## 3 Theoretical Aspects Regarding the Kinematic and Dynamic Analysis of the ASPIRE Robot in MSC.ADAMS
### Angular Velocities  
To meet the angular constraints, the markers orientation of a flexible body during deformation is instantly evaluated. As the body deforms, the marker rotates at relatively small angles to the reference frame. Like translational deformations, these angles are obtained using a modal overlap, similar to Equation (3),  
\[[\theta_{p}]=[\Phi_{p}^{*}][q], \tag{10}\]  
where \([\Phi_{p}^{*}]\) is the part of form functions matrix that corresponds to the degrees of freedom of rotation of the node \(P\). The matrix size of \([\Phi_{p}^{*}]\) is \(3\times M\), where \(M\) is the number of form functions. The \(J\) marker orientation in relation to the base is represented by the Euler transformation matrix, \([{}^{G}A^{I}]\). This matrix is the product of the three transformation matrices:  
\[[{}^{G}A^{J}]=\left[{}^{G}A^{B}\right]\cdot\left[{}^{B}A^{P}\right]\cdot\left[{ }^{P}A^{J}\right], \tag{11}\]  
where \([{}^{G}A^{B}]\) is the matrix for transforming the coordinates from the local reference system attached in \(B\) to the body to basic or global coordinates; \([{}^{B}A^{P}]\) is the transformation matrix during the change of orientation by deforming the flexible body of the pole \(P\) coordinates in relation to the local reference system attached to the body; and \([{}^{P}A^{J}]\) is the constant transformation matrix, defined by the user when placing the marker \(J\) on the flexible body. Matrix \([{}^{B}A^{P}]\) is defined by the cosine directors \(\theta_{pz}\), \(\theta_{py}\) of a vector with small angles, \(\theta_{p}\):  
\[\left[{}^{B}A^{P}\right]=\left[ \begin{array}{ccc}1&-\theta_{pz}&\theta_{py} \\ \theta_{pz}&1&-\theta_{pz}\\ -\theta_{py}&\theta_{pz}&1\end{array} \right]=I+\widetilde{\theta}_{p}, \tag{12}\]  
where the sign \(\sim\) represents the antisymmetric operator. ### Angular Velocities  
The angular velocity of a marker, \(J\), of a flexible body is the sum of the angular velocities of the body and the angular velocities during deformation:  
\[[{}^{G}\omega_{B}^{J}]=[{}^{G}\omega_{B}^{P}]=[{}^{G}\omega_{B}^{B}]+[{}^{B} \omega_{B}^{P}]=[{}^{G}\omega_{B}^{B}]+[{\Phi}_{P}^{*}][\overset{\bullet}{q}], \tag{13}\]  
The equations governing a flexible body are derived from [PERSON]'s equations:  
\[\left\{ \begin{array}{c}\frac{d}{dt}\left(\frac{\partial I}{\partial x} \right)-\frac{\partial I}{\partial\xi}+\frac{\partial F}{\partial\xi}+\left[ \frac{\partial F}{\partial\xi}\right]^{T}\lambda-Q=0\\ \Gamma=0\end{array} \right., \tag{14}\]  
where \(L\) is the Lagrangian, defined by the relationship \(L=T-V\), where \(T\) and \(V\) represents the kinetic and potential energy, respectively; \(F\) represents the energy dissipation function, defined by the relationship:\[[F]=\frac{1}{2}[\overset{\bullet}{q}]^{T}[D][\overset{\bullet}{q}]. \tag{15}\]  
\(\Gamma\) represents the constraint equations; \(\lambda\) represents [PERSON]'s multipliers for constraints; \(\xi\) represents the generalized coordinates defined in the relationships Equation (4); \(Q\) represents the generalized applied forces (applied forces projected on \(\xi\)); and \([D]\) represents the damping matrix. The velocity in relation to Equation (5) can be expressed in terms of the derivative with respect to time of the state vector, \(\xi\). \[[v_{p}]=\Big{[}[I]-[^{G}A^{B}]\big{(}[s_{p}]+[u_{p}]\big{)}[B]+[^{G}A^{B}][ \Phi_{p}]\Big{]}\overset{\bullet}{[\xi]}. \tag{16}\]  
The kinetic energy for a flexible body is given by:  
\[T=\frac{1}{2}\int_{V}\rho v^{T}vdV\approx\frac{1}{2}\sum_{p}\big{[}[m_{p}][v_{ p}^{T}][v_{p}]+[^{G}\omega_{p}^{B}]^{T}[I_{p}][^{G}\omega_{p}^{B}]\big{]}, \tag{17}\]  
where \(\big{[}m_{p}\big{]}\) and \([I_{p}]\) are the nodal mass matrix and the nodal inertia tensor of the node \(P\). \([I_{p}]\) is sometimes a negligible amount, as, for example, bars, beams, housings are used in the model with flexible components. By substituting \(v_{p}\) and \(\omega_{p}\) and simplifying, we obtain the kinetic energy in the generalized coordinates and the matrix of the generalized masses.