Combinatorial Method for Checking Stability in Tensegrity Structures

Tensegrity structures have a great value in the academia and in industry, in particular for adjustable tensegrity structures that can sustain external forces when deployed. The main problem with the latter systems is checking their stability during deployment. One of the most famous methods for checking stability was developed 20 years ago by two mathematicians [1]. They showed that if the tensegrity structure is redundant then the check is simple. But if it is a determinate tensegrity structure then there is a need to calculate the velocities of all the joints and then after matrix multiplications a scalar is obtained. If the scalar is negative then it is concluded that the tensegrity system is unstable without knowing which element causes the problem and what should be done in order to stabilize it. This paper proves that if the structure is a minimal rigid determinate structure, named Assur Graph, then there is a simple method for checking the stability. The proposed method suggests to remove a cable, calculate the curvature radius of one its inner joint and then conclude whether the structure is stable or not. In case that it was concluded that the system is unstable, then to shorten the cable so it becomes stable. The main topic from the combinatorial method being used in this paper is the special properties of Assur Graphs, in particular their singular positions. It is proved that from all the determinate structures only the Assur Graphs have these special singular properties, upon which the proposed method and the proof relies on.