# A Performance Evaluation Approach for Satellite Attitude Control System in Tracking Mode
## 3 Satellite Attitude Control System Evaluation Indicator Set Construction  
### Performance Evaluation Indicator Set  
According to the efficiency analysis of the satellite attitude control system in tracking mode in Section 2.1, the control performance indicator set of the satellite attitude control system in tracking mode is constructed, as shown in Figure 2, including six performance indicators such as attitude pointing accuracy, pointing stability and maneuvering average angular velocity. According to the operating principles and control model of the attitude control system of the earth observation satellite, the definition and calculation methods of the performance indicators were studied, and the calculation methods for each indicator in the performance evaluation indicator system are as follows:  
1. Attitude pointing accuracy. Satellite attitude pointing accuracy refers to the deviation between the expected and actual pointing directions after the satellite undergoes three-axis rotation. Assuming that the initial unit vector \(\delta\), after rotating the initial unit vector \(\delta\) by an angle of \(\theta=(\alpha,\beta,\gamma)\) in Eulerian angle order, it is turned to a predetermined position to become the vector \(\delta^{\prime}\). In fact, various errors are generated during the rotation in tracking mode, and the initial unit vector \(\delta\) cannot be turned to a predetermined position to become vector \(\delta^{\prime}\), but rather to vector \(\delta^{\prime\prime}\). Then we have [15; 16]:  
\[\delta^{\prime\prime}=\rho(\theta,T)\cdot\delta \tag{1}\]  
\[\rho(\theta,T)=Rot(x,\alpha+\Delta\alpha)Rot(y,\beta+\Delta\beta)Rot(z,\gamma+ \Delta\gamma) \tag{2}\]  
Figure 2: Attitude tracking mode performance evaluation indicator set. In Equation (2), \(T\) represents the error generated by the rotation process, \(\rho(\theta,T)\) represents the total rotation matrix, \(\Delta\alpha,\ \Delta\beta,\ \Delta\gamma\) represent the three-axis error caused by the error, which can be described as:  
\[\sigma=\delta^{\prime\prime}-\delta^{\prime}=[\rho(\theta,T)-\rho(\theta)] \cdot\delta \tag{3}\]  
In Equation (3), \(\rho(\theta)\) represents the ideal rotation matrix, the pointing error \(\sigma=\left[\sigma_{x},\sigma_{y},\sigma_{z}\right]\) is obtained and the attitude pointing accuracy is \(\sigma_{0}=\max(\sigma)\). Relative attitude pointing accuracy. Relative attitude pointing accuracy refers to the ratio between the attitude pointing accuracy of a satellite's attitude maneuver process and the maneuver angle when the satellite undergoes an attitude maneuver. In the paper, the maximum error of the three axes is selected as the relative attitude pointing accuracy \(\hat{\sigma}\):  
\[\hat{\sigma}=\max\left(\frac{\sigma_{x}}{\alpha},\frac{\sigma_{y}}{\beta}, \frac{\sigma_{z}}{\gamma}\right) \tag{4}\]  
Pointing stability. The pointing stability is the rate at which the attitude angle changes with time due to disturbances during the time the satellite remains attitude stabilized. The three-axis attitude stability is during the time period that the satellite remains attitude stabilized [15]:  
\[\omega^{\prime}=\frac{\sum\limits_{i=1}^{n}\omega_{i}}{n}=\left[\omega_{x}{}^{ \prime},\omega_{y}{}^{\prime},\omega_{z}{}^{\prime}\right] \tag{5}\]  
In Equation (5), \(\omega^{\prime}\) represents the attitude stabilization of the three axes, \(\omega_{i}\) represents the instantaneous angular velocity within steady-state time, n represents the time n seconds after the end of the maneuvering process. During steady-state time, the attitude stabilization is \(\Delta\omega=\max(\omega^{\prime})\). Maneuvering angular velocity. Maneuver angular velocity refers to the rate of change of attitude angles during the time period when a satellite is undergoing an attitude maneuver. As shown in Figure 3, the 3D vector can be regarded as the unit vector \(\delta\) rotated to obtain, then the attitude angle change of the satellite after the maneuver can be regarded as finding the angle \(\theta\) between two 3D vectors.