An Accurate and Efficient Gaussian Fit Centroiding Algorithm for Star Trackers

Abstract

This paper presents a novel centroiding algorithm for star trackers. The proposed algorithm, which is referred to as the Gaussian Grid algorithm, fits an elliptical Gaussian function to the measured pixel data and derives explicit expressions to determine the centroids of the stars. In tests, the algorithm proved to yield accuracy comparable to that of the most accurate existing algorithms, while being significantly less computationally intensive. Hence, the Gaussian Grid algorithm can deliver high centroiding accuracy to spacecraft with limited computational power. Furthermore, a hybrid algorithm is proposed in which the Gaussian Grid algorithm yields an accurate initial estimate for a least squares fitting method, resulting in a reduced number of iterations and hence reduced computational cost. The low computational cost allows to improve performance by acquiring the attitude estimates at a higher rate or use more stars in the estimation algorithms. It is also a valuable contribution to the expanding field of small satellites, where it could enable low-cost platforms to have highly accurate attitude estimation.

Appendix

In this Appendix, the expressions to calculate the values for A _{i, j} and B _{i, j} are given for the 3×3, 5×5, and 7×7 pixel window sizes (Table 7). The expressions are a function of the weights w _i used in the cost function (Table 7). These weights are assigned as follows:

Table 7 Weights grid

3×3 Pixel Window

The values for A _{i, j} and B _{i, j} for a 3×3 pixel window are calculated as follows:

$$\begin{array}{@{}rcl@{}} B_{0} &=& 0\\ B_{1} &=& 1\\ B_{-1} &=& -1\\ A_{0} &=& 4\\ A_{1} &=& 2\\ A_{-1} &=& 2\\ C_{0} &=& 2\\ C_{1} &=& -1\\ C_{-1} &=&-1\\ D &=& 6 \end{array} $$

5×5 Pixel Window

The values for A _{i, j} and B _{i, j} for a 5×5 pixel window are calculated as follows:

$$\begin{array}{@{}rcl@{}} A_{0} &= & 2(w_{0}w_{1}w_{2}+w_{0}w_{-1}w_{-2})-4w_{0}w_{1}w_{-1} -18(w_{0}w_{1}w_{-2}+w_{0}w_{-1}w_{2})\\ &&-64w_{0}w_{2}w_{-2}\\ A_{1} &=& 2w_{0}w_{-1}w_{1}+6w_{1}w_{-1}w_{-2}-4w_{0}w_{1}w_{2}+12w_{0}w_{1}w_{-2}-18w_{1}w_{2}w_{-1}\\&&-48w_{1}w_{2}w_{-2}\\ A_{2} &=& 2w_{0}w_{1}w_{2}+6w_{0}w_{2}w_{-1}+12(w_{1}w_{2}w_{-1}+w_{2}w_{-1}w_{-2})+32w_{0}w_{2}w_{-2}\\&&+36w_{1}w_{2}w_{-2}\\ A_{-1} &=& 2w_{0}w_{1}w_{-1}+6w_{1}w_{2}w_{-1}-4w_{0}w_{-1}w_{-2}+12w_{0}w_{2}w_{-1}-18w_{1}w_{-1}w_{-2}\\&&-48w_{2}w_{-1}w_{-2}\\ A_{-2} &=& 2w_{0}w_{-1}w_{-2}+6w_{0}w_{-2}w_{1}+12(w_{1}w_{2}w_{-2}+w_{1}w_{-1}w_{-2})+32w_{0}w_{2}w_{-2}\\&&+36w_{2}w_{-1}w_{-2}\\ B_{0} &=& 3(w_{0}w_{1}w_{2}-w_{0}w_{-1}w_{-2})+9(w_{0}w_{1}w_{-2}-w_{0}w_{-1}w_{2})\\ B_{1} &=& -w_{0}w_{1}w_{-1}-4w_{0}w_{1}w_{2}-9(w_{1}w_{2}w_{-1}+w_{1}w_{-2}w_{-1})-12w_{0}w_{1}w_{-2}\\ B_{2} &=& w_{0}w_{1}w_{2}-3w_{0}w_{-1}w_{2}-18(w_{1}w_{-2}w_{2}+w_{-1}w_{-2}w_{2})-32w_{0}w_{2}w_{-2}\\ B_{-1} &=& w_{0}w_{1}w_{-1}+4w_{0}w_{-1}w_{-2}+9(w_{1}w_{2}w_{-1}+w_{1}w_{-1}w_{-2})+12w_{0}w_{2}w_{-1}\\ B_{-2} &=& -w_{0}w_{-1}w_{-2}+3w_{0}w_{1}w_{-2}+18(w_{1}w_{2}w_{-2}+w_{2}w_{-1}w_{-2})+32w_{0}w_{2}w_{-2} \end{array} $$

$$\begin{array}{@{}rcl@{}} C_{0} &=& -32 \ast w_{2}\ast w_{-2}-9 \ast(w_{1}\ast w_{-2}+w_{2} \ast w_{-1})+w_{-1}\ast w_{-2}+w_{1}\ast w_{2}\\&&-2\ast w_{1}\ast w_{-1}\\ C_{1} &=& -2 \ast w_{0} \ast w_{2}+w_{0} \ast w_{-1}+6 \ast w_{0} \ast w_{-2}-9 \ast w_{2} \ast w_{-1}-24 \ast w_{2} \ast w_{-2}\\&&+3 \ast w_{-1} \ast w_{-2}\\ C_{2} &=& w_{0} \ast w_{1}+3 \ast w_{0} \ast w_{-1}+16 \ast w_{0} \ast w_{-2}+18 \ast w_{1} \ast w_{-2}\\&&+6 \ast (w_{1} \ast w_{-1}+w_{-1} \ast w_{-2})\\ C_{-1} &=& w_{0} \ast w_{1}+6 \ast w_{0} \ast w_{2}-2 \ast w_{0} \ast w_{-2}+3 \ast w_{1} \ast w_{2}-9 \ast w_{1} \ast w_{-2}\\&&-24 \ast w_{2} \ast w_{-2}\\ C_{-2} &=& 18*w_{2}*w_{-1}+6*(w_{1}*w_{-1}+w_{1}*w_{2})+16*w_{0}*w_{2}+w_{0}*w_{-1}\\&&+3*w_{0}*w_{1}\\ D &=& -128 \ast w_{0} \ast w_{2} \ast w_{-2}-72 \ast w_{2} \ast (w_{1} \ast w_{-2}+w_{-1} \ast w_{-2})\\ &&-18 \ast (w_{0} \ast w_{1} \ast w_{-2}+w_{2} \ast w_{-1} \ast w_{0}+w_{1} \ast w_{-2} \ast w_{-1}+w_{1} \ast w_{2} \ast w_{-1})\\ &&-2 \ast w_{0} \ast (w_{1} \ast w_{2}+w_{1} \ast w_{-1}+w_{-1} \ast w_{-2}) \end{array} $$

7×7 Pixel Window

The values for A _{i, j} and B _{i, j} for a 7×7 pixel window are calculated as follows:

$$\begin{array}{@{}rcl@{}} A_{-3} &=& 12w_{0}w_{1}w_{-3}+60w_{0}w_{2}w_{-3}+162w_{0}w_{3}w_{-3}+6w_{0}w_{-1}w_{-3}\\&&+12w_{0}w_{-2}w_{-3}+20w_{1}w_{2}w_{-3}+96w_{1}w_{3}w_{-3}+32w_{1}w_{-1}w_{-3}\\&&+36w_{1}w_{-2}w_{-3}+30w_{2}w_{3}w_{-3}+90w_{2}w_{-1}w_{-3}+80w_{2}w_{-2}w_{-3}\\ &&+192w_{3}w_{-1}w_{-3}+150w_{3}w_{-2}w_{-3}+2w_{-1}w_{-2}w_{-3}\\ A_{-2} &=& 2w_{0}w_{-1}w_{-2}-18w_{0}w_{-2}w_{-3}+12w_{1}w_{2}w_{-2}+60w_{1}w_{3}w_{-2}\\&&+12w_{1}w_{-1}w_{-2}-48w_{1}w_{-2}w_{-3}+20w_{2}w_{3}w_{-2}+36w_{2}w_{-1}w_{-2}\\&&-100w_{2}w_{-2}w_{-3}+80w_{3}w_{-1}w_{-2}-180w_{3}w_{-2}w_{-3}-4w_{-1}w_{-2}w_{-3}\\ &&+6w_{0}w_{1}w_{-2}+32w_{0}w_{2}w_{-2}+90w_{0}w_{3}w_{-2}\\ A_{-1} &=&2w_{0}w_{1}w_{-1}+12w_{0}w_{2}w_{-1}+36w_{0}w_{3}w_{-1}-4w_{0}w_{-1}w_{-2}-18w_{0}w_{-1}w_{-3}\\&&+6w_{1}w_{2}w_{-1}+32w_{1}w_{3}w_{-1}-18w_{1}w_{-1}w_{-2}-64w_{1}w_{-1}w_{-3}\\ &&+12w_{2}w_{3}w_{-1}-48w_{2}w_{-1}w_{-2}-150w_{2}w_{-1}w_{-3}-100w_{3}w_{-1}w_{-2}\\&&-288w_{3}w_{-1}w_{-3}+2w_{-1}w_{-2}w_{-3}\\ A_{0} &=& 2w_{0}w_{1}w_{2}+12w_{0}w_{1}w_{3}-4w_{0}w_{1}w_{-1}-18w_{0}w_{1}w_{-2}-48w_{0}w_{1}w_{-3}\\&&+6w_{0}w_{2}w_{3}-18w_{0}w_{2}w_{-1}-64w_{0}w_{2}w_{-2}-150w_{0}w_{2}w_{-3}-48w_{0}w_{3}w_{-1}\\&&-150w_{0}w_{3}w_{-2}-324w_{0}w_{3}w_{-3}+2w_{0}w_{-1}w_{-2}+12w_{0}w_{-1}w_{-3}\\&&+6w_{0}w_{-2}w_{-3}\\ A_{1} &=& 36w_{0}w_{1}w_{-3}+2w_{1}w_{2}w_{3}-18w_{1}w_{2}w_{-1}-48w_{1}w_{2}w_{-2}-100w_{1}w_{2}w_{-3}\\&&-64w_{1}w_{3}w_{-1} -150w_{1}w_{3}w_{-2}-288w_{1}w_{3}w_{-3}+6w_{1}w_{-1}w_{-2}\\&&+32w_{1}w_{-1}w_{-3}+12w_{1}w_{-2}w_{-3}-4w_{0}w_{1}w_{2} -18w_{0}w_{1}w_{3}+2w_{0}w_{1}w_{-1}\\&&+12w_{0}w_{1}w_{-2} \end{array} $$

$$\begin{array}{@{}rcl@{}} A_{2} &=& 2w_{0}w_{1}w_{2}-18w_{0}w_{2}w_{3}+6w_{0}w_{2}w_{-1}+32w_{0}w_{2}w_{-2}+90w_{0}w_{2}w_{-3}\\&&-4w_{1}w_{2}w_{3} +12w_{1}w_{2}w_{-1}+36w_{1}w_{2}w_{-2}+80w_{1}w_{2}w_{-3}-48w_{2}w_{3}w_{-1}\\&&-100w_{2}w_{3}w_{-2}-180w_{2}w_{3}w_{-3} +12w_{2}w_{-1}w_{-2}+60w_{2}w_{-1}w_{-3}\\&&+20w_{2}w_{-2}w_{-3}\\ A_{3} &=& 6w_{0}w_{1}w_{3}+12w_{0}w_{2}w_{3}+12w_{0}w_{3}w_{-1}+60w_{0}w_{3}w_{-2}+162w_{0}w_{3}w_{-3}\\&&+2w_{1}w_{2}w_{3}+32w_{1}w_{3}w_{-1}+90w_{1}w_{3}w_{-2}+192w_{1}w_{3}w_{-3}+36w_{2}w_{3}w_{-1}\\&&+80w_{2}w_{3}w_{-2}+150w_{2}w_{3}w_{-3}+20w_{3}w_{-1}w_{-2}+96w_{3}w_{-1}w_{-3}\\&&+30w_{3}w_{-2}w_{-3}\\ B_{-3} &=& 6w_{0}w_{1}w_{-3}+60w_{0}w_{2}w_{-3}+243w_{0}w_{3}w_{-3}-3w_{0}w_{-1}w_{-3}\\&&-12w_{0}w_{-2}w_{-3}+30w_{1}w_{2}w_{-3}+192w_{1}w_{3}w_{-3}-18w_{1}w_{-2}w_{-3}\\&&+75w_{2}w_{3}w_{-3}+45w_{2}w_{-1}w_{-3}+192w_{3}w_{-1}w_{-3}+75w_{3}w_{-2}w_{-3}\\ &&-3w_{-1}w_{-2}w_{-3}\\ B_{-2} &=& 3w_{0}w_{1}w_{-2}+32w_{0}w_{2}w_{-2}+135w_{0}w_{3}w_{-2}-w_{0}w_{-1}w_{-2}+27w_{0}w_{-2}w_{-3}\\&&+18w_{1}w_{2}w_{-2}+120w_{1}w_{3}w_{-2}+48w_{1}w_{-2}w_{-3}+50w_{2}w_{3}w_{-2}\\&&+18w_{2}w_{-1}w_{-2}+50w_{2}w_{-2}w_{-3}+80w_{3}w_{-1}w_{-2}+8w_{-1}w_{-2}w_{-3}\\ B_{-1} &=& w_{0}w_{1}w_{-1}+12w_{0}w_{2}w_{-1}+54w_{0}w_{3}w_{-1}+4w_{0}w_{-1}w_{-2}+27w_{0}w_{-1}w_{-3}\\&&+9w_{1}w_{2}w_{-1}+64w_{1}w_{3}w_{-1}+9w_{1}w_{-1}w_{-2}+64w_{1}w_{-1}w_{-3}\\&&+30w_{2}w_{3}w_{-1}+75w_{2}w_{-1}w_{-3}-50w_{3}w_{-1}w_{-2}-5w_{-1}w_{-2}w_{-3}\\ B_{0} &=& -15w_{0}w_{-2}w_{-3}+3w_{0}w_{1}w_{2}+24w_{0}w_{1}w_{3}+9w_{0}w_{1}w_{-2}+48w_{0}w_{1}w_{-3}\\&&+15w_{0}w_{2}w_{3}-9w_{0}w_{2}w_{-1}+75w_{0}w_{2}w_{-3}-48w_{0}w_{3}w_{-1}-75w_{0}w_{3}w_{-2}\\&&-3w_{0}w_{-1}w_{-2}-24w_{0}w_{-1}w_{-3}\\ B_{1} &=& -75w_{1}w_{3}w_{-2}-9w_{1}w_{-1}w_{-2}-64w_{1}w_{-1}w_{-3}-30w_{1}w_{-2}w_{-3}-4w_{0}w_{1}w_{2}\\&&-27w_{0}w_{1}w_{3}-w_{0}w_{1}w_{-1}-12w_{0}w_{1}w_{-2}-54w_{0}w_{1}w_{-3}+5w_{1}w_{2}w_{3}\\&&-9w_{1}w_{2}w_{-1}+50w_{1}w_{2}w_{-3}-64w_{1}w_{3}w_{-1}\\ B_{2} &=& w_{0}w_{1}w_{2}-27w_{0}w_{2}w_{3}-3w_{0}w_{2}w_{-1}-32w_{0}w_{2}w_{-2}-135w_{0}w_{2}w_{-3}\\&&-8w_{1}w_{2}w_{3}-18w_{1}w_{2}w_{-2}-80w_{1}w_{2}w_{-3}-48w_{2}w_{3}w_{-1}-50w_{2}w_{3}w_{-2}\\&&-18w_{2}w_{-1}w_{-2}-120w_{2}w_{-1}w_{-3}-50w_{2}w_{-2}w_{-3}\\ B_{3} &=& 3w_{0}w_{1}w_{3}+12w_{0}w_{2}w_{3}-6w_{0}w_{3}w_{-1}-60w_{0}w_{3}w_{-2}-243w_{0}w_{3}w_{-3}\\&&+3w_{1}w_{2}w_{3}-45w_{1}w_{3}w_{-2}-192w_{1}w_{3}w_{-3}+18w_{2}w_{3}w_{-1}-75w_{2}w_{3}w_{-3}\\&&-30w_{3}w_{-1}w_{-2}-192w_{3}w_{-1}w_{-3}-75w_{3}w_{-2}w_{-3} \end{array} $$