# A New Concept Compliant Platform with Spatial Mobility and Remote Actuation
## 2 Description of the New Concept Platform
### Direct Position Analysis
After this substitution, a new three-legged structure is obtained, as depicted in Figure 4.  
The new mechanism will have a null DoF because the whole set of input positions has already been assigned. In fact, after assigning a value to the three angles \(\alpha_{i}\) and to the three displacements \(d_{i}\), the axes \(w_{i}\), the lengths \(|Q_{i}P_{i}|\), and the positions of points \(Q_{i}\) can be uniquely identified. The resulting structure consists of a mobile platform that is connected to the base via three links only arranged in a parallel configuration, each one having one revolute R and one spherical S joint at its ends incident to the base and the platform, respectively. This structure has received the attention of several eminent scholars in the field and has been extensively studied [52; 53; 54]. From the geometric and kinematic point of view, the study of the possible assembly modes of this zero-DoF structure gives the same solution as for problem of finding the assembly modes of a six-DoF, so called 6-3-type fully-parallel mechanism, belonging to the class of [PERSON] platform mechanisms, once six elongation values are assigned to the six linear actuators along its legs. [PERSON] and [PERSON] solved the direct position analysis in 1990 [55] and found results that were coherent with [PERSON]'s works [2]. According to this method (see also [56]), three closed-loop vector equations:  
\[ \begin{split}\overrightarrow{P_{1}P_{2}}&=- \overrightarrow{Q_{1}P_{1}}-\overrightarrow{P_{2}Q_{2}}-\overrightarrow{Q_{2}Q _{1}}\\ \overrightarrow{P_{2}P_{3}}&=-\overrightarrow{Q_{2}P _{2}}-\overrightarrow{P_{3}Q_{3}}-\overrightarrow{Q_{3}Q_{2}}\\ \overrightarrow{P_{3}P_{1}}&=-\overrightarrow{Q_{3}P _{3}}-\overrightarrow{P_{1}Q_{1}}-\overrightarrow{Q_{3}Q_{1}}\\ \end{split}  \tag{2}\]  
Figure 4: The reduced parallel structure: points \(Q_{i}\), axes \(w_{i}\) and points \(P_{i}\). can be written, where \(\overrightarrow{P_{1}P_{2}}\), \(\overrightarrow{P_{2}P_{3}}\), and \(\overrightarrow{P_{3}P_{1}}\) are the edges of the upper platform. For the sake of the present investigation \(\overrightarrow{P_{1}P_{2}}\), \(\overrightarrow{P_{2}P_{3}}\), and \(\overrightarrow{P_{3}P_{1}}\) have constant modules because of the symmetry of the upper platform. Therefore, given the constant length \(l\) of the upper platform edges, Equation (2) can be rewritten as:  
\[\overrightarrow{P_{1}P_{2}}\cdot\overrightarrow{P_{1}P_{2}}=l^{2},\qquad \quad\overrightarrow{P_{2}P_{3}}\cdot\overrightarrow{P_{2}P_{3}}=l^{2},\qquad \quad\overrightarrow{P_{3}P_{1}}\cdot\overrightarrow{P_{3}P_{1}}=l^{2} \tag{3}\]  
from which three scalar conditions are obtained where the three unknown are the rotation angles \(\theta_{i}\) (with \(i=1,\ldots,3\)) of the virtual bars \(Q_{i}P_{i}\) around the axes \(w_{i}\). It is now essential to remind that once the three rotations \(\alpha_{i}\) of the CSFHs and the three displacements \(d_{i}\) of the Type (f) cables have been assigned, the zero-DoF structure illustrated in Figure 4 is completely configured because the positions of the points \(A_{i}\), \(H_{i}\), and \(Q_{i}\) can be easily calculated. Therefore, the solutions of the problem expressed by the set of Equation (3) are also the solution of the assembly configuration for the PRBM mechanism depicted in Figure 3a. The system (3) consists of three second-order algebraic equations in three unknowns \(\theta_{i}\) (with \(i=1,\ldots,3\)), where any equation contains two unknown variables only. Therefore, the solutions can be achieved by firstly eliminating one variable \(\hat{\theta}\) from two equations where \(\hat{\theta}\) appears, so obtaining one equation in the other two variables, and then, by eliminating one of the other two variables, say \(\hat{\theta}\), from the remaining two equations. The result is a 16 th-order polynomial equation in the remaining variable \(\theta\), with 16 real and complex solutions for \(\theta\).