## 2 Geometric Parameters  
In order to analyze whether the shown type of mechanism can be successfully applied in practice for the flight simulator, foremost, the geometry must be defined. The geometry should be defined with a minimum number of parameters that fully define the mechanism with all constraints. Changing the values of these parameters within the optimization process should provide better characteristics for the flight simulator. As shown in Figure 3, the first step is to define two reference frames. A global reference frame (\(O_{b}x_{b}y_{b}z_{b}\)) is fixed to the center of the base and a moving platform reference frame (\(O_{p}x_{p}y_{p}z_{p}\)) is fixed to the center of the platform. Control values are defined by vector \(\mathbf{x}= \begin{bmatrix}x&y&z&\psi&\theta&\varphi\end{bmatrix} ^{T}\), and they represent external coordinates of the moving platform's local reference frame in reference to the global frame fixed to the nonmoving base of the platform. In other words, translation of the platform's frame in reference to the global frame is defined by \(\mathbf{T}= \begin{bmatrix}x_{b,O_{p}}&y_{b,O_{p}}&z_{b,O_{p}}\end{bmatrix} ^{T}\), while orientation is defined using Euler angles (\(\psi,\theta,\varphi\)), with yaw, pitch, and roll about for \(z,y,\) and \(x\) global axes, respectively. The rotary actuator is fixed to the base of the mechanism, and it rotates the lower lever. Figure 4 shows point \(B_{i}\); this is the point of connection between the rotary actuator and the lower lever, which means that it is the intersection point of the actuator's axis of rotation  
Figure 3: Global and platform’s reference frame. (defined with \(i_{\mathbf{m},i}\)) and the lower lever. The lower lever has a constant length (from point \(B_{i}\) to point \(A_{i}\)), and it is equal to parameter \(a\), while the constant length of the upper lever that connects points \(B_{i}\) and \(P_{i}\) is equal to parameter \(s\). These are the first two geometric parameters. In reference to the horizontal plane on which the base lies, the actuator's axis of rotation is tilted by the \(\varepsilon\) angle, the third geometric parameter. Point \(P_{i}\) is a point on the platform around which the spherical joint rotates, the sphere's center. Point \(A_{i}\) is the center of the spherical joint connecting the upper and lower lever. Due to the six legs of this mechanism \(i=1,2,\ldots,6\). The vector \(\mathbf{p_{i}}= \begin{bmatrix}x_{Pi}&y_{Pi}&z_{Pi}\end{bmatrix} ^{T}\) is the position vector of a point \(P_{i}\) in the reference frame of the platform. A vector \(\mathbf{b_{i}}= \begin{bmatrix}x_{Bi}&y_{Bi}&z_{Bi}\end{bmatrix} ^{T}\) is the position vector of a point \(B_{i}\) in the global reference frame. Coordinates of points \(B_{i}\) and \(P_{i}\) must be determined in corresponding reference frames in order to fully define the mechanism's base and platform. The intuitive way to do this is to use polar coordinates, noticing that points \(B_{i}\) and \(P_{i}\) lie on constant diameter circles and that there are three axes of symmetry between these points (point \(B_{1}\) can be mapped to point \(B_{2}\), while the axis of symmetry is at a certain angle to the \(x\) axis of global frame). As shown in Figure 5, the axis of symmetry between points \(B_{1}\) and \(B_{2}\) is defined by angle \(\beta_{s,i}\). This angle has a constant value for all considered geometries, and its purpose is to define three segments for points \(B_{i}\). The axis of symmetry of points \(B_{1}\) and \(B_{2}\) is rotated by \(30^{\circ}\) in reference to the \(x\)-axis, then the axis of symmetry of points \(B_{3}\) and \(B_{4}\) is rotated by \(150^{\circ}\), and the axis of symmetry of points \(B_{5}\) and \(B_{6}\) is rotated by \(-90^{\circ}\). These three axes of symmetry make the distribution of points uniform. Angle \(\beta_{s,i}\) is not a geometrical parameter for optimization, but it is necessary for determining the coordinates of points \(B_{i}\) with other parameters.