## 3 Analysis of the Composite Docking Test Platform
### Attitude Error Analysis and Modeling  
#### 3.2.1 Attitude Angle Error Analysis and Modeling  
Attitude angle error refers to the error in the spatial attitude of the passive docking mechanism. The passive docking mechanism was mounted on a flange plate at the end of the passive motion simulator, while the transformation of the attitude angle was achieved by the rotation of the spherical joint. Therefore, the rotation angle error of the spherical joint and coupling motion error of each rotation angle were the primary causes of the attitude angle error. The motion error of the conversion mechanism simultaneously affects the  
Figure 2: The space docking coordinate system.  
attitude angle of the docking mechanism when the composite docking test platform was free to convert between vertical and horizontal docking modes.  
As shown in Figure 3, coordinate system-oxyz is established whose center was located at the centroid of the passive docking mechanism.  
The docking mechanism rotates \(\alpha\), \(\beta\), and \(\gamma\) around \(z\), \(x\), and \(y\) to successively reach \(\alpha\)\(y\)\(z^{\prime}\); therefore, the transformation matrix of the attitude angle is:  
\[\begin{bmatrix}x\\ y\\ z\end{bmatrix}=\left[\begin{array}{ll}\cos\gamma\cos\alpha&\sin\alpha&-\cos \gamma\sin\alpha\\ \sin\beta\sin\alpha\cos\gamma+\cos\beta\sin\gamma&-\sin\beta\cos\alpha&-\sin \beta\sin\alpha\sin\gamma+\cos\beta\cos\gamma\\ -\cos\beta\sin\alpha\cos\gamma+\sin\beta\sin\gamma&\cos\beta\cos\alpha&\cos \beta\sin\alpha\sin\gamma+\sin\beta\cos\gamma\end{array}\right]\begin{bmatrix}x ^{\prime}\\ y^{\prime}\\ z^{\prime}\end{bmatrix} \tag{1}\]  
According to Formula (1) [45], under ideal conditions, the change in attitude angle attained by rotating the passive docking mechanism corresponded to the change in rotation angle at the spherical joint. Consequently, we determined a relationship between the attitude angle and the rotation angle of the spherical joint as follows:  
\[\left\{\begin{array}{l}\alpha=\arctan(\tan\psi\sec\theta)\\ \beta=\arcsin(\sin\theta\cos\psi)\\ \gamma=\arctan(\frac{\tan\theta-\tan\theta\sin\psi}{1+\tan\theta\tan\sin\psi} )\end{array}\right. \tag{2}\]  
Among these were the generalized rotation angles of the simulator of passive motion as it rotated about the \(x\), \(y\), and \(z\) axes. Since the origins of the moving and stationary coordinate systems coincided and the axis directions coincided with the axes of the original coordinate system, Formula (2) is obtained by putting it into the space Euler angle rotation formula.  
In addition, when the rotation angles of the spherical joint are in errors, the attitude angles are as follows:  
\[\left\{\begin{array}{l}\alpha=\arctan[\tan(\psi+\Delta\psi)\sec(\theta+ \Delta\theta)]\\ \beta=\arcsin[\sin(\theta+\Delta\theta)\cos(\psi+\Delta\psi)]\\ \gamma=\arctan\left[\frac{\tan(\varphi+\Delta\varphi)-\tan(\theta+\Delta \theta)\sin(\varphi+\Delta\varphi)}{1+\tan(\varphi+\Delta\varphi)\tan(\theta+ \Delta\theta)\sin(\psi+\Delta\varphi)}\right]\end{array}\right. \tag{3}\]  
Frictional wear during the operation of the spherical joint can induce non-orthogonality between the \(X_{2}\), \(Y_{2}\), and \(Z_{2}\) axes depicted in Figure 2; nevertheless, Equation (3) defined the attitude error of the passive docking mechanism without taking this wear into account. We determined from the equation that the mistake has no effect, thus reducing the difficulty of assessing coupling motion errors and providing the technical foundation for future theoretical research.