# A New Concept Compliant Platform with Spatial Mobility and Remote Actuation
## 2 Description of the New Concept Platform
### Direct Position Analysis  
Thanks to the construction of the PRBM depicted in Figure 2b, it is possible to identify some fundamental characteristics of the platform, as they have been introduced in the literature. For example, it is clear that this platform is not a fully-parallel platform, because two DoF are needed for each leg. With reference to Figure 3a, the \(i\)th leg (with \(i=1,\ldots,0\)) will be composed of the pseudo rigid links \(C_{i}A_{i}\) and \(A_{i}P_{i}\), which respectively correspond to Type (g) and Type (d) links of the original compliant structure. For the generic \(i\)th leg, one DoF is firstly required to define the angular position of Type (g) link \(C_{i}A_{i}\). An angle \(\alpha_{i}\) can be introduced to measure the rotation of \((C_{i}A_{i})\) with respect to its position in the undeformed configuration of the CSFH joint. Since the latter corresponds to a revolute joint with a known rotation axis, the angle \(\alpha_{i}\) uniquely identifies the position of the center \(A_{i}\) of the spherical pair. Secondly, another DoF is needed to assess the position of the upper Type (d) \(A_{i}P_{i}\) link of the \(i\)th leg. Indeed, an assigned pull command on the Type (f) cable has the effect of bringing points \(E_{i}\) and \(B\) closer to each other. This effect could be similarly obtained by introducing in the PRBM a virtual linear actuator being active along the line \(BE_{i}\). Any contraction of the linear actuator \(BE_{i}\) corresponds to a pull command of the (f) rope, while an extension of the linear actuator would represent a reduction of the cable tension. Since mobility is granted by the elasticity of the structure, its overall configuration will depend on the whole set of tensions that are assigned to the six cables. Furthermore, it is worth noticing that positive tensions (push) are not considered here, and therefore, the maximum elongation of the three virtual linear actuators will depend on the whole balance between the tensions that are applied to the cables and the structural elasticity. For the linear actuator \(BE_{i}\), the pulled distance \(d_{i}\) can be introduced as the shortening of the original length \(|BE_{i}|\) with respect to the distance between points \(B\) and \(E_{i}\) in the undeformed structure configuration. Once the input parameters \(\alpha_{i}\) and \(d_{i}\) have been assigned, the chain \(BA_{i}E_{i}\) behaves as a rigid structure, which may rotate about the axis \(w_{i}\) passing through points \(A_{i}\) and \(B\) (see Figure 3b). Therefore, point \(E_{i}\) describes a circle \(\Gamma_{E_{i}}\) laying on a plane \(\pi_{H_{i}}\) that is orthogonal to \(w_{i}\) and whose  
Figure 3: (**a**) The circles \(\Gamma_{E_{i}}\) and \(\Gamma_{i}\) as loci of points \(E_{i}\) and \(P_{i}\), respectively, for the \(i\)th leg; (**b**) the virtual bar \(Q_{i}P_{i}\) rotating along the axis \(w_{i}\) parallel to the line \(A_{i}B\). center \(H_{i}\) is on line \(w_{i}\). Since \(A_{i}P_{i}\) can be regarded as a rigid link, point \(P_{i}\) is also forced to lay on a circle \(\Gamma_{i}\) that lays on a plane \(\pi_{Q_{i}}\) parallel to \(\pi_{H_{i}}\). The center \(Q_{i}\) of \(\Gamma_{i}\) is on the line \(w_{i}\). Considering the three loci \(\Gamma_{i}\) (with \(i=1,\ldots,3\)) of the possible positions of the points \(P_{i}\), it is clear that three points of the upper platform, coincident with \(P_{1}\), \(P_{2}\), and \(P_{3}\), will belong to the three circles \(\Gamma_{1}\), \(\Gamma_{2}\), and \(\Gamma_{3}\), respectively. However, exactly the same constraint could be imposed by introducing a virtual link \(Q_{i}P_{i}\), represented in Figure 3b, that rotates about the \(w_{i}\) axis. Therefore, three new links \(Q_{i}P_{i}\), with \(i=1,\ldots,3\), can be used to get rid of the three whole chains made of links \(C_{i}A_{i}\), \(A_{i}P_{i}\), and the linear actuators \(BE_{i}\).