# A Star Recognition Method Based on the Adaptive Ant Colony Algorithm for Star Sensors

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## Abstract

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## 1. Introduction

## 2. The Star Recognition Method Based on AAC Algorithm

#### 2.1. The Principle of the AAC Algorithm

_{ij}denotes the distance from food source i to food source j, namely the distance of the path (i,j), where i,j ∈ G. C

_{i}(t) denotes the number of ants at time t at the food source i, $N=\sum _{i=1}^{M}{C}_{i}(t)$.

_{ij}(t) denotes the amount of pheromone on the path (i,j) at time t. The amount of pheromone is equal on every path at initial time, set τ

_{ij}(0) is a constant. The table u

_{k}(k = 1,2,…,N) is used to store the visited food source. During the movement of the ant k, the ant chooses the path according to the transition probability, which is calculated by using the amount of pheromone and heuristic information of ant path, ${p}_{\mathit{ij}}^{k}(t)$ denotes the transition probability of the ant from food source i to food source j at time t.

_{k}= {G − u

_{k}} indicates the food sources for the ant k to choose. α and β are two constants that respectively define the influence of heuristic component and the trail information on decision of ants [15]. η

_{ij}(t) is the heuristic function; η

_{ij}(t) = 1/g

_{ij}indicates the expected time of the ant from food source i to food source j. With the movement of the ant colony, the pheromone released previously will gradually desalt. After time T has passed, all ants will complete one circulation, and the amount of pheromone will be updated by the following formula on the path (i,j) at the time t + T:

_{ij}(t) is the increment of pheromone on the path (i,j) during this circulation, Δτ

_{ij}(0) = 0 at initial time. $\mathrm{\Delta}{\tau}_{\mathit{ij}}^{k}(t)$ is the number of pheromone which is released by the ant k on the path (i,j) in this circulation [17].

_{k}is the total length that the ant k participates in this circulation. T denotes the ant k passes the path (i,j) in this circulation.

#### 2.2. The Construction of the Guidance-Star Database Based on the AAC Algorithm

- Select guidance stars. In order to guarantee existing guidance stars in random FOV, the screening of all stars is carried out. According to the spectral response curve of sensitive chip, optical lens and sensitivity of the star sensor, some stars will be selected from the basic star catalog to cover the overall celestial sphere as far as possible [23].
- For any guidance star i, a circle is drawn that is centered on the guidance star i, with radius r, which is used to guarantee at least three stars in the circle for the Guidance-star database construction based on AAC algorithm. If the guidance star j meets the formula (4), it will be put into the star point set.$$r>\theta =\text{arccos}(\frac{\overline{i}\u2022\overline{j}}{\left|\overline{i}\right|\left|\overline{j}\right|})$$Where, θ is the angular distance of star i̅ and j̅ star in the celestial coordinate system, arccos() means anti-cosine function, i̅ · j̅ means dot product between i̅ and j̅.
- For the stars in each star point set, the angular distance of any pair of star points in the star sets is calculated and the star point closest to the center of the star point set as the starting point is taken. Then the optimal (shortest) path of this star point set is retrieved with the AAC. Suppose the star point P is the starting point, then the path in Figure 1 shows the optimal path.
- At last, the angular distance values of the path optimization for each star point set are stored in an ascending-order database as a record. At the same time, the coordinate information of the stars is stored as well. Finally, the Guidance-star database is constructed. Under the condition of guaranteeing a successful rate of identification, and referring to the triangle identification algorithm (which only uses three angular distance paths), and also considering the memory load of the Guidance-star database, the only two angular distance paths and three star points’ sequence are involved. The Guidance-star database will be constructed by this method, without redundant data and with a smaller capacity than the triangle identification method, which needs some redundant triangles. Because this database stores angular distances with ascending order, the quick look-up algorithm and the binary search algorithm can be used to further enhance the identification speed.

#### 2.3. The Star Recognition Based on the AAC Algorithm

_{1}, s

_{2}, ..., s

_{Q}}. Using the adaptive ant colony algorithm to identify the star map IMG, the steps are as follows:

- The average gray value D
_{avg}of point set S is calculated. Suppose the centroid position of element s_{i}is (x_{i},y_{i}), whose pixel value is Vx_{i}y_{i}, the intensity Ds_{i}of element s_{i}is calculated by the following formula. Here, a circle average is taken for the star intensity of which the radius is three pixels.$${D}_{{s}_{i}}=\sum _{p={x}_{i}-2}^{{x}_{i}+2}\sum _{q={y}_{i}-2}^{{y}_{i}+2}{V}_{\mathit{pq}}+{V}_{{x}_{i}-3,{y}_{i}}+{V}_{{x}_{i},{y}_{i}-3}+{V}_{{x}_{i}+3,{y}_{i}}+{V}_{{x}_{i},{y}_{i}+3}$$So, the D_{avg}is$${D}_{\mathit{avg}}=\frac{1}{Q}\cdot \sum _{j=1}^{Q}{D}_{{s}_{j}}$$Selecting the light points whose value D and angular distance ϕ between itself and the center point of the star map meet the following formula (5) to form the light point set T, T = {t_{1}, t_{2}, ..., t_{R}}, R is the size of set T.$$(D>{D}_{\mathit{avg}})\cup (\varphi +r\le \frac{\mathit{Fov}}{2})$$Here, Fov is the FOV of the star sensor. - Centering at every light point of the point set T, a circle is drawn with a radius equal to angular distance r, centered on this light point. Figure 2 is the results of step (1) and step (2) for a star map. There are four light points (P
_{0}, P_{1}, P_{2}and P_{3}), which meet the formula (5). - Select a light point (P
_{0}), which is nearest to the center of the star map from P_{0}, P_{1}, P_{2}and P_{3}. There is a circle drawn with the center as light point P_{0}and the radius equal to special angular distance r. The light points that belong to the circle form the light point set M, and the angular distance of every two light points is found in M. Then the path optimization of M is accomplished by the AAC. Figure 3 is the result of optimization. - The angular distance values of P
_{0}P_{01}and P_{01}P_{02}are stored and compared respectively with angular distance d_{1}and angular distance d_{2}of the Guidance-star database. If the inequality (6) is true, the result of identification is successful.$$\sqrt{{({\mathrm{P}}_{0}{P}_{01}-{d}_{1})}^{2}+{({\mathrm{P}}_{01}{P}_{02}-{d}_{2})}^{2}}<{\theta}_{\text{th}}$$Here, θ_{th}is the minimal threshold of identification, the default value is the mean square deviation of system error.Otherwise, the light point P_{1}is selected and a circle is drawn with the center as the light point P_{1}and the radius being the angular distance r. Then step (3) and step (4) are repeated. If the light points P_{2}and P_{3}do not meet the inequality (6) until no other light points are selected, the result of identification is unsuccessful.

## 3. Hybrid Simulation Result and Analysis

## 4. Conclusions

## Acknowledgments

## References

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**Figure 2.**The results after step 1) and step 2) of the adaptive ant colony algorithm used to identify a star map.

Serial number | Angular distance 1 | Angular distance 2 | Star point 1 | Star point 2 | Star point 3 |
---|---|---|---|---|---|

Degree/° | (right ascension/°, declination/°) | ||||

2558 | 1.9001 | 1.9385 | (185.55,24.774) | (187.23,25.913) | (185.07,26.002) |

2559 | 1.9009 | 2.0574 | (185.08,26.620) | (187.16,26.227) | (185.55,24.774) |

2560 | 1.9013 | 2.1628 | (187.19,25.899) | (185.07,26.002) | (184.87,28.157) |

© 2010 by the authors; licensee Molecular Diversity Preservation International, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

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**MDPI and ACS Style**

Quan, W.; Fang, J.
A Star Recognition Method Based on the Adaptive Ant Colony Algorithm for Star Sensors. *Sensors* **2010**, *10*, 1955-1966.
https://doi.org/10.3390/s100301955

**AMA Style**

Quan W, Fang J.
A Star Recognition Method Based on the Adaptive Ant Colony Algorithm for Star Sensors. *Sensors*. 2010; 10(3):1955-1966.
https://doi.org/10.3390/s100301955

**Chicago/Turabian Style**

Quan, Wei, and Jiancheng Fang.
2010. "A Star Recognition Method Based on the Adaptive Ant Colony Algorithm for Star Sensors" *Sensors* 10, no. 3: 1955-1966.
https://doi.org/10.3390/s100301955