Instability results from purely rotational stiffness for general tensegrity structure with rigid bodies

The stability and stiffness analysis of tensegrities is significant for their applications in robotic areas. Generally, tensegrity structures' stiffness is divided into a non-negative definite material stiffness and an indefinite geometric stiffness. The stability of tensegrities is closely related to the geometric stiffness caused by the prestress of cables. However, the nodal geometric stiffness derived from nodal static equilibrium equations hides the root of instability for tensegrity structures. This article develops a new derivation of the total stiffness for a general tensegrity structure based on rigid bodies' static equilibrium equations. And then, it subdivides the geometric stiffness into a non-negative definite stiffness and an indefinite stiffness. The indefinite stiffness is purely rotational stiffness and the only root of instability for tensegrity structures, which has remarkable geometrical significance. By analyzing such indefinite stiffness, a set of sufficient and necessary conditions are derived to guarantee tensegrity structures' stability. Some numerical examples are presented to verify the effectiveness and versatility of the proposed approach.