# Numerical Calculation for the Line-of-Sight Attitudes of Multi-Address Transceivers without 2:1 Transmissions for Space Laser Communication Networking

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## Abstract

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## 1. Introduction

- (1)
- A new method of establishing a reflector coordinate system is proposed, which realizes the conversion of projections between a reflector and LOS in different coordinate systems.
- (2)
- A mathematical method for the mutual conversion between the attitudes of a multi-reflector is proposed, which can realize the numerical solution of the attitudes of a reflector and the LOS at any position. The attitude of all the reflectors and the LOS can be calculated simultaneously by a single gyro.
- (3)
- According to the spatial transformation relation of Snell’s law of reflection, the Snell transformation matrix is established. Through the Snell transformation of LOS in different reflector coordinate systems, the doubled coupling effects produced in the numerical solution of the LOS attitude of a multi-reflector are eliminated.

## 2. Scheme for Space Laser Communication Networking

#### 2.1. Networking Principle

#### 2.2. Indoor Experiment for Laser Communication Networking

#### 2.3. Multi-Reflector Scheme for Laser Communication Networking

## 3. Calculations for the Attitudes of the LOS of the Multi-Reflectors

#### 3.1. Establishing the Coordinate System

_{4×4}Y

_{4}Z

_{4}was established such that it coincided with the initial azimuth axis, pitch axis, and roller of the base. The barycenter of the base was the origin of this system. The coordinate system coinciding with the coordinate system of the base when the multi-address transceivers were in their initial steady states was defined as the reference coordinate system OX

_{1}Y

_{1}Z

_{1}. When the base’s coordinate system moved, the reference coordinate system was considered the static coordinate system to measure the rotation Euler angle of the base’s coordinate system.

_{4i}X

_{4i}Y

_{4i}Z

_{4i}(i = 1,2,…,6) were established on two parallel planes, one for each of the three coordinate systems. The coordinate system O

_{4}X

_{4}Y

_{4}Z

_{4}rotated for 0°, 60°, 120°, 180°, 240°, and 300°, respectively, around the Z

_{4}axis, thereby transforming into six coordinate systems, namely, O

_{41}X

_{41}Y

_{41}Z

_{41}, O

_{42}X

_{42}Y

_{42}Z

_{42}, O

_{43}X

_{43}Y

_{43}Z

_{43}, O

_{44}X

_{44}Y

_{44}Z

_{44}, O

_{45}X

_{45}Y

_{45}Z

_{45}, and O

_{46}X

_{46}Y

_{46}Z

_{46}. The transformed coordinate systems of each reflector are illustrated in Figure 5.

**n**was established coinciding with the initial position of a reflector’s LOS. When the coordinate system rotated in space, the projections of the said vector

**n**were observed in different coordinate systems. Accordingly, the attitude angles of the vector

**n**in different coordinate systems were calculated. Further, the reflector’s attitude could be adjusted according to the calculated attitude angles of the LOS.

#### 3.2. Mathematical Model of the Attitudes of the LOS of the Multi-Reflector

**r**

_{1}be the projection of the initial vector

**n**in the reference coordinate system OX

_{1}Y

_{1}Z

_{1}and

**r**

_{41}be the projection of the initial vector

**n**in the reflector coordinate system O

_{41}X

_{41}Y

_{41}Z

_{41}. Then, when the reference coordinate system OX

_{1}Y

_{1}Z

_{1}rotated for an angle α around the axis Z

_{1}, a transformed coordinate system O

_{2}X

_{2}Y

_{2}Z

_{2}was obtained, with the transformation matrix ${\mathit{C}}_{1}^{2}$. When the coordinate system O

_{2}X

_{2}Y

_{2}Z

_{2}rotated for an angle β around its axis Y

_{2}, a coordinate system OX

_{3}Y

_{3}Z

_{3}was obtained, with the transformation matrix ${\mathit{C}}_{2}^{3}$. When the coordinate system O

_{3}X

_{3}Y

_{3}Z

_{3}rotated for an angle γ around the axis Y

_{3}of the reference coordinate system, a coordinate system OX

_{4}Y

_{4}Z

_{4}(which is the coordinate system of the base) was obtained, with the transformation matrix ${\mathit{C}}_{3}^{4}$. Accordingly, the transformation matrix ${\mathit{C}}_{1}^{4}$ from the reference coordinate system to the base’s coordinate system was as follows:

_{41}X

_{41}Y

_{41}Z

_{41}. Accordingly, the direction cosine transformation relationship between

**r**and

_{1}**r**was as follows:

_{4}_{i}be the rotation angle and i be the serial number of the corresponding reflector. Then, the following is obtained:

**r**

_{4i}be the vector of each reflector’s LOS. Then, the LOS vector of a reflector without a 2:1 transmission was indicated as follows:

_{1}. Accordingly, the geometrical relationship between the two angles is:

_{1}is the LOS attitude of each reflector. In order to eliminate the doubled optical coupling effect, we need to establish a reflector coordinate system with a double LOS. The vector

**r**

_{4i}in six reflector coordinate systems is projected into the reflector coordinate system with double LOS to obtain the vector ${\mathit{r}}_{4i}^{1}$. According to Euler’s theorem and Snell’s law of reflection, the transformation matrix between the two coordinate systems is established.

#### 3.3. Numerical Calculations for the Attitudes of the LOS of the Multi-Reflector

#### 3.3.1. Quaternion-Based Numerical Calculation Model

**i**,

**j**, and

**k**are unit vectors for the different directions in three dimensions.

**r**

_{1}and

**r**

_{4}could be expressed in the form of quaternion, as follows:

**C**′ represents the quaternion transformation matrix, which was transformed into a transformation matrix from

**r**

_{4}to

**r**

_{1}, such that the following holds:

**ω**is the angular velocity of the base disturbance measured using a MEMS gyro.

**ω**(t) denotes the real-time angular velocity measured and updated using gyros,

**q**(t) represents the real-time calculation data of the quaternion at time t, and

**q**(t + T) is the quaternion generated through iteration using the stated algorithm at the next time.

#### 3.3.2. Direct Calculation Method

_{y}is the rotating angular velocity of the LOS. Substituting Formula (21) into Formula (20) for the numerical iteration would then generate the quaternion updated in real-time. In this manner, the attitude of the reflector’s LOS was calculated.

## 4. Simulation Experiment

_{4}X

_{4}Y

_{4}Z

_{4}coincided with the reference coordinate system, during the rotation of the former coordinate system for an angle a around Y

_{1}, its axis X

_{4}and the axis X

_{1}of the latter coordinate system formed an included angle, which was the cone angle a of the conical motion. At this time, axis X4 was the sideline OL of the conical motion. The base’s coordinate system O

_{4}X

_{4}Y

_{4}Z

_{4}rotated around the axis X

_{1}at an angular velocity ω, and the step size for the calculation was h. The equivalent unit vector of this conical motion was as follows:

_{41}X

_{41}Y

_{41}Z

_{41}, O

_{42}X

_{42}Y

_{42}Z

_{42}, O

_{43}X

_{43}Y

_{43}Z

_{43}, O

_{44}X

_{44}Y

_{44}Z

_{44}, O

_{45}X

_{45}Y

_{45}Z

_{45}, and O

_{46}X

_{46}Y

_{46}Z

_{46}. We obtained the LOS attitude curves and calculation error curves of six reflectors under the ideal conical motion, as shown in Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17 and Figure 18.

^{–3}µrad, for the conical motion (b) was over 10

^{–2}µrad, and for the conical motion (c) was higher than 0.2 µrad.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Wang, C.; Zhang, T.; Tong, S.; Li, Y.; Jiang, L.; Liu, Z.; Shi, H.; Liu, J.; Jiang, H. Pointing and tracking errors due to low-frequency deformation in inter-satellite laser communication. J. Mod. Opt.
**2019**, 66, 430–437. [Google Scholar] [CrossRef] - Chaudhry, A.U.; Yanikomeroglu, H. Laser intersatellite links in a starlink constellation: A classification and analysis. IEEE Vehic. Technol. Mag.
**2021**, 16, 48–56. [Google Scholar] [CrossRef] - Li, R.; Lin, B.; Liu, Y.; Dong, M.; Zhao, S. A survey on laser space network: Terminals, links, and architectures. IEEE Access
**2022**, 10, 34815–34834. [Google Scholar] [CrossRef] - Chen, S.; Liang, Y.; Sun, S.; Kang, S.; Cheng, W.; Peng, M. Vision, requirements, and technology trend of 6G: How to tackle the challenges of system coverage, capacity, user data-rate and movement speed. IEEE Wirel. Commun.
**2020**, 27, 218–228. [Google Scholar] [CrossRef][Green Version] - Edwards, B.; Randazzo, T.; Babu, N.; Murphy, K.; Albright, S.; Cummings, N.; Ocasio-Perez, J.; Potter, W.; Roder, R.; Zehner, S.A. Challenges, Lessons Learned, and Methodologies from the LCRD Optical Communication System AI&T. In Proceedings of the 2022 IEEE International Conference on Space Optical Systems and Applications (ICSOS), Kyoto City, Japan, 28–31 March 2022; pp. 22–31. [Google Scholar]
- Chishiki, Y.; Yamakawa, S.; Takano, Y.; Miyamoto, Y.; Araki, T.; Kohata, H. Overview of optical data relay system in JAXA. In Proceedings of the Free-Space Laser Communication and Atmospheric Propagation XXVIII, San Francisco, CA, USA, 13–18 February 2016; pp. 114–118. [Google Scholar]
- Yamakawa, S.; Chishiki, Y.; Sasaki, Y.; Miyamoto, Y.; Kohata, H. JAXA’s optical data relay satellite programme. In Proceedings of the 2015 IEEE International Conference on Space Optical Systems and Applications (ICSOS), New Orleans, LA, USA, 26–28 October 2015; pp. 1–3. [Google Scholar]
- Zhang, Y.L.; An, Y.; Wang, C.H.; Jiang, L.; Zhan, J.T.; Liu, X.Z.; Jiang, H.L. Research on rotating paraboloid based surface in space laser communication network. Acta Opt. Sin.
**2015**, 35, 86–90. [Google Scholar] [CrossRef] - Zhang, T.; Mao, S.; Fu, Q.; Cao, G.; Su, S.; Jiang, H. Networking optical antenna of space laser communication. J. Laser Appl.
**2017**, 29, 012013. [Google Scholar] [CrossRef] - Wang, L.; Zhang, L.; Meng, L.; Bai, Y. A calculation method for line-of-sight stable attitude of networked optical transceiver based on depth feedforward neural network. In Proceedings of the 2022 3rd International Conference on Computer Vision, Image and Deep Learning & International Conference on Computer Engineering and Applications (CVIDL & ICCEA), Changchun, China, 20–22 May 2022; pp. 32–35. [Google Scholar]
- Kennedy, P.J.; Kennedy, R.L. Direct versus indirect line of sight (LOS) stabilization. IEEE Trans. Control Syst. Technol.
**2003**, 11, 3–15. [Google Scholar] [CrossRef] - Hamilton, A. Strapdown optical stabilization system for EO sensors on moving platforms. Design and Engineering of Optical Systems. SPIE
**1996**, 2774, 631–645. [Google Scholar] - Mao, Y.; Tian, J.; Ma, J.G. Realization of LOS (Line of Sight) stabilization based on reflector using carrier attitude compensation method. In XX International Symposium on High-Power Laser Systems and Applications 2014. SPIE
**2015**, 9255, 1003–1008. [Google Scholar] - Walter, R.E.; Danny, H.; Donaldson, J. Stabilized inertial measurement system (SIMS). Laser Weapons Technology III. SPIE
**2002**, 4724, 57–68. [Google Scholar] - Schneeberger, T.J.; Barker, K.W. High-altitude balloon experiment: A testbed for acquisition, tracking, and pointing technologies. Acquisition, Tracking, and Pointing VII. SPIE
**1993**, 1950, 2–15. [Google Scholar] - Gilmore, J.P.; Luniewicz, M.F.; Sargent, D. Enhanced precision pointing jitter suppression system. Laser and Beam Control Technologies. SPIE
**2002**, 4632, 38–49. [Google Scholar] - Hilkert, J.M. Inertially stabilized platform technology concepts and principles. IEEE Contr. Syst. Mag.
**2008**, 28, 26–46. [Google Scholar] [CrossRef] - Hilkert, J.M. A comparison of inertial line-of-sight stabilization techniques using mirrors. In Proceedings of the Acquisition, Tracking, and Pointing XVIII, Orlando, FL, USA, 12–16 April 2004; pp. 13–22. [Google Scholar]
- Masten, M.K. Inertially stabilized platforms for optical imaging systems. IEEE Control. Syst. Mag.
**2008**, 28, 47–64. [Google Scholar] - Wu, Y.; Litmanovich, Y.A. Strapdown attitude computation: Functional iterative integration versus taylor series expansion. Gyroscopy Navig.
**2020**, 11, 263–276. [Google Scholar] [CrossRef] - Zhao, C.; Fan, J.; Liu, N. Simulation Research on Attitude Solution Method of Micro-Mini Missile. J. Syst. Simul.
**2019**, 31, 2877–2884. [Google Scholar] [CrossRef] - Wu, S.; Radice, G.; Gao, Y.; Sun, Z. Quaternion-based finite time control for spacecraft attitude tracking. Acta Astronaut.
**2011**, 69, 48–58. [Google Scholar] [CrossRef] - Savage, P.G. Strapdown inertial navigation integration algorithm design part 1: Attitude algorithms. J. Guid. Contr. Dynam.
**1998**, 21, 19–28. [Google Scholar] [CrossRef] - Shi, K.; Liu, M. Strapdown inertial navigation quaternion fourth-order runge-kutta attitude algorithm. J. Detect. Contr.
**2019**, 41, 61–65. [Google Scholar] - Zhang, Z.; Geng, L.; Fan, Y. Performance Analysis of Three Attitude Algorithms for SINS. Acad. J. Comput. Inform. Sci.
**2022**, 5, 1–5. [Google Scholar] [CrossRef] - Lee, J.G.; Yoon, Y.J.; Mark, J.G.; Tazartes, D.A. Extension of strapdown attitude algorithm for high-frequency base motion. J. Guid. Contr. Dynam.
**1990**, 13, 738–743. [Google Scholar] [CrossRef] - Jiang, Y.F.; Lin, Y.P. Improved strapdown coning algorithms. IEEE Trans. Aerosp. Electr. Syst.
**1992**, 28, 484–490. [Google Scholar] [CrossRef]

**Figure 3.**Transceivers for indoor laser communication networking. (

**a**) The master optical transceiver; (

**b**) The slave optical transceivers.

**Figure 4.**An illustration of the different types of multi-address transceivers. (

**a**) Transceiver with 2:1 transmissions; (

**b**) Transceiver without 2:1 transmissions.

**Figure 7.**The LOS attitude curves under the condition of conical motion in an ideal state in the reflector coordinate system O

_{41}X

_{41}Y

_{41}Z

_{41}. (

**a**) a = 1°, ω = 60°/s; (

**b**) a = 3°, ω = 60°/s; (

**c**) a = 1°, ω = 180°/s.

**Figure 8.**The error curve of the calculation of the indirect numerical calculation method in the reflector coordinate system O

_{41}X

_{41}Y

_{41}Z

_{41}. (

**a**) a = 1°, ω = 60°/s; (

**b**) a = 3°, ω = 60°/s; (

**c**) a = 1°, ω = 180°/s.

**Figure 9.**The LOS attitude curves under the condition of conical motion in an ideal state in the reflector coordinate system O

_{42}X

_{42}Y

_{42}Z

_{42}. (

**a**) a = 1°, ω = 60°/s; (

**b**) a = 3°, ω= 60°/s; (

**c**) a = 1°, ω = 180°/s.

**Figure 10.**The error curve of the calculation of the indirect numerical calculation method in the reflector coordinate system O

_{42}X

_{42}Y

_{42}Z

_{42}. (

**a**) a = 1°, ω = 60°/s; (

**b**) a = 3°, ω = 60°/s; (

**c**) a = 1°, ω = 180°/s.

**Figure 11.**The LOS attitude curves under the condition of conical motion in an ideal state in the reflector coordinate system O

_{43}X

_{43}Y

_{43}Z

_{43}. (

**a**) a = 1°, ω = 60°/s; (

**b**) a = 3°, ω = 60°/s; (

**c**) a = 1°, ω = 180°/s.

**Figure 12.**The error curve of the calculation of the indirect numerical calculation method in the reflector coordinate system O

_{43}X

_{43}Y

_{43}Z

_{43}. (

**a**) a = 1°, ω = 60°/s; (

**b**) a = 3°, ω = 60°/s; (

**c**) a = 1°, ω = 180°/s.

**Figure 13.**The LOS attitude curves under the condition of conical motion in an ideal state in the reflector coordinate system O

_{44}X

_{44}Y

_{44}Z

_{44}. (

**a**) a = 1°, ω = 60°/s; (

**b**) a = 3°, ω = 60°/s; (

**c**) a = 1°, ω = 180°/s.

**Figure 14.**The error curve of the calculation of the indirect numerical calculation method in the reflector coordinate system O

_{44}X

_{44}Y

_{44}Z

_{44}. (

**a**) a = 1°, ω = 60°/s; (

**b**) a = 3°, ω = 60°/s; (

**c**) a = 1°, ω = 180°/s.

**Figure 15.**The LOS attitude curves under the condition of conical motion in an ideal state in the reflector coordinate system O

_{45}X

_{45}Y

_{45}Z

_{45}. (

**a**) a = 1°, ω = 60°/s; (

**b**) a = 3°, ω = 60°/s; (

**c**) a = 1°, ω = 180°/s.

**Figure 16.**The error curve of the calculation of the indirect numerical calculation method in the reflector coordinate system O

_{45}X

_{45}Y

_{45}Z

_{45}. (

**a**) a = 1°, ω = 60°/s; (

**b**) a = 3°, ω = 60°/s; (

**c**) a = 1°, ω = 180°/s.

**Figure 17.**The LOS attitude curves under the condition of conical motion in an ideal state in the reflector coordinate system O

_{46}X

_{46}Y

_{46}Z

_{46}. (

**a**) a = 1°, ω = 60°/s; (

**b**) a = 3°, ω = 60°/s; (

**c**) a = 1°, ω = 180°/s.

**Figure 18.**The error curve of the calculation of the indirect numerical calculation method in the reflector coordinate system O

_{46}X

_{46}Y

_{46}Z

_{46}. (

**a**) a = 1°, ω = 60°/s; (

**b**) a = 3°, ω = 60°/s; (

**c**) a = 1°, ω = 180°/s.

**Figure 19.**The error curve of the calculation of the direct numerical calculation method. (

**a**) a = 1°, ω = 60°/s; (

**b**) a = 3°, ω = 60°/s; (

**c**) a = 1°, ω = 180°/s.

**Table 1.**The numerical calculation error (µrad) values (1 σ) obtained using the indirect calculation method.

Coordinate Systems of the Reflectors | Attitudes | (a) a = 1°, ω = 60°/s | (b) a = 1°, ω = 180°/s | (c) a = 3°, ω = 180°/s |
---|---|---|---|---|

O_{41}X_{41}Y_{41}Z_{41} | Azimuth | 4.6 × 10^{−5} | 1.6 × 10^{−4} | 2.0 × 10^{−2} |

Pitch | 1.8 × 10^{−4} | 5.9 × 10^{−4} | 6.3 × 10^{−2} | |

O_{42}X_{42}Y_{42}Z_{42} | Azimuth | 4.6 × 10^{−5} | 1.7 × 10^{−4} | 2.7 × 10^{−2} |

Pitch | 1.5 × 10^{−4} | 6.9 × 10^{−4} | 1.8 × 10^{−1} | |

O_{43}X_{43}Y_{43}Z_{43} | Azimuth | 4.6 × 10^{−5} | 2.0 × 10^{−4} | 4.4 × 10^{−2} |

Pitch | 1.3 × 10^{−4} | 1.2 × 10^{−3} | 2.9 × 10^{−1} | |

O_{44}X_{44}Y_{44}Z_{44} | Azimuth | 4.6 × 10^{−5} | 2.5 × 10^{−4} | 5.2 × 10^{−2} |

Pitch | 1.5 × 10^{−4} | 1.3 × 10^{−3} | 3.2 × 10^{−1} | |

O_{45}X_{45}Y_{45}Z_{45} | Azimuth | 4.6 × 10^{−5} | 2.3 × 10^{−4} | 4.5 × 10^{−2} |

Pitch | 2.1 × 10^{−4} | 1.4 × 10^{−3} | 3.1 × 10^{−1} | |

O_{46}X_{46}Y_{46}Z_{46} | Azimuth | 4.6 × 10^{−5} | 1.8 × 10^{−4} | 2.9 × 10^{−2} |

Pitch | 2.3 × 10^{−4} | 1.1 × 10^{−3} | 2.1 × 10^{−1} |

**Table 2.**The numerical calculation error (µrad) value (1 σ) obtained using the direct calculation method.

Conical Motions. | Attitudes | Calculation Error (µrad) |
---|---|---|

(a) a = 1°, ω = 60°/s | Azimuth | 2.7 |

Pitch | 5.3 | |

(b) a = 3°, ω = 60°/s | Azimuth | 77.3 |

Pitch | 145.4 | |

(c) a = 3°, ω = 180°/s | Azimuth | 89.4 |

Pitch | 145.6 |

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**MDPI and ACS Style**

Wang, L.; Zhang, L.; Meng, L.; Bai, Y. Numerical Calculation for the Line-of-Sight Attitudes of Multi-Address Transceivers without 2:1 Transmissions for Space Laser Communication Networking. *Electronics* **2023**, *12*, 1575.
https://doi.org/10.3390/electronics12071575

**AMA Style**

Wang L, Zhang L, Meng L, Bai Y. Numerical Calculation for the Line-of-Sight Attitudes of Multi-Address Transceivers without 2:1 Transmissions for Space Laser Communication Networking. *Electronics*. 2023; 12(7):1575.
https://doi.org/10.3390/electronics12071575

**Chicago/Turabian Style**

Wang, Lihui, Lizhong Zhang, Lixin Meng, and Yangyang Bai. 2023. "Numerical Calculation for the Line-of-Sight Attitudes of Multi-Address Transceivers without 2:1 Transmissions for Space Laser Communication Networking" *Electronics* 12, no. 7: 1575.
https://doi.org/10.3390/electronics12071575