# Mathematical Methods for an Accurate Navigation of the Robotic Telescopes

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

- Preliminary sky identification in the CCD-frames in a series, which allows finding the consistency between all objects in such CCD-frames in a series.
- Automatic selection of the reference astronomical objects (stars) [25] in the CCD-frame, which have fixed positional celestial coordinates in the sky.

#### 2.1. Preliminary Sky Identification

_{1(i)}, y

_{1(i)}, x

_{2(i)}, y

_{2(i)}are the coordinates of measurements of the same i-th object (estimates of the object’s coordinates) on the first and second identified frames in the coordinate system of the base frame of the series.

_{ack}was used. The number of acknowledgments is the number of acknowledgment circular areas (strobes) to which at least one measurement of another frame belongs (associated). The acknowledgment circular area (strobe) has a predetermined radius R

_{rej}and center with the measurement coordinates of the first frame with the shift values (1), (2) added to them. In the general case, frames are quite rarefied and diverse in the sense that their individual parts are not like each other. Under this assumption, it is not possible to test all hypotheses about the combination of measurements of two frames. It is enough to find the first hypothesis, in which the number of acknowledgments will be higher than the predetermined allowable number of acknowledgments N

_{min_ack}(Figure 1).

_{reg}× M

_{reg}. From each such area, the same predetermined number of the brightest objects is selected N

_{mea_reg}. Thus, the selected measurements will be evenly distributed over the frame, which will help to minimize the probability of errors in the preliminary sky identification. Such a selection of measurements will allow, for example, to exclude from consideration many bright false measurements caused by the charge flow of a large star or a bright satellite track.

- The frame is divided into a set of equal regions M
_{reg}× M_{reg}. Sets of the brightest measurements in frame are formed based on an equal predetermined number N_{mea_reg}of measurements with the highest brightness estimates corresponding to the hypothetical objects selected from each region. - Selecting of the next measurement from a preselected set of the brightest measurements in the first frame. There should be no more than three such measurements. If, during the process, this step is reached for the fourth time (trying to select the fourth measurement), an emergency exit is performed with a message about identification failure. This is usually associated with large errors in estimating the anchoring coordinates of center in the identified frame.
- The investigated measurement of the first frame is put in correspondence with the next measurement of the second frame from a preselected set of measurements of the second frame (a cycle is organized according to the investigated measurements of the second frame). For this, a conditional estimate of the shift parameters is preliminarily calculated by the pair hypothesis, according to Equations (1) and (2).
- For each selected pair (steps 2 and 3), the weight of the next hypothesis about the correspondence of pairs of measurements of the first and second frames (measurement of the frame and the star catalog) to the same object is estimated. For this, each measurement of the first frame is compared with each measurement of the second frame. Additionally, the shift parameters (1) and (2) are added to the measurement coordinates of the first frame. Based on the deviations between the measurements of the first and second frames, a fact that the measurements of the second frame fall into the acknowledgment area (strobe) is determined.
- If a sufficient number of measurements of the second frame fell into the strobe, then it is considered that the hypothesis about the combination of pairs of measurements of the first and second frames is confirmed (go to step 6). If not, then the hypothesis about the shift parameters is considered false and a transition is made (to step 3) to the next measurement of the second frame. When the preselected set of measurements of the second frame is exhausted, a transition is made to the next measurement of the first frame (to step 2). If this set is also exhausted, a message is displayed about the impossibility of identifying the measurements of the first and second frames.
- The final estimate of the shift parameters (3) and (4) is calculated.

#### 2.2. Full Sky Identification

_{1}, y

_{1}), B(x

_{2}, y

_{2}), C(x

_{3}, y

_{3}). The catalog equatorial coordinates of these stars correspond to the ideal coordinates A(ξ

_{1}, η

_{1}), B(ξ

_{2}, η

_{2}), C(ξ

_{3}, η

_{3}), respectively. The ideal coordinates of an object with its coordinates in the CS of the CCD-frame are related by the reduction equation [30]:

_{bl50}are used in turn. For the triple of measurements with coordinates (x

_{1(k)}, y

_{1(k)}), (x

_{2(k)}, y

_{2(k)}), (x

_{3(k)}, y

_{3(k)}) in the CS of the CCD-frame to form a triangle, which covers a significant part of the frame, for the other two points of the triple, the following conditions are experimentally introduced. The second point of the triple must be no closer than ${k}_{h}$ of the frame’s angular size ${R}_{CCD}$ from the first one:

- For a set of measurements of a CCD-frame, when forming the triplets of primary sky identification, the following sequence of operations is performed.
- Formation of a set Ω
_{bl50}of the brightest measurements in a CCD-frame, consisting of N_{bl50}applicants when choosing triplets of primary identification. To ensure a stability of the identification results, the frame is divided into ${M}_{reg}^{2}$ parts. The specified number of frame measurements N_{bl50}is divided by the number of frame fragments, and in each such fragment, the brightest frame measurements ${N}_{bl50}/{M}_{reg}^{2}$ are selected. - Formation of an additional set Ω
_{bl100}of the brightest measurements in a CCD-frame, consisting of N_{bl100}elements evenly distributed in a frame (by analogy with 1a). The set Ω_{bl100}is used to confirm the hypotheses of primary identification (formation of a weight of the next hypothesis about the correspondence of triples in frame and the astronomical catalog).

- For a set of measurements of the astronomical catalog, when forming the triplets of primary sky identification, the following sequence of operations is performed.
- Formation of a set Ω
_{star100}of catalog measurements, considering the uniform distribution of stars in the investigated area of the sky. - Formation of an additional set Ω
_{star200}of catalog measurements, consisting of N_{star200}elements, which are used to confirm the hypotheses of primary identification.

- Enumeration and confirmation of hypotheses of the primary sky identification.
- Enumerating the measurements of a set Ω
_{bl50}as elements of triples of the primary sky identification. The measurements that make up the triple of the primary sky identification must satisfy the conditions (9) and (10). - Enumeration of a set Ω
_{star100}of catalog measurements as elements of triples of the primary sky identification from the astronomical catalog side. - Comparison of triples of measurements for the primary sky identification from the frame and catalog sides based on the corresponding angles of triangles, the values of which are calculated according to Equations (11)–(16).
- Confirmation of the hypothesis about the parameters of frame and catalog identification, which corresponds to the considered triplets of the primary sky identification. The hypothesis is recognized as true if during the identification process of the sets Ω
_{bl100}and Ω_{star200}the formed admissible pairs exceed the predefined value v_{min_ident}. When the identification hypothesis is confirmed, further enumeration stops.

#### 2.3. Automatic Selection of the Reference Stars

_{fr}-th CCD-frame;

- Frame fragmentation for uniform distribution of the reference star candidates in a CCD-frame.
- Selection of measurements from the frame and catalog for their mutual identification.
- Rejection of candidates for the reference stars:
- Identification of the selected measurements from the frame and catalog with the formation of identified pairs.
- Calculation of the plate constants (19) (at each next step with a higher degree model).
- Rejection of identified pairs by the total deviation ${\Delta}_{\alpha \delta ijk}$ (21) between estimates of equatorial coordinates in an identified pair (22).
- Final calculation of the plate constants.
- The UML-diagram of the developed mathematical methods for the sky identification is presented in Figure 5.

#### 2.4. Accuracy Indicators of Estimates of the Angular Position and Brightness of the Reference Stars

_{1}-th measurement from a frame and j

_{2}-th measurement from a catalog, forming the i-th identified pair;

_{1}-th measurement from a frame and j

_{2}-th measurement from a catalog, forming the i-th identified pair;

_{1}is an index of the measurement from a frame in the internal numeration;

_{2}is an index of the measurement from a catalog in the internal numeration.

_{mea}is the number of measurements used to analyze the accuracy of estimates of the angular position of objects.

## 3. Results

#### 3.1. Real Astronomical Data Sources

#### 3.2. Reference Data Sources

#### 3.3. Accuracy of the Developed Mathematical Methods for the Sky Identification

- Mean deviation (27)–(29);
- Max. deviation module;
- Min. deviation module;
- Standard deviation of estimates (30)–(32).

#### 3.4. Implementation in the CoLiTec Software

## 4. Discussion

_{star_group}, the number of fragments into which the frame is divided along each coordinate when selecting the reference stars M

_{reg}, and the maximum allowable distance between neighboring measurements of a group of pixels r

_{mea_group}.

## 5. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

- Radius R
_{rej}of the acknowledgment circular area (strobe) is R_{rej}= 20 pixels; - Minimum allowable number of acknowledgments N
_{min_ack}= 70%; - Number of equal regions M
_{reg}×M_{reg}, on which frame is divided into is M_{reg}×M_{reg}= 4×4; - Number N
_{mea_reg}of measurements with the highest brightness estimates in frame is ${N}_{mea\_reg}={N}_{mea}/{M}_{reg}^{2}=3$; - Number N
_{bl50}of measurements (candidates) in frame for the role of elements of triplets (vertices of triangles) of the primary sky identification is N_{bl50}= 50; - Number N
_{bl100}of elements of the set Ω_{bl100}of measurements in frame used to confirm the hypotheses of the primary sky identification is N_{bl100}= 100; - Ratio of the number of elements of the sets Ω
_{bl100}and Ω_{bl50}of measurements in frame was assumed to be equal to k_{blob}= N_{bl100}/N_{bl50}= 2; - Number of regions M
_{reg}, on which frame is divided into is M_{reg}= 4; - Number N
_{star100}of stars (candidates) in astrometric catalog for the role of elements of triplets (vertices of triangles) of the primary sky identification is N_{start100}= 100; - Number N
_{star200}of stars of the set Ω_{star200}of measurements in astrometric catalog used to confirm the hypotheses of the primary sky identification is N_{star200}= 200; - Ratio of the number of elements of the sets Ω
_{star200}and Ω_{star100}of measurements in frame was assumed to be equal to k_{star}= N_{star200}/N_{star100}= 2; - Maximum allowable minimal distance between the second and first points of the triple of the primary sky identification, expressed in the angular measurements of a CCD-frame is k
_{h}= 0.1; - Under the condition of a rectangular (not square) frame, to determine the minimum distance between the second and first points of the triple, the value k
_{h}is multiplied by the average value of the frame size for both coordinates; - Maximum allowable deviation of values of the corresponding angles of the triangles (from a CCD-frame and the astrometric catalog sides) of the primary sky identification is Δᵞ = 60′.
- Limiting maximum value of the distance between the elements of an identified pair, at which it is considered valid is Δr
_{ident}= 10 pixels; - Minimum allowable ratio of the number of allowed pairs to the set Ω
_{bl100}size is v_{min_ident}= 0.7.

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**Figure 1.**Determining the shift parameters between measurements in frame (gray dot) and catalog or other frame (black dot): (

**a**) Correct identification; (

**b**) Wrong identification.

**Figure 2.**Various kinds of destabilizing factors in CCD-frame: (

**a**) bright track of the satellite; (

**b**) charge flow in images of the brightest stars.

**Figure 4.**(

**a**) The brightness measurements in a frame; (

**b**) Uniform distribution of the reference stars in a frame.

**Figure 6.**Histograms of the distributions of deviations of the equatorial coordinates of the reference stars in: (

**a**) right ascension; (

**b**) declination.

**Table 1.**The main parameters of deviations of the angular positions and brightness of the observed reference stars.

Processed Measurements | 30,391 | 28,872 | 27,352 |
---|---|---|---|

Rejection percentage of the worst measurements, % | 0 | 5 | 10 |

Mean deviation of RA, arcsec | 0.003 | 0.002 | 0.001 |

Mean deviation of DE, arcsec | 0.002 | 0.001 | 0.001 |

Mean deviation of brightness, mag. | 0.03 | 0.03 | 0.03 |

Max. deviation module of RA, arcsec | 0.32 | 0.15 | 0.13 |

Max. deviation module of DE, arcsec | 0.33 | 0.14 | 0.12 |

Min. deviation module of brightness, mag. | 0.002 | 0.001 | 0.001 |

Max. deviation module of brightness, mag. | 3.51 | 0.51 | 0.36 |

Standard deviation of RA, arcsec | 0.08 | 0.08 | 0.07 |

Standard deviation of DE, arcsec | 0.07 | 0.07 | 0.06 |

Standard deviation of brightness, mag. | 0.38 | 0.38 | 0.37 |

Processing Results | Number |
---|---|

Astronomical observations | >700,000 |

Discoveries of the Solar System objects (SSOs) | >1600 |

Discoveries of the Comets | 5 |

Discoveries of the Near-Earth objects (NEOs) | 5 |

Discoveries of the Trojan asteroids of Jupiter | 21 |

Discoveries of the Centaurs | 1 |

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Savanevych, V.; Khlamov, S.; Briukhovetskyi, O.; Trunova, T.; Tabakova, I.
Mathematical Methods for an Accurate Navigation of the Robotic Telescopes. *Mathematics* **2023**, *11*, 2246.
https://doi.org/10.3390/math11102246

**AMA Style**

Savanevych V, Khlamov S, Briukhovetskyi O, Trunova T, Tabakova I.
Mathematical Methods for an Accurate Navigation of the Robotic Telescopes. *Mathematics*. 2023; 11(10):2246.
https://doi.org/10.3390/math11102246

**Chicago/Turabian Style**

Savanevych, Vadym, Sergii Khlamov, Oleksandr Briukhovetskyi, Tetiana Trunova, and Iryna Tabakova.
2023. "Mathematical Methods for an Accurate Navigation of the Robotic Telescopes" *Mathematics* 11, no. 10: 2246.
https://doi.org/10.3390/math11102246