## 3 Modeling and Simulations of the Assembly Error Effect on the Stability Accuracy
### Definitions of Coordinates and Kinematics Analysis of the System  
To stabilize LOS, the angular deviations must be compensated for by the two motor-driven frames. To realize that, establishment of coordinates and relevant kinematics analysis is essential. To describe the motion relationships among the components in the system, their coordinates and motions should be defined separately. Those definitions of the commonly used azimuth-pitch two-axis mechanism are as follows, and shown in Figure 5.  
* The inertial coordinate  
This is denoted as \(\{i\}\) with axes of \((o_{i}x_{i}y_{i}z_{i})\). The center point of the Earth is taken as the coordinate origin \(o_{i}\); \(o_{i}x_{i}\) is along with the Earth's rotation axis; \(o_{i}y_{i}\) and \(o_{i}z_{i}\) are orthogonal to each other in the equatorial plane of the Earth, pointing to two fixed stars. \(o_{i}x_{i}\) and \(o_{i}y_{i}\) form a right-hand coordinate with \(o_{i}z_{i}\) perpendicularly.  
* The coordinate of the system base  
This is denoted as \(\{b\}\) with \((o_{p}x_{i}y_{p}z_{i})\). To simplify the modeling, the system is treated as fixed to the UAV, neglecting effects of the vibration damper, which is not the focus of this research. Hence, the coordinates of the UAV and the system base are unified as \(\{b\}\). \(o_{b}\) is the geometric center point of the location where the azimuth frame is assembled on the base; \(o_{b}x_{b}\), \(o_{b}y_{b}\) and \(o_{b}z_{b}\) are along the heading,  
Figure 4: Meaning of pointing error of a sample \(\Delta\theta_{si}\).  
Figure 5: The coordinate and motion definitions of the system.  
left-moving, and rising directions of the UAV. Then, the linear and angular velocity vectors of the system base in the inertial coordinate \(\{i\}\) are denoted as \(\overrightarrow{v}_{ib}=\left[v_{ibx},v_{iby},v_{ibz}\right]^{T}\) and \(\overrightarrow{\omega}_{ib}=\left[\omega_{ibx},\omega_{iby},\omega_{ibz} \right]^{T}\).  
The coordinate of the azimuth frame and axis  
This is denoted as \(\{a\}\) with \(\left(o_{a}x_{a}y_{a}z_{a}\right)\). The coordinates of the azimuth frame and axis are unified as \(\{a\}\) for simplification. \(o_{a}\) is the geometric center where the pitch frame is connected to the azimuth frame. \(o_{a}x_{a}\), \(o_{a}y_{a}\) and \(o_{a}z_{a}\) form a coordinate, parallel to \(\{b\}\) when the system has no azimuth rotation; the compensation azimuth rotation is around \(o_{a}z_{a}\). The angular velocity vector of the azimuth in \(\{i\}\) is denoted as \(\overrightarrow{\omega}_{ia}=\left[\omega_{iax},\omega_{iby},\omega_{iaz} \right]^{T}\); the rotation angle and angular velocity of azimuth compensation driven by the azimuth motor are denoted as \(\theta_{a}\) and \(\dot{\theta}_{a}\), which are around \(o_{a}z_{a}\) in \(\{a\}\).  
The coordinate of the pitch frame and axis  
This is denoted as \(\{p\}\) with \(\left(o_{p}x_{p}y_{p}z_{p}\right)\). \(o_{p}\) is treated as the start point of the LOS, and \(o_{p}x_{p}\) is treated as the LOS for simplification; \(\{p\}\) is parallel to \(\{a\}\) when the system has no pitch rotation; the compensation pitch rotation is around \(o_{p}y_{p}\); the angular velocity vector of the pitch in \(\{i\}\) is denoted as \(\overrightarrow{\omega}_{ip}=\left[\omega_{ipx},\omega_{ipy},\omega_{ipz} \right]^{T}\); the rotation angle and angular velocity of pitch compensation around \(o_{p}y_{p}\) in \(\{p\}\) are denoted as \(\theta_{p}\) and \(\dot{\theta}_{p}\).  
The coordinates of the gyroscopes  
These include \(\{ga\}\) with \(\left(o_{ga}x_{ga}y_{ga}z_{ga}\right)\) for the azimuth gyroscope, and \(\{gp\}\) with \(\left(o_{gp}x_{gp}y_{gp}z_{gp}\right)\) for the pitch one. \(\{ga\}\) and \(\{gp\}\) are ideally parallel to \(\{p\}\). \(o_{ga}\) and \(o_{gp}\) are the original points where the gyroscopes are assembled. The sensitive axis for LOS angular velocity of \(\omega_{ipz}\) is \(o_{ga}z_{ga}\) of \(\{ga\}\), and that for \(\omega_{ipy}\) is \(o_{gp}y_{gp}\) of \(\{gp\}\).