of such structures. Here, the most common approaches to tensegrity dynamic modeling used so far are presented, giving the most important results about their dynamic behavior under external action.Also, the main underlying problems are identified which allow the authors to give a clear picture of the main research lines currently open, as well as the most relevant contributions in each of them, which is in fact the second main objective of this paper. From the extensive literature available on the subject, four main areas have been identified: design and form-finding methods which deal with the problem of finding stable configurations, shape changing algorithms which deal with the problem of finding stable trajectories between them and, also control algorithms which take into account the dynamic model of the tensegrity structure and possible external perturbations to achieve the desired goal and performance.Finally, some applications of such structures are presented emphasizing the increasing interest of the scientific community on tensegrity structures.

Abstract-Tensegrity structures appeared in the science community about half a century ago, but they have already been applied to several heterogeneous research fields, such as architecture, civil engineering, space and even biology. Such structures keep a stable volume in space due to an intricate balance of forces between a disjoint set of rigid elements (bars) and a continuous set of tensile elements (cables).The use of tensegrity structures in robotics is still new and there exist only a handful of works about this subject. Some of their main features such as light weight, flexibility, energetic efficiency and redundancy, make them interesting candidates for both mobile robots and manipulators. In this paper, a new method to detect and avoid both internal collisions between the structure members and external collisions with the environment is presented. In this way, we are providing a fundamental tool to develop more complete form-finding procedures and pathplanning strategies for tensegrity structures.

Abstract