# Form-Finding Analysis of a Class 2 Tensegrity Robot
## 3 Static Analysis  
For a tensegrity robot, the kinematic position analysis does not provide sufficient conditions to determine whether the robot can adopt a particular geometric configuration. This is because tensegrity robots can only adopt geometrical configurations of equilibrium, i.e., geometric configurations in which the forces exerted by the flexible elements, only tension forces, can oppose to both the forces exerted by the rigid elements, only compression forces, and the external forces applied to the robot. Consequently, it is necessary to complement the forward position kinematic analysis with the static analysis of the robot to determine if a particular geometric configuration is a geometric configuration of equilibrium. In general, if a rigid body is subjected to the action of multiple forces, \(\vec{f}_{1}\), \(\vec{f}_{2}\),..., \(\vec{f}_{n}\), as shown in Figure 3, it is possible to find an equivalent force-torque system that imprints the same physical effect on the rigid body as the original force system [26] (Figure 4). The equivalent force, \(\vec{f}_{e}\), and the equivalent moment, \(\vec{m}_{e_{O}}\), acting on the rigid body B, are given, respectively, by:  
\[\vec{f}_{e} = \sum_{i=1}^{n}\vec{f}_{i} \tag{18}\] \[\vec{m}_{e_{O}} = \sum_{i=1}^{n}\vec{m}_{O_{i}} \tag{19}\]  
Figure 4: Equivalent force system. Figure 3: Original force system. If the equivalent force-torque system described in the Equations (18) and (19) equals zero, the external forces form a zero-equivalent system, and the rigid body is said to be in equilibrium. The conditions of the equilibrium above can be expressed in general terms as:  
\[\vec{f}_{e} = \sum_{i=1}^{n}\vec{f}_{i}=\vec{0} \tag{20}\] \[\vec{m}_{e_{0}} = \sum_{i=1}^{n}\vec{m}_{O_{i}}=\vec{0} \tag{21}\]  
Equations (20) and (21) are taken as a basis in the development of the static analysis of the class 2 tensegrity robot. Additionally, the origin of the \(\Sigma_{0}\) framework, attached to the center of the rotation axis of the first joint joined to the fixed base of the robot, is used as a point of reference for calculating displacements, forces and moments contained in the elements of the robot. Similarly, the \(\Sigma_{M}\) frame, attached to the center of the moving platform, is used to describe the desired geometric configuration, i.e., the coordinates of the moving reference system as well as the lengths of the flexible elements. Additionally, it is considered that the weight of the moving platform is concentrated in the centroid, G. Based on the previous considerations, the problem of static analysis can be posed as follows:  
From the forward and inverse kinematics of position analyses are obtained:  
* The lengths \(\mathrm{L_{C_{1}}}\) and \(\mathrm{L_{C_{2}}}\) correspond to cables \(\mathrm{C_{1}}\) and \(\mathrm{C_{2}}\), respectively. * The lengths \(\mathrm{L_{r_{1}}}\) and \(\mathrm{L_{r_{2}}}\) correspond to springs \(\mathrm{r_{1}}\) and \(\mathrm{r_{2}}\), respectively. * The position and orientation of the reference frameworks \(\Sigma_{M}\), \(\Sigma_{M_{1}}\), \(\Sigma_{M_{2}}\), \(\Sigma_{R_{1}}\) and \(\Sigma_{R_{2}}\) are with respect to the reference framework \(\Sigma_{W}\). Furthermore, the total mass of the robot is known. The static analysis requires knowing the set of forces applied by the flexible elements on the moving platform to ensure that the robot is in a geometric equilibrium configuration, according to the conditions described in Equations (20) and (21). When replacing the fixed base, as well as the flexible elements by the forces and moments that the aforementioned elements exert on the moving platform, it results in the free-body diagram used for the static analysis shown in Figure 5.  
Forces \(\vec{f}_{\mathrm{C_{1}}}\) and \(\vec{f}_{\mathrm{C_{2}}}\) refer to the forces exerted by cables \(\mathrm{C_{1}}\) and \(\mathrm{C_{2}}\), respectively, on the moving platform.