## 3 Theoretical Aspects Regarding the Kinematic and Dynamic Analysis of the ASPIRE Robot in MSC.ADAMS  
For the dynamic study of the robotic systems, the most used software is MSC.ADAMS [44], due to the accuracy of results obtained. Thus, the use of this software in various research fields in the industry is noticeable, among which can be mentioned exoskeleton robotic systems or flexible robotic units for minimally invasive surgery [45; 46; 47; 48; 49; 50; 51; 52]. MSC.ADAMS uses the kinematics of markers in three essential areas: kinematics, dynamics and optimization of mobile mechanical systems. Marker kinematics deals with the study of the position, orientation, velocity and acceleration of markers [53].  
The instantaneous position vector of a marker that is attached to a node, noted \(P\), on a flexible body is the sum of three vectors (Figure 2).  
\[\overset{\rightarrow}{r}_{p}=\overset{\rightarrow}{x}+\overset{\rightarrow}{s }_{p}+\overset{\rightarrow}{u}_{p}, \tag{1}\]  
where \(\overset{\rightarrow}{x}\) is the position vector of the origin of the local reference system with the origin in \(B\), attached to the flexible body, relative to the origin of fixed reference system, \(G\); \(\overset{\rightarrow}{s}_{p}\) represents the position vector of the point \(P\) in the initial position on the undeformed body in relation to the local reference system attached to the flexible body; and \(\overset{\rightarrow}{u}_{p}\) is the position vector of the point \(P\), the new position of \(P\) on the deformed body, in relation to the initial position of the point \(P\).  
Figure 1: (a) Kinematic scheme; (b) physical prototype of ASPIRE [43].  
Figure 2: The position vector of the point \(P\) of a deformed flexible body relative to a reference system attached to the flexible body and the fixed base G.  
We write Equation (1) in a matrix form expressed in relation to the basic coordinate system:  
\[[r_{p}]=[x]+\ [^{G}A^{B}]\big{(}[s_{p}]+[u_{p}]\big{)}, \tag{2}\]  
where \([x]\) is matrix associated with the position vector \(\overrightarrow{x}\), having as elements the generalized coordinates \(x\), \(y\) and \(z\) of the origin B of the system associated with the flexible body.  
\([s_{p}]\) is a matrix associated with the position vector \(\overrightarrow{s}_{p}\) having as elements the constant coordinates of the point \(P\) in relation to the local system of the body.  
\([^{G}A^{B}]\) is the transformation matrix from the local reference system in \(B\) to the basic system, \(G\), being composed of cosine directors of the origin of the local reference system in relation to the fixed one. The orientation is calculated using [PERSON]'s angles, \(\psi\), \(\theta\) and \(\varphi\), which are used to define the generalized coordinates of the flexible body.  
\([u_{p}]\) is a matrix associated with the position vector of point \(P^{\prime}\) with respect to \(P\), expressed by coordinates related to the local coordinates system attached to the body. Matrix \([u_{p}]\) can be written:  
\[[u_{p}]=[\Phi_{p}]\cdot[q], \tag{3}\]  
where \([\Phi_{p}]\) is the portion of the form function matrix that corresponds to the translation degrees of freedom of the node \(P\), being a \(3\times M\) matrix type, where \(M\) is the number of form functions. Coordinates of form functions \(q_{i}\), \((i=1...M)\) also represent the generalized coordinates of the flexible body.  
Therefore, the complete matrix form of the generalized coordinates of the flexible body is:  
\[\xi=\left\{\begin{array}{c}x\\ y\\ z\\ \psi\\ \theta\\ \varphi\\ q_{i},\ (i=1...M)\end{array}\right\}=\left\{\begin{array}{c}X\\ \Psi\\ q\end{array}\right\}, \tag{4}\]