## 3 Analysis of the Composite Docking Test Platform
### Attitude Error Analysis and Modeling
#### 3.2.2 Displacement Error Analysis and Modeling of the Double-Deck Sliding Platform  
A double-deck sliding platform and a converter mechanism comprise the active motion simulator. The double-deck sliding platform was built in a cross configuration,  
Figure 3: Attitude transformation of the passive docking mechanism establishing coordinate system oxyz. 2. Space Space docking coordinate system. allowing for translational mobility on both the \(X\) and \(Y\) axes. The two degrees of freedom were independent of one another. As illustrated in Figure 4, when the docking mode was changed from vertical to horizontal, the primary motion simulator moved down the V-groove track in the \(X\) direction of the \(O_{XYZ}\) coordinate system. The movement along the \(X_{5}\) and \(Y_{5}\) axes in the \(O_{5}X_{5}Y_{5}Z_{5}\) coordinate system was accomplished by the motor pulling the gear rotation to drive the pinion movement. The V-shaped wheel and the roller finished off the guiding limit. During motion, resistance and resistance torque were generated by the friction between the V wheel and the roller and the guide rail, as well as the system's inertia. The displacement errors along the \(X_{5}\) and \(Y_{5}\) axes of the double-deck sliding platform, as depicted in Figure 5, were caused primarily by transmission errors of the rack and pinion, deformations in the guide rail, and measurement errors of the grating ruler. The transmission faults of the rack and pinion were mostly caused by manufacture and installation problems, deformations in the gear teeth under the contact force experienced during meshing transmission, etc. Changes in the gap between the indicative grating and the scale grating caused by structural errors in the system, installation errors in the roller and lug, and friction and wear on the slide rails affected the precision of the grating ruler. Here, the rotary motion was converted to linear motion by the rack and pinion drive mechanism; furthermore, the rack and sliding platform were fixed together, and the linear displacement of the rack along the meshing line was the displacement of the sliding platform along the corresponding coordinate axis. Therefore, the linear displacement of the rack in transmission was directly reflected in the displacement error of the sliding platform; given this, the rack displacement error model was:  
\[\Delta S=S^{\prime}-S \tag{4}\]  
Figure 4: Attitude transformation of the active motion simulator. Figure 5: Structure of the double-deck sliding platform. 1. Indicative grating, 2. Scale grating, 3. V-type wheel, 4. Slide rail seat, 5. Roller, 6. Rack, 7. Pinion. where \(S\) and \(S^{\prime}\) indicate the theoretical and actual displacements of the gear when the gear rotated at an angle of \(\theta_{1}\), respectively. Furthermore, \(r_{b}\) represents the radius of the gear pitch circle, and \(DB\) and \(DH\) represent the elastic and contact deformation. The subscripts 1 and 2 were used to denote gears 1 and 2 quantities, respectively. We have:  
\[\Delta S=(r_{b}\theta_{1}-DB_{1}-DH_{1}+E_{1}+E_{2}-DB_{2}-DH_{2})-r_{b}\theta_ {1}=(E_{1}+E_{2})-(DB_{1}+DH_{1}+DB_{2}+DH_{2}) \tag{5}\]  
where \(E_{1}\) and \(E_{2}\) represent the tooth profile error and tooth pitch error, respectively, of the gear and rack at the meshing point. Suppose that:  
\[DB_{1}+DH_{1}+DB_{2}+DH_{2}=\delta \tag{6}\]  
and then Equation (5) can be expressed as:  
\[\Delta S=E-\delta \tag{7}\]  
where \(E\) is the algebraic sum of \(E_{1}\) and \(E_{2}\), and \(\delta\) is the sum of the deformation of meshing teeth along the meshing line and is the position function of meshing point. The deformation of the contact surface between the slide rail and the roller was caused by the gravities of conversion mechanism, docking mechanism, upper slide rail, and their own impacting force experienced during the docking process.