## 2 Geometric Parameters
The value of this angle for all legs is defined with vector \(\mathbf{\beta_{s}}= \begin{bmatrix}\frac{1}{6}\pi&\frac{1}{6}\pi&\frac{5}{6}\pi& \frac{5}{6}\pi&-\frac{1}{2}\pi&-\frac{1}{2}\pi\end{bmatrix} ^{T}\). Figure 4: Geometric parameters of one leg. Angle \(\gamma_{b}\) and radius \(R_{b}\) are the next two variable geometric parameters that are used to define the position of the point \(B_{i}\) relative to the previously defined axis of symmetry. In order to define both points \(B_{1}\) and \(B_{2}\), located on different sides relative to the axis of symmetry, a vector \(\mathbf{\sigma_{I}}= \begin{bmatrix}-1&1&-1&1&-1&1\end{bmatrix} \) is introduced. This vector also has a constant value for all considered geometries. Its purpose is also to define different rotation directions of even and odd indexed actuators. Finally, the position of all six points \(B_{i}\) is defined as follows:  
\[x_{B,i}=R_{b}\cos\left(\delta_{b,i}\right),\quad y_{B,i}=R_{b}\sin\left( \delta_{b,i}\right),\quad z_{B,i}=0,\quad\delta_{\mathbf{b}}=\mathbf{\beta_{s}}+\gamma _{b}\,\mathbf{\sigma_{I}} \tag{1}\]  
Similar to points \(B_{i}\), the position of all six points \(P_{i}\) in reference to the platform's reference frame is defined by the following two variable geometric parameters, radius \(R_{p}\) and angle \(\gamma_{p}\), as shown in Figure 6. There are again three axes of symmetry between points, the same as in the case of the base. \[x_{p,i}=R_{p}\cos\left(\delta_{p,i}\right),\quad y_{p,i}=R_{p}\sin\left( \delta_{p,i}\right),\quad z_{p,i}=0,\quad\delta_{\mathbf{p}}=\mathbf{\beta_{s}}+\gamma _{p}\,\mathbf{\sigma_{I}} \tag{2}\]  
Figure 5: Geometric parameters of the mechanism’s base. Figure 6: Geometric parameters of the mechanism’s platform. It is now necessary to define the position of points \(A_{i}\), which depends on the previously mentioned vector, \(i_{m,i}\) which is the unit vector of the actuator's axis of rotation. As shown in Figure 7, the actuator's axis of rotation does not have to be normal to the line passing through points \(B_{1}\) and \(B_{2}\) but this line is a referent line for the definition of the orientation of lower levers in the horizontal plane (when lower levers are in the home position). This referent line is defined with the angle \(\
u_{II,i}\) measured from the \(x\) axis of the global reference frame. The valueof this angle is defined with a vector \(\
u_{II}=\beta_{s}-\frac{\pi}{2}\sigma_{I}\) for each lower lever. Finally, the angle between the lower lever's referent line and the actual orientation of the lower lever in the home position is the next geometric parameter \(\tau_{II}\), which is the same for all six lower levers. Then, the angle between the lower lever and the \(x_{b}\) axis is given by the following equation:  
\[\beta_{II}=\
u_{II}+\tau_{II}\sigma_{I} \tag{3}\]  
The projection of the unit vector of the actuator's axis of rotation in the horizontal plane is normal to the lower lever. Based on that and the already defined tilt angle \(\varepsilon\), the unit vector of the actuator's axis of rotation is:  
\[i_{m,i}= \begin{bmatrix}\cos\left(\beta_{II,i}+\frac{\pi}{2}\sigma_{I,i}\right) \ \cos\varepsilon\\ \sin\left(\beta_{II,i}+\frac{\pi}{2}\sigma_{I,i}\right)\ \cos\varepsilon\\ \sin\varepsilon\end{bmatrix}  \tag{4}\]  
Point \(A_{0i}\) (shown in Figure 4) is the referent point that represents a point \(A_{i}\) in the home position and its position vector \(a_{0i}= \begin{bmatrix}x_{A_{0i}}&y_{A_{0i}}&z_{A_{0i}}\end{bmatrix} ^{T}\) is defined with:  
\[x_{A_{0i}}=a\ \cos\beta_{II,i}+x_{B,i},\quad y_{A_{0i}}=a\ \sin\beta_{II,i}+y_{B,i}, \quad z_{A_{0i}}=z_{B,i} \tag{5}\]  
To define a point \(A_{i}\) and position vector \(\mathbf{a}_{i}= \begin{bmatrix}x_{Ai}&y_{Ai}&z_{Ai}\end{bmatrix} ^{T}\) in the global reference frame, it is first necessary to define or find angle \(a_{i}\) (also shown in Figure 4).