## 2 Geometric Parameters
Then, based on Rodrigues' rotation formula, coordinates of the vector \(\mathbf{a}_{i}\) can be expressed as follows:  
\[\mathbf{a}_{i}=\mathbf{b}_{i}+\mathbf{B}_{i}\mathbf{A}_{0i}+(1-\cos\alpha_{i})\ \mathbf{R}\mathbf{m}\mathbf{1}_{\mathbf{B}_{i}\mathbf{A}_{i}}+(\sin\alpha_  
With all previously defined points, the geometry of the mechanism is fully defined. Still, it is necessary to define two more unit vectors for each leg in order to be able to model the constraints of spherical joints. These unit vectors of spherical joints are essential for optimization, as their orientation is used for the calculation of the physical limitations of this mechanism. This physical limitation is modeled as a nonlinear constraint because at a certain maximal angle of the joint, there is contact between different parts of the joint, and further movement is denied. It is expected that the optimization algorithm adapts the orientation of these vectors in order to get the largest possible workspace.  
Figure 8 shows a projection of the unit vector of the upper joint axis \(i_{pp,i}\) in the plane defined by \(x_{p}\) and \(y_{p}\). It is also shown that angle \(\tau_{jp}\) is used to define its orientation. This angle is the next geometric parameter. The orientation of the unit vector of the upper joint axis in this plane (angle between vector projection and \(x_{p}\) axis) is defined as follows:  
\[\beta_{jp}=\beta_{s}+\tau_{jp}\,\sigma_{I} \tag{8}\]  
A previously analyzed unit vector is used for the calculation of the physical limitation of the spherical joint that connects the platform and upper lever. The last unit vector is \(i_{II}\), which defines the axis of the spherical joint that connects the upper and lower lever. This unit vector is already shown in Figure 7. In the home position, its projection to the horizontal plane has the same direction as the projection of the actuator's axis of rotation. Unit vectors of joints are again shown in Figure 9; their orientation in planes of the platform and the base are already discussed, but two more and two last geometric parameters are introduced for total orientation. Vectors are tilted in reference to the corresponding planes by angles \(\mu_{jp}\) and \(\mu_{II}\). The same principle is already used for the actuators, and the following equation is obtained:  
\[i_{pp,i}=\left[\cos\beta_{jp,i}\,\cos\mu_{jp}\quad\sin\beta_{jp,i}\,\cos\mu_{ ip}\quad\sin\mu_{jp}\right]^{T} \tag{9}\]  
Figure 8: Axes of platform and upper lever joint. Because one part of the joint between the lower and upper lever is fixed to the lower lever, it rotates together with the lower lever, and it is necessary to define the referent vector of the axis in home position \(i_{II0,i}\) with:  
\[i_{II0,i}= \begin{bmatrix}\cos\left(\beta_{II,i}+\frac{\pi}{2}\sigma_{II}\right) \ \cos(\varepsilon+\mu_{II})\\ \sin\left(\beta_{II,i}+\frac{\pi}{2}\sigma_{II}\right)\ \cos(\varepsilon+\mu_{II})\\ \sin(\varepsilon+\mu_{II})\end{bmatrix}  \tag{10}\]  
Then, again based on [PERSON] rotation formula, the vector \(i_{II,i}\) can be expressed as follows:  
\[i_{II,i}=i_{II0,i}+\left(1-\cos\alpha_{i}\right)\mathbf{Rm1}_{i_{II}}+\left(\sin \alpha_{i}\right)\mathbf{Rm2}_{i_{II}}, \tag{11}\]  
where \(\mathbf{Rm2}_{i_{II}}\) and \(\mathbf{Rm1}_{i_{II}}\) are:  
\[\mathbf{Rm2}_{i_{II}}=i_{\mathbf{m},i}\times i_{II0,i},\quad\mathbf{Rm1}_{i_{II}}=i_{\mathbf{m },i}\times\mathbf{Rm2}_{i_{II}} \tag{12}\]  
To optimize the geometry, eleven geometric parameters that determine the geometry of the mechanism for flight simulators are precisely defined. They unambiguously describe platform geometry. It should be noted that the geometry could be defined in other ways. Still, because no restrictions are imposed besides symmetry between joints (mirroring of, i.e., point \(B_{1}\) to point \(B_{2}\)), any usable configuration can be obtained. It is necessary only once to implement the previously described algorithm for calculating the position and axes vectors based on geometric parameters.