## 2 Self-Control: Geometrically Non-Linear Analysis
The following values of self-stress were applied: \(-0.421637\cdot S_{0}\) (struts), \(0.182574\cdot S_{0}\) (cables of the upper triangle), \(0.333333\cdot S_{0}\) (diagonal cables), \(0.105409\cdot S_{0}\) (cables of the lower triangle), where \(S_{0}\) is a multiplier of the pre-stressing force. Figure 3: Structure No. 1: influence of self-stress (\(S_{0}\)) on the internal final force in a cable (\(S_{0}\)), with four levels of loading, according to third order theory. Figure 2: Structure No. 1: influence of self-stress (\(S_{0}\)) on the vertical displacement (\(\delta\)), with two levels of loading, according to the second and third order theory. The diagrams presented below show the influence of external load \(P\) on nodal displacement \(\delta\) (Figure 5) and internal forces in two selected elements of the structure: a cable (\(S_{f1}\)) (Figure 6) and a strut (\(S_{f2}\)) (Figure 7). The calculations were performed according to the geometrically non-linear theory. The analyzed tensegrity module exhibits similar behavior as Structure No. 1. Both self-stress and external loading stiffen the structure, but the influence of external loading is bigger when the values of pre-stressing forces are small. Figure 4: Tensegrity module: a three-strut simplex (dimensions units (mm)). Figure 5: Structure No. 2: influence of self-stress on the vertical displacement (\(\delta\)), with two levels of loading, according to the third order theory. Figure 6: Structure No. 2: influence of self-stress on the internal force in a cable (\(S_{f1}\)), with two levels of loading, according to the third order theory. An analogous analysis was conducted for Structure No. 3, a tensegrity plate-like structure. The structure (Figure 8) was built from fourteen three-strut simplex modules, rotated clockwise and counter-clockwise and connected together in nodes. The structure was loaded with concentrated forces \(P\) applied to the nodes of the upper surface. Figure 8: Tensegrity plate-like structure (dimensions units (mm)). (**a**) 3D view; (**b**) Top view. Figure 7: Structure No. 2: influence of self-stress on the internal force in a strut (\(S_{\text{2}}\)), with two levels of loading, according to the third order theory. In the analysis, the same values of self-stress were applied as in the case of the three-strut simplex module. The diagrams presented below show the influence of external load \(P\) on the displacement of the central upper node \(\delta_{A}\) (Figure 9) and internal forces in two selected elements of the structure: a cable (\(S_{f1}\)) (Figure 10) and a strut (\(S_{f2}\)) (Figure 11). The calculations were performed according to the geometrically non-linear theory. The analyzed tensegrity plate-like structure exhibits a similar behavior as two previously-presented structures. Both self-stress and external loading stiffen the structure, but the influence of external loading is bigger when the values of pre-stressing forces are small. Similar analyses were performed for other tensegrity modules and more complex multi-module structures. All of them led to analogous results. All these analyses show that the properties of tensegrity structures do not depend only on self-stress. Tensegrities have the ability of self-control, which is realized by self-stiffening of the structure under the applied external load, which acts in the direction that is consistent with the infinitesimal mechanism mode. This feature is particularly important in the case of non-typical loading, which is hard to foresee. Figure 10: Structure No. 3: influence of self-stress on the internal force in a cable (\(S_{f1}\)), with two levels of loading, according to the third order theory. Figure 9: Structure No. 3: influence of self-stress on the vertical displacement (\(\delta_{A}\)), with two levels of loading, according to the third order theory.