## 2 Model Creation and Data Processing  
Relations describing the position kinematics by means of position vectors \(\mathbf{p_{i}}\), where \(i\epsilon\)<1;7> at individual open kinematic chain segments have also been derived in [25; 26]. This is the case of the robotic arm's forward kinematics. The relations serve the purpose of calculating the position of a point that would enable the attachment of either an effector or another device. Thus, the position of a point \([p_{xi},p_{yi},p_{zi}]\) on the module will be defined by the function of the rotation positions \(\varphi_{i}\), \(\theta_{i}\) and the segment size of the structure at hand-- \(|\mathbf{r}_{i}|\); \(p_{i}=\theta_{yi}\), \(\theta_{i}\), \(\tau_{j}\)). The basis of the arm [1; 2] is an autonomous, cylinder-shapedmodule. Such modules have a single degree of freedom, namely the rotational one. The rotation happens around the module's main axis and is not limited in the interval under consideration ranging from 0 to 360 degrees. The rotation may occur continuously in either direction of rotation without limitation. The modules and joints are contemplated to be perfectly solid bodies. The individual modules' axes of rotation cross each other at the point where they are joined by passive joints, Figure 1. The distances of section points in space are quantified by the \(\mathbf{r}_{i}\) vector (a kinematic chain segment), and this vector has its own Cartesian system of \(S_{i}[O_{i},x_{i},y_{i},z_{i}]\). The segment's rotation around the \(z_{i}\) axis is defined by the rotation matrix \(\mathbf{R}_{zi}(\varphi_{i})\)[3; 4; 5]. The segment's rotation around the \(y_{i}\) axis is defined by the rotation matrix \(\mathbf{R}_{yi}(\vartheta_{i})\). Thus, we can write the following for \(\mathbf{R}_{zi}(\varphi_{i})\):  
\[\mathbf{R}_{zi}(\varphi_{i})=\left[ \begin{array}{ccc}\cos(\varphi_{i})&-\sin( \varphi_{i})&0\\ \sin(\varphi_{i})&\cos(\varphi_{i})&0\\ 0&0&1\end{array} \right] \tag{1}\]  
Moreover, the following for \(\mathbf{R}_{yi}(\vartheta_{i})\):  
\[\mathbf{R}_{yi}(\vartheta_{i})=\left[ \begin{array}{ccc}\cos(\vartheta_{i})&0& \sin(\vartheta_{i})\\ 0&1&0\\ -\sin(\vartheta_{i})&0&\cos(\vartheta_{i})\end{array} \right] \tag{2}\]  
The \(\mathbf{r}_{1}\) vector is connected to the base perpendicularly to the plane with the \(x_{1},y_{1}\) axes. \(\mathbf{p}_{1}\equiv\mathbf{r}_{1}\), because the coordinate system is orthogonal. According to Figure 1, the following applies to the position vector \(\mathbf{p}_{1}\):  
\[\mathbf{p}_{1}=\mathbf{R}_{z1}(\varphi_{1})\times\mathbf{r}_{1} \tag{3}\]  
According to Figure 1, the following applies to the position vectors \(\mathbf{p}_{2},\mathbf{p}_{3},\mathbf{p}_{4}\), to \(\mathbf{p}_{i}\):  
\[\mathbf{p}_{2}=\mathbf{p}_{1}+\mathbf{R}_{z1}(\varphi_{1})\times\mathbf{R}_{y2}(\vartheta_{2} )\times\mathbf{R}_{z2}(\varphi_{2})\times\mathbf{r}_{2} \tag{4}\]  
\[\mathbf{p}_{3}=\mathbf{p}_{2}+\mathbf{R}_{z1}(\varphi_{1})\times\mathbf{R}_{y2}(\vartheta_{2} )\times\mathbf{R}_{z2}(\varphi_{2})\times\mathbf{R}_{y3}(\vartheta_{3})\times\mathbf{R}_{ z3}(\vartheta_{3})\times\mathbf{r}_{3} \tag{5}\]  
\[\mathbf{p}_{4}=\mathbf{p}_{3}+\mathbf{R}_{z1}(\varphi_{1})\times\mathbf{R}_{y2}(\vartheta_{2} )\times\mathbf{R}_{z2}(\varphi_{2})\times\mathbf{R}_{y3}(\vartheta_{3})\times\mathbf{R}_{ z3}(\vartheta_{3})\times\mathbf{R}_{y4}(\vartheta_{4})\times\mathbf{R}_{z4}(\vartheta_{4}) \times\mathbf{r}_{4} \tag{6}\]  
Figure 1: Robotic arm vector model composed of the universal rotational modules (URM) modules.