## 2 Materials and Methods
### The Task Kinematics
#### 2.2.1 Inverse Task Kinematics  
After Figure 5, we remark that angles \(\bar{\alpha}\), \(\bar{\beta}\) are measured _w.r.t._ in the base frame \(\Sigma_{0}\) at the platform, being positive in the conventional counter-clockwise sense after the right-hand rule, and the vertical rotation angle \(\bar{\beta}\) is the quadrant complement of the elevation angle \(\beta\), which is measured from the horizontal plane. Moreover, the azimuth angles \(\alpha\) and \(\alpha_{0}\) are positive in the clockwise sense and measured by a compass _w.r.t._ the North; therefore,  
\[\alpha+\bar{\alpha}=\alpha_{0};\qquad\qquad\qquad\qquad\qquad\beta+\bar{\beta} =\pi/2 \tag{10}\]  
In (9a), the rotation from the world frame \(\Sigma_{w}\) to the final frame \(\Sigma_{a}\) is constructed by only two consecutive rotations, with one of them constant and defined by the particular orientation toward the North. This constant rotation \(R_{w}^{0}\), from frame \(\Sigma_{w}\) to \(\Sigma_{0}\), is expressed by the rotation of a negative angle \(-\alpha_{0}\) about the \(z\) axis:  
\[R_{w}^{0}=R_{z,-\alpha_{0}}=R_{z,\alpha_{0}}^{T} \tag{11}\]  
To uniquely define the variable rotation \(R(\cdot)\in SO(3)\), dependent on generalized coordinate of the robot platform and aiming at parametrizing with \(\bar{\alpha}\) and \(\bar{\beta}\), of the antenna _w.r.t._ in the base frame \(\Sigma_{0}\). Thus, consider \(R(\bar{\alpha},\bar{\beta})\triangleq R_{z,\bar{\alpha}}R_{y,\bar{\beta}}R_{ z,-\bar{\alpha}}\in SO(3)\) performed after three basic rotations as follows:  
1. A rotation \(R_{z,\bar{\alpha}}\) of an angle \(\bar{\alpha}\) about the vertical \(z_{0}\)-axis (positive in the counter clockwise sense). Figure 5: Reference frames and transformation elements for the antenna/Gough–Stewart robot (**left**); and sketch of the antenna’s attitude definition (**right**). Notice that the azimuth is positive in a clockwise sense, while the rotation angles are defined anti-clockwise positive according to the right-hand rule. A rotation \(R_{y,\bar{\beta}}\) of an angle \(\bar{\beta}\) about the current horizontal \(y\)-axis; such that the new \(z\)-axis points in the antenna's direction \(\lambda_{a}\). 3. A \"negative\" rotation \(R_{z,-\bar{\alpha}}\) of the same angle \(\bar{\alpha}\) about the new \(z_{p}\)-axis; such that the antenna plate is _constraint_ to turn like a rolling coin in order to avoid unfeasible physical configurations in the platform. The overall rotation (9a), after (11) and \(R_{z,-\alpha_{0}}\)\(R_{z,\bar{\alpha}}=R_{z,\bar{\alpha}-a_{0}}=R_{z,-\bar{\alpha}}=R_{z,\bar{ \alpha}}^{T}\), becomes: \[R_{w}^{a}(y)=R_{w}^{0}R(y) =R_{z,\bar{\alpha}}^{T}\ R_{y,\bar{\beta}}\ R_{z,\bar{\alpha}}^{T}\] (12) \[= \begin{bmatrix}s_{a}s_{\bar{\alpha}-a_{0}}+c_{a}s_{\bar{\beta}} c_{\bar{\alpha}-a_{0}}&s_{a}c_{\bar{\alpha}-a_{0}}-c_{a}s_{\bar{\beta}}s_{ \bar{\alpha}-a_{0}}&c_{a}c_{\bar{\beta}}\\ c_{\bar{\alpha}}s_{\bar{\alpha}-a_{0}}-s_{\bar{\alpha}}s_{\bar{\beta}}c_{ \bar{\alpha}-a_{0}}&c_{\bar{\alpha}}c_{\bar{\alpha}-a_{0}}+s_{\bar{\alpha}}s_{ \bar{\beta}}s_{\bar{\alpha}-a_{0}}&-s_{\bar{\alpha}}c_{\bar{\beta}}\\ -c_{\bar{\beta}}c_{\bar{\alpha}-a_{0}}&c_{\bar{\beta}}s_{\bar{\alpha}-a_{0}}& s_{\bar{\beta}}\end{bmatrix} \]  
**Remark 1**. : _The aim is to precisely point the antenna to the downlink LEO signal; thus, it is convenient to make a clear and intuitive design of the antenna-pointing direction to the LEO satellite position. Such a point vector should only depend on the task coordinates \(\lambda_{a}=\lambda_{a}(y)\). This implies that the FTK shall uniquely depend on elements of the rotation matrix \(R\) and not the distance \(d\): \(y=\text{FTK}(R(t))\). In this way, the inverse mapping for the platform position \(d\) can be any arbitrary function. Now, notice that antenna's \(z\)-axis unit vector is nothing but \(\lambda_{a}^{(a)}=\lambda_{a}^{(p)}=k\), independent of \(d\), which, in world-frame coordinates, becomes \(\lambda_{a}^{(w)}=R_{w}^{a}(t)k\) given by the third column above.