The paper focuses on the static behavior of double-layered tensegrity grids. Due to the specific characteristics, like the self-stress states and infinitesimal mechanisms, tensegrities can be used as deployable structures. For such structures, the possibility of the control of the behavior is very important. The main purpose of the work is to prove that the control of tensegrity structures with mechanisms is possible. The stiffness of such structures is found to depend not only on the geometry and material properties, but also on the initial prestress level and external load. In the case, when mechanisms do not exist, structures are insensitive to the initial prestress. It is possible to control the occurrence of mechanisms by changing the support conditions of the structure. Grids built with modified Simplex modules are considered. Two-stage analysis is performed. Firstly, the presence of the characteristic tensegrity features is examined and then, on that basis, the structures are classified into one of two classes. Next, the influence of the level of initial prestress on the behavior of structures under static load is analyzed. To evaluate this behavior, a geometrically non-linear model is used.