## 2 Error Analysis of the Center of Mass Method  
The CoM method is adopted by most star sensors to estimate the star centroid position with subpixel accuracy since it is easy to implement and exhibits better real-time performance. Therefore, the error of the CoM method is utilized for all discussions in this paper. When the CoM method is applied to digital images captured by the star sensor, the model of the method is described as  
\[\hat{x}_{g}=\frac{\sum_{i=1}^{n_{w}}x_{i}I_{i}}{\sum_{i=1}^{n_{w}}I_{i}},\hat{y }_{g}=\frac{\sum_{i=1}^{n_{w}}y_{i}I_{i}}{\sum_{i=1}^{n_{w}}I_{i}} \tag{1}\]  
where \((\hat{x}_{g},\hat{y}_{g})\) is the actual star centroid position, \(n_{w}\) is the number of validated discrete pixels in the sampling window, and \((x_{i},y_{i})\) and \(I_{i}\) are the geometric center coordinates and the detected signal intensity, respectively, of the \(i\)th pixel.  
From Equation (1), we can see that the centroiding accuracy of the CoM method is associated with three factors: the size of the sampling window, the coordinates of the validated pixel, and the corresponding detected signal intensity. The use of both the sampling window and the geometric center coordinates is the nature of the CoM method, while the uncertainty in the detected signal intensity is related to random noise caused by dark current noise, radiation noise, readout noise, etc. As a result, the error of the CoM method includes two parts, namely, the systematic error and the random error. The systematic error is derived from discrete approximation and sampling window truncation in the centroiding process, while the random error is introduced as image sensor noise.  
Firstly, the random error resulting from uncertainty in the detected \(I_{i}\) is analyzed in this section. Since there are various noises in the sensor image, the measured signal intensity \(I_{i}\) of the \(i\)th pixel includes two components: the starlight intensity \(E_{i}\) and the noise intensity \(\sigma_{I}\). According to the statement in Reference [15], the random error \(\sigma_{rand}\) in the \(x\) direction of the CoM method can be given as  
\[\sigma_{rand}^{2}=\frac{\left[\sum_{i=1}^{n_{w}}\left(x_{i}-x_{0}\right)\right] ^{2}\sigma_{I}^{2}}{I_{0}^{2}}=\left(\frac{\sigma_{I}}{I_{0}}\right)^{2}\sum _{i=1}^{n_{w}}\left[\left(x_{i}-x_{0}\right)\right]^{2} \tag{2}\]  
where  
\[I_{0}=\sum\
olimits_{i=1}^{n_{w}}I_{i},\ E_{0}=\sum\
olimits_{i=1}^{n_{w}}E_ {i},\ x_{0}=\sum\
olimits_{i=1}^{n_{w}}x_{i}E_{i}/E_{0}. \tag{3}\]  
Through Equation (2), it can be seen that the \(\sigma_{rand}\) value is determined by the signal-to-noise ratio (SNR), so improving the SNR, such as by optimizing the circuits or choosing low-noise image sensors, is an effective method to reduce the random error [20]. Scholars have proposed some methods to eliminate the random error, and further details concerning these are described in [12; 21].  
In this paper, we focus on the systematic error in the CoM method. The frequency field method and simulations are utilized to present an explicit analysis of the two types of error sources that bring about the systematic error, namely, the discrete approximation error and the sampling window truncation error.