## 2 Analysis Methods
### Equation of Motion of a Constrained Rigid Body System
#### 2.1.1 Equation of Motion
The constraint equations of the hinge joint between body \(i\) and body \(j\) are expressed in Equation (4) and described in Figure 1.  
\[\mathbf{C}(\mathbf{u},t)_{\text{(hinge, 5\times 1)}}=\left\{ \begin{array}{c} \mathbf{C}_{\text{trans},(3\times 1)}\\ \mathbf{C}_{\text{rot},(2\times 1)}\end{array} \right\}=\left\{ \begin{array}{c} \mathbf{R}^{i}+\left(\mathbf{T}^{i}\right)^{\text{T}}\bar{\mathbf{u}}^{i}- \mathbf{R}^{j}-\left(\mathbf{T}^{j}\right)^{\text{T}}\bar{\mathbf{u}}^{j}\\ \left(\bar{\mathbf{n}}^{i}\right)^{\text{T}}\mathbf{T}^{i}\left(\mathbf{T}^{j} \right)^{\text{T}}\bar{\mathbf{n}}^{j,1}\\ \left(\bar{\mathbf{n}}^{i}\right)^{\text{T}}\mathbf{T}^{i}\left(\mathbf{T}^{j} \right)^{\text{T}}\bar{\mathbf{n}}^{j,2}\end{array} \right\}=0_{(5\times 1)} \tag{4}\]  
where \(t\) is instantaneous time; \(\mathbf{R}\) is the displacement vector from the origin to the center of gravity of the body; \(\mathbf{T}\) is the transformation matrix; \(\bar{\mathbf{u}}\) is the displacement vector from the center of gravity to the joint; \(\bar{\mathbf{n}}^{i}\) is the axis of the rotation vector in the local coordinates of the body \(i\), \(\bar{\mathbf{n}}^{i,1}\) and \(\bar{\mathbf{n}}^{j,2}\) are normal vectors in the local coordinates of body \(j\). The Jacobian matrix of constraint equations and \(\mathbf{Q}_{d}\) can be derived by first and second differentiating the constraint equations with respect to time, respectively. \[\mathbf{C}_{d\text{(hinge, 5\times 1)}}= \begin{bmatrix}\mathbf{I}_{3\times 3}&- \left(\mathbf{T}^{i}\right)^{\text{T}}\bar{\mathbf{u}}^{i}\bar{\mathbf{G}}^{i}& -\mathbf{I}_{3\times 3}&\left(\mathbf{T}^{j}\right)^{\text{T}}\bar{ \mathbf{u}}^{j}\bar{\mathbf{G}}^{j}\\ 0_{2\times 3}&\left(\bar{\mathbf{n}}^{i}\right)^{\text{T}}\mathbf{T}^{i}\left( \mathbf{T}^{j}\right)^{\text{T}}\bar{\mathbf{n}}^{j,n}\mathbf{T}^{j}\left( \mathbf{T}^{i}\right)^{\text{T}}\bar{\mathbf{G}}^{i}&0_{2\times 3}&-\left(\bar{ \mathbf{n}}^{i}\right)^{\text{T}}\mathbf{T}^{i}\left(\mathbf{T}^{j}\right)^{ \text{T}}\bar{\mathbf{n}}^{j,n}\bar{\mathbf{G}}^{j}\end{bmatrix}  \tag{5}\]  
\[\mathbf{Q}_{d\text{(hinge, 5\times 1)}}= \begin{bmatrix}\left(\mathbf{T}^{i} \right)^{\text{T}-i}\bar{\mathbf{u}}^{i}\bar{\mathbf{G}}^{i}\bar{\mathbf{ \mathbf{\phi}}}^{i}-\left(\mathbf{T}^{i}\right)^{\text{T}}\left(\bar{\mathbf{ \omega}}^{i}\right)^{\text{2}}\bar{\mathbf{u}}^{i}-\left(\mathbf{T}^{j} \right)^{\text{T}-j}\bar{\mathbf{u}}^{j}\bar{\mathbf{G}}^{j}\bar{\mathbf{ \phi}}^{j}+\left(\mathbf{T}^{j}\right)^{\text{T}}\left(\bar{\mathbf{\omega}}^{ j}\right)^{\text{2}}\bar{\mathbf{u}}^{j}\\ -\left(\bar{\mathbf{n}}^{i}\right)^{\text{T}}\left(\begin{matrix}\mathbf{T}^{i} \mathbf{T}^{j}\bar{\mathbf{n}}^{j,n}\mathbf{T}^{j}\left(\mathbf{T}^{i}\right) ^{\text{T}}\bar{\mathbf{G}}^{i}\bar{\mathbf{\phi}}^{i}-\mathbf{T}^{i}\left( \mathbf{T}^{j}\right)^{\text{T}}\bar{\mathbf{n}}^{j,n}\tilde{\mathbf{G}}^{j} \\ +\left(\bar{\mathbf{\omega}}^{-i}\right)^{\text{2}}\mathbf{T}^{i}\left(\mathbf{T} ^{j}\right)^{\text{T}}\bar{\mathbf{n}}^{j,n}+\mathbf{T}^{i}\left(\mathbf{T} ^{j}\right)^{\text{T}}\left(\bar{\mathbf{\omega}}^{-j}\right)^{\text{2}}\bar{ \mathbf{n}}^{j,n}-2\bar{\mathbf{\omega}}^{i}\mathbf{T}^{i}\left(\mathbf{T}^{j }\right)^{\text{T}}\bar{\mathbf{\omega}}^{i,n}\end{matrix}\right)\end{bmatrix}  \tag{6}\]  
where \(\bar{\mathbf{u}}\) is a skew-symmetric matrix of \(\bar{\mathbf{u}}\), \(\bar{\mathbf{G}}\) is a matrix that converts Euler angles to the angular velocity vector, \(\bar{\mathbf{n}}\) is the skew-symmetric matrix of \(\bar{\mathbf{n}}\), and \(\dot{\bar{\mathbf{\theta}}}\) is the angular velocity vector. \(\bar{\mathbf{\omega}}\) and \(n\) are as follows:  
\[\bar{\mathbf{\omega}}=\mathbf{T}\Big{(}\dot{\mathbf{T}}\Big{)}^{\text{T}},\;n= \begin{bmatrix}1&2\end{bmatrix} ^{\text{T}} \tag{7}\]  
In the time integration process, the two-loop procedure was applied to satisfy the errors of the constraint equations at the displacement, velocity, and acceleration levels within the tolerance [30; 31; 32].