# A New Concept Compliant Platform with Spatial Mobility and Remote Actuation
## 2 Description of the New Concept Platform
### Deduction of the Pseudo-Rigid Body Equivalent Mechanism  
A compliant mechanism can be generally modeled as its equivalent so-called pseudo-rigid body mechanism (PRBM). The PRBM is a mechanism with only ordinary kinematic pairs, which presents approximately the same motion as the original compliant mechanism in the neighborhood of the neutral configuration. For this purpose, each elastic joint is replaced by an ordinary kinematic pair, which can offer mobility that is compatible with the selective compliance characteristics of that  
Figure 1: A view of the compliant platform and of all its elements: platform (**a**); platform hole (**b**); type \(S\) elastic joints (**c**); type-U elastic joints (**e**); upper (**d**) and lower (**g**) links; upper (**f**) and lower cables (**i**); conjugate surface flexure hinge (CSFH) hinges (**h**); base link (**k**). elastic joint. Figure 2 shows the original compliant mechanism (Figure 2a) and its corresponding spatial PRBM (Figure 2b). With reference to Figures 1 and 2, the elastic joints can be distinguished according to the following characteristics. * A Type (e) elastic joint is delimited by a toroidal surface obtained by revolving a circular arc around a vertical axis \(\gamma_{E}\). As a consequence, the neck cross-sectional area is normal to \(\gamma_{E}\) and is positioned in correspondence to the symmetry middle plane of the joint surface. The relative motion between the two connected pseudo-rigid parts has at least two degrees of freedom (DoF), namely those due to the bending along two orthogonal planes passing through \(\gamma_{E}\). Furthermore, one more DoF could be added depending on whether the neck section allows the two opposite end sections to rotate significantly, one with respect to the other, about \(\gamma_{E}\). For Type (e) joints, the choice of the diameter of the neck section is decisive for assuming whether the amount of torsional rotation might play a significant role. If so, Type (e) can be replaced by a three-DoF spherical S joint, otherwise a two DoF (Hooke's) universal U will be used. In the latter case, the flexure works approximately as a Cardan joint whose center is placed at the center of elastic weights of the elastic joint itself. * A Type (c) elastic joint is also delimited by a portion of a toroidal surface obtained by the rotation of a circular arc around an axis of revolution \(\gamma_{C}\
eq\gamma_{E}\). For this type of joint, the rotations about \(\gamma_{C}\) are considered to be always non-negligible. In fact, they are provided with an even more reduced middle cross-section that gives rise to possible torsion rotations about \(\gamma_{c}\). Therefore, the joints (c) substantially have the function of spherical elastic joints, the equivalent center of the spherical kinematic pair being placed at the center of elastic weights of the joint itself. For this reasons, Type (c) joints will be replaced by spherical joints S.
* Type (h) elastic joints belong to the class of the conjugate surface flexure hinge (CSFH). This kind of flexure has been patented (see Section 7) and extensively studied for several aspects: their theoretical behavior [38; 39], design [40; 41], dynamic simulation [42], fabrication [43; 44; 45], and applications [46; 47; 48; 49; 50; 51]. Type (h) CSFH hinges selectively provide rotations around the CSFH axis  
Figure 2: The compliant platform (**a**) and its corresponding pseudo-rigid body mechanism (**b**), with the equivalent revolute joints R, spherical joints S, prismatic pairs P, a triple spherical joint \(S_{3}\), and the cable directions \(\xi_{1}\), \(\xi_{2}\), and \(\xi_{3}\). with a very good accuracy, because of the presence of a portion of conjugate surfaces. Therefore, they are replaced by classical revolute joints R.
* Type (f) cables are the three moving elements for Type (d) upper links out of a total of six cables that are used to operate the six-DoF platform. For example, a simultaneous pull command on the three Type (f) cables will move the platform downward.