## 2 Error Analysis of the Center of Mass Method
### Discrete Approximation Error
The detected star is viewed as a point source for star sensors, and it usually covers a certain pixel region in the actual star images due to the defocus measure. Therefore, a Gaussian function is a reasonable approximation to describe the signal blurring effect [7; 22], and the 2-D signal intensity distribution of the star spot can be written as  
\[I(x,y,x_{0},y_{0})=\frac{I_{0}}{2\pi\sigma_{PSF}}\exp[-\frac{(x-x_{0})^{2}+(y- y_{0})^{2}}{2\sigma_{PSF}{}^{2}}]. \tag{16}\]  
where \(I_{0}\) is the total energy of the starlight, \((x_{0},y_{0})\) is the ideal star centroid position, and \(\sigma_{PSF}\) is the Gaussian width. For the 1-D case, Equation (16) can be reduced to  
\[I(x,x_{0})=\frac{I_{0}}{\sqrt{2\pi}\sigma_{PSF}}\exp[-\frac{(x-x_{0})^{2}}{2 \sigma_{PSF}{}^{2}}]. \tag{17}\]  
Since \(I(x,0)\) is an even function, the function \(f_{e}(x)\), which is the convolution result of \(I(x,0)\) and \(p(x)\), is also even. Thus, we have  
\[\sigma_{\hat{x}_{g}}=\frac{1}{\pi}\frac{\sum_{n=1}^{\infty}\sin(2\pi nx_{0})F_ {e^{\prime}}(n)}{F_{e}(0)+2\sum_{n=1}^{\infty}\cos(2\pi nx_{0})F_{e}(n)}. \tag{18}\]  
From Equation (5), it follows that  
\[ \begin{array}{ll}F_{e}(s)&=\mathcal{F}[f_{e}(x)]=\mathcal{F}[I(x,0)\otimes rect (x)]\\ &=I_{0}\exp[-2(\pi s\sigma_{PSF})^{2}]\sin(\pi s)/\pi s.\end{array}  \tag{19}\]  
where \(\mathcal{F}[]\) denotes the Fourier transform operation. Then  
\[ \begin{array}{ll}F_{e}{}^{{}^{\prime}}(n)&=I_{0}\exp[-2(\pi\sigma_{PSF}n)^{2 }]\cos(\pi n)/n\\ &=(-1)^{n}I_{0}\exp[-2(\pi n\sigma_{PSF})^{2}]/n.\end{array}  \tag{20}\]  
Therefore, the theoretical expression of the systematic error with a Gaussian distribution of starlight energy can be written as  
\[\sigma_{\hat{x}_{g}}=\frac{1}{\pi}\sum_{n=1}^{\infty}\left(-1\right)^{n}\exp[- 2(\pi n\sigma_{PSF})^{2}]\sin(2\pi nx_{0})/n. \tag{21}\]Actually, when the condition \(\sigma_{PSF}>0.2\) is satisfied, \(\exp[-2(\pi\sigma_{PSF})^{2}]\gg\exp[-2(\pi n\sigma_{PSF})^{2}]\left(n\geq 2\right)\); then Equation (21) is simplified to  
\[\sigma_{\hat{x}_{g}}=\frac{1}{\pi}\exp[-2(\pi\sigma_{PSF})^{2}]\sin(2\pi x_{0 }). \tag{22}\]  
Equation (22) indicates that the relationship between \(\sigma_{\hat{x}_{g}}\) and \(x_{0}\) can be described as a sinusoid function under sampling frequency limitation. Moreover, the amplitude of \(\sigma_{\hat{x}_{g}}\) decreases exponentially as the Gaussian width \(\sigma_{PSF}\) increases. To obtain the relationship in an actual situation, numerical simulations were performed in the following part. During the experiment, the sampling window was fixed at \(3\times 3\) pixels, and the ideal point spread function (IPSF) model [23] was utilized to simulate a space discretized digital gray star image spot, that is, the signal intensity detected by the \(i\)th pixel was  
\[I_{i}=\int_{x_{i}-1/2}^{x_{i}+1/2}I(x,x_{0})dx. \tag{23}\]  
After drawing a group of relationship curves between \(\sigma_{\hat{x}_{g}}\) and \(x_{0}\) under the different Gaussian widths, the corresponding 3-D numerical simulation results are illustrated in Figure 2.  
In Figure 2, there is an approximately sinusoid relationship when the Gaussian width \(\sigma_{PSF}\) is at a small value (\(\sigma_{PSF}<0.3\) pixels), and the numerical simulation results are also consistent with the expression in Equation (22). The relationship curve tends to be linear when the Gaussian width \(\sigma_{PSF}\) is relatively large (\(\sigma_{PSF}>0.6\) pixels), which results from the appearance of truncation error caused by the limitation of the sampling window size; this will be discussed in detail in the next section. Figure 2: Simulated relationship between the systematic error and the ideal centroid location under different Gaussian widths.