## 2 Employed Methodology of Kinematics and Dynamics of Three-like Multibody Systems  
This paper recalls the classic definition of multibody structures interpreted as systems composed of interconnected non-deformable bodies. An additional assumption states that relative displacements can only occur at joints. All joints are single-degree-of-freedom massless elements. One can assume displacements of the joint as the generalized coordinates of the system. The joint displacements are significant in magnitude (especially at the rotational joints). Following [26; 27; 28], one can employ the classic concept of the kinematical chain (Figure 1a). Suppose body \(\#i\) is the generic body of the system; \(\overline{x}^{i}\), is the vector of the absolute position of its centre of mass, Ci (with respect to the origin of the base-fixed frame); and \(\mathbf{T}^{i}\) is the absolute orientation matrix of the body-fixed coordinate system (with respect to the base-fixed frame). One can write the vector and the matrix as the sum and product of the relative subparts. \[\overline{x}^{i}=\sum_{j:j\leq i}\left(\ p^{k}\cdot\overline{a}^{k}+\overline{ d}^{ji}\right)=\sum_{j:j\leq i}\overline{j}^{ji};\mathbf{T}^{i}=\prod_{j:j\leq i} \mathbf{R}^{j}, \tag{1}\]  
Next, their time derivatives allow us to write the velocity and acceleration formulae. This leads to [26; 27; 28]:  
\[\dot{\overline{x}}^{i}=\sum_{k\leq i}(\dot{p}^{k}\cdot\overline{a}^{k}+ \overline{\omega}^{k}\times\overline{l}^{ij}); \tag{2a}\] \[\overline{\omega}^{i}=\sum_{k\leq i}\dot{\phi}^{k}\cdot\overline{ e}^{k},\] (2b) \[\dot{\overline{x}}^{i}=\sum_{k:k\leq i}(\dot{p}^{k}\cdot\overline{ a}^{k}\ +\dot{\overline{\omega}}^{k}\ \times\overline{l}^{ji}\ +2\dot{p}^{k}\cdot \overline{\omega}^{k}\ \times\overline{a}^{k}\ +\overline{\omega}^{k}\ \times\ (\overline{\omega}^{k}\ \times \overline{l}^{ij}));\] (2c) \[\dot{\overline{\omega}}^{i}=\sum_{k:k\leq i}\left(\dot{\phi}^{k}\cdot \overline{e}^{k}+\dot{\phi}^{k}\cdot\overline{\omega}^{k}\times\overline{e}^{k }\right), \tag{2d}\]  
where \(\overline{a}^{j}\) is the unit vector of direction of translation line at joint \(\#j\) (unit vector collinear to the line of the translation; it is the zero vector for rotational joints); \(\overline{\psi}\) is the unit vector of direction of rotation at joint \(\#j\) (collinear to the rotation axis; it is the zero vector when one examines translational joints); \(p^{j}\) is the magnitude of translations at the joint \(\#j\); and \(\varphi^{j}\) is the value of the rotation angle at this joint \(\#j\). Figure 1: Elements of multibody structure: joints, bodies, and numbering (**a**); geometrical distances at _body i_ and interactions acting on the body (**b**). To obtain the dynamics equations, one should cut all joints of the system and introduce the joint's interactions to replace the cut connections (Figure 1b). Then, Newton/Euler dynamics equations are written for each of the free-body diagrams of all bodies of the system [26; 27; 28]:  
\[m^{i}\overset{\cdot\overline{i}}{\overline{x}}=\overline{f}^{i}+ \overline{f}^{i,e}-\sum_{j\in i^{+}}\overline{f}^{j}\ ; \tag{3a}\] \[\overline{\omega}^{i}\ \times(\ \overline{i}^{\overline{i}}\ \cdot \overline{\omega}^{i})\ +\overline{i}^{\overline{i}}\ \cdot\overset{\cdot \overline{i}}{\overline{\omega}}^{i}\ =\ \overline{i}^{i,e}+\overline{i}^{i}\ \times\ \overline{f}^{i}\ + \overline{i}^{i}-\ \sum_{j\in i^{+}}\overline{i}^{j}-\ \sum_{j\in i^{+}} \overline{i}^{j}\times\overline{f}^{j}, \tag{3b}\]  
where \(m^{i}\) is a mass of body \(\#i\); \(\overline{i}^{\overline{i}}\) is its tensor of moments of inertia (calculated about the centre of mass of the body \(\#i\)); \(\overline{f}^{i}\), \(\overline{i}^{\overline{i}}\) are the joint force and torque, respectively, at the cut joint \(\#i\) that act on body \(\#i\), \(\overline{f}^{i,e}\) is a net external force that acts at the mass centre of body \(\#i\); and \(\ \overline{i}^{i,e}\) is the net external torque that acts on body \(\#i\).