  This paper presents the equilibrium and stiffness study of clustered
tensegrity structures (CTS) considering pulley sizes. We first derive the
geometric relationship between clustered strings and pulleys, where the nodal
vector is chosen as the generalized coordinate. Then, the equilibrium equations
of the clustered tensegrity structure with pulleys based on the Lagrangian
method are given. Since the stiffness of a structure is usually weakened when
using clustering strings, we formulate the tangent stiffness matrix equations
for analysis. It is also shown that as pulley sizes go to zero, the governing
equations of the clustered tensegrity system with pulleys yield to the
classical clustered tensegrity structure without pulleys, which is consistent
with the existing literature. Three examples are demonstrated to validate the
given theory. The proposed method allows one to conduct equilibrium, stiffness,
and robustness studies of cluster tensegrity structures with pulleys.
Nevertheless, the approach developed in this paper is not limited to the
tensegrity structures. It can also be applied to a wide range of applications
with pulley-rope systems, such as drilling rigs, ocean platform anchors, and
cargo cranes.
