The equilibrium and form-finding of general tensegrity systems with rigid bodies

We develop a general approach to study the equilibrium and form-finding of any general tensegrity systems with rigid bodies. The equilibrium equations are derived in an explicit form in terms of a nodal coordinate and orientation parameter as the minimal coordinate. The nodal vector consists of nodes (either free or pinned) in the pure bar-string tensegrity network and nodes on the rigid bodies (those connected to the pure bar-string tensegrity network). Based on the Lagrangian method, the nonlinear statics of the general tensegrity system in terms of the minimal coordinate is first given. Then, we linearize the statics equation and obtain its equivalent form, in terms of the force vector of the compressive and tensile members, to analyze structure equilibrium configurations and prestress modes. To study the system's stability and have a comprehensive insight into the materials and structure members, we present the tangent stiffness matrix as a combination of the structure's prestress, material, and geometric information. It is also shown that without rigid bodies, the governing equations of the general tensegrity system yield to the classical tensegrity structure (pure string-bar network). Form-finding of general tensegrity is implemented based on solving the nonlinear equilibrium equation, where the modification of tangent stiffness matrix and line search algorithm is used. Numerical examples demonstrate the capability of our developed method in finding the feasible prestress modes, conducting form-finding and prestress designs, and checking the structural robustness of any tensegrity systems with rigid bodies.