# Application of Pulsar-Based Navigation for Deep-Space CubeSats

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Signal Time-of-Arrival

#### 2.2. Pulsar Phase

#### 2.3. Signal Model

- (1)
- The probability of detecting one photon in a time interval $\Delta t$ is given by:$P\left({N}_{t+\Delta t}-{N}_{t}=1\right)=\lambda \left(t\right)\Delta t$ with $\Delta t$ approaching zero.
- (2)
- The probability of detecting more than one photon in $\Delta t$ is given by:$P\left({N}_{t+\Delta t}-{N}_{t}>1\right)=0$ with $\Delta t$ approaching zero.
- (3)
- Non-overlapping increments are independent, with ${N}_{t}$ as the increment of the stochastic process:${N}_{l,q}={N}_{q}-{N}_{l}$ with $q\ge l$.

#### 2.4. Epoch Folding

- The photons’ time tags during the set observation window are collected.
- They are folded back into a single time interval equal to one pulse period.
- The period duration is divided into some equal-length bins.
- The number of photons in each bin is counted.
- The computed photon counts are normalized, and the empirical pulsar profile is derived.

#### 2.5. Phase Delay Estimation

## 3. Measurement Modeling

#### 3.1. Photon Path

#### 3.2. Timing Model

#### 3.3. Time Conversion

#### 3.4. Atomic Clocks

#### 3.5. Signal-to-Noise Ratio

## 4. Simulation

#### 4.1. New Horizons Mission Case

`nh_recon_pluto_od122_v01`containing the data required for the simulation was then retrieved, and the initial state is the one at midday of 7 December 2014. This has been also reported in Table 1.

#### 4.2. LUMIO Mission Case

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Thornton, C.L.; Border, J.S. Radiometric Tracking Techniques for Deep-Space Navigation; John Wiley & Sons: Hoboken, NJ, USA, 2003. [Google Scholar]
- Franzese, V.; Topputo, F. Deep-Space Optical Navigation Exploiting Multiple Beacons. J. Astronaut. Sci.
**2022**, 69, 368–384. [Google Scholar] [CrossRef] - Franzese, V.; Topputo, F. Optimal beacons selection for deep-space optical navigation. J. Astronaut. Sci.
**2020**, 67, 1775–1792. [Google Scholar] [CrossRef] - Andreis, E.; Franzese, V.; Topputo, F. Onboard orbit determination for deep-space CubeSats. J. Guid. Control Dyn.
**2022**, 45, 1466–1480. [Google Scholar] [CrossRef] - Franzese, V.; Topputo, F. Celestial Bodies Far-Range Detection with Deep-Space CubeSats. Sensors
**2023**, 23, 4544. [Google Scholar] [CrossRef] [PubMed] - Franzese, V.; Di Lizia, P.; Topputo, F. Autonomous Optical Navigation for the Lunar Meteoroid Impacts Observer. J. Guid. Control Dyn.
**2019**, 42, 1579–1586. [Google Scholar] [CrossRef] - Pugliatti, M.; Franzese, V.; Topputo, F. Data-Driven Image Processing for Onboard Optical Navigation Around a Binary Asteroid. J. Spacecr. Rocket.
**2022**, 59, 943–959. [Google Scholar] [CrossRef] - Pugliatti, M.; Piccolo, F.; Rizza, A.; Franzese, V.; Topputo, F. The vision-based guidance, navigation, and control system of Hera’s Milani Cubesat. Acta Astronaut.
**2023**, 210, 14–28. [Google Scholar] [CrossRef] - Zoccarato, P.; Larese, S.; Naletto, G.; Zampieri, L.; Brotto, F. Deep Space Navigation by Optical Pulsars. J. Guid. Control Dyn.
**2023**, 210, 1–12. [Google Scholar] [CrossRef] - Chen, P.T.; Zhou, B.; Speyer, J.L.; Bayard, D.S.; Majid, W.A.; Wood, L.J. Aspects of pulsar navigation for deep space mission applications. J. Astronaut. Sci.
**2020**, 67, 704–739. [Google Scholar] [CrossRef] - Lohan, K.; Putnam, Z. Characterization of Candidate Solutions for X-Ray Pulsar Navigation. IEEE Trans. Aerosp. Electron. Syst.
**2022**. [Google Scholar] [CrossRef] - Fang, H.; Su, J.; Li, L.; Zhang, L.; Sun, H.; Gao, J. An analysis of X-ray pulsar navigation accuracy in Earth orbit applications. Adv. Space Res.
**2021**, 68, 3731–3748. [Google Scholar] [CrossRef] - Zheng, S.; Zhang, S.; Lu, F.; Wang, W.; Gao, Y.; Li, T.; Song, L.; Ge, M.; Han, D.; Chen, Y.; et al. In-orbit demonstration of X-ray pulsar navigation with the Insight-HXMT satellite. Astrophys. J. Suppl. Ser.
**2019**, 244, 1. [Google Scholar] [CrossRef] [Green Version] - Huang, L.; Shuai, P.; Zhang, X.; Chen, S. Pulsar-based navigation results: Data processing of the X-ray pulsar navigation-I telescope. J. Astron. Telesc. Instruments Syst.
**2019**, 5, 018003. [Google Scholar] [CrossRef] - Ely, T.; Bhaskaran, S.; Bradley, N.; Lazio, T.J.W.; Martin-Mur, T. Comparison of Deep Space Navigation Using Optical Imaging, Pulsar Time-of-Arrival Tracking, and/or Radiometric Tracking. J. Astronaut. Sci.
**2022**, 69, 385–472. [Google Scholar] [CrossRef] - Wang, Y.; Zheng, W.; Zhang, S.; Ge, M.; Li, L.; Jiang, K.; Chen, X.; Zhang, X.; Zheng, S.; Lu, F. Review of X-ray pulsar spacecraft autonomous navigation. Chin. J. Aeronaut.
**2023**. [Google Scholar] [CrossRef] - Deng, X.; Hobbs, G.; You, X.; Li, M.; Keith, M.; Shannon, R.; Coles, W.; Manchester, R.; Zheng, J.; Yu, X.; et al. Interplanetary spacecraft navigation using pulsars. Adv. Space Res.
**2013**, 52, 1602–1621. [Google Scholar] [CrossRef] [Green Version] - Lorimer, D.R.; Kramer, M. Handbook of Pulsar Astronomy; Cambridge University Press: Cambridge, UK, 2005; Chapter 8. [Google Scholar]
- Downs, G.S. Interplanetary Navigation Using Pulsating Radio Sources; Technical Report N74-34150; NASA: Pasadena, CA, USA, 1974.
- Sheikh, S.I.; Pines, D.J.; Ray, P.S.; Wood, K.S.; Lovellette, M.N.; Wolff, M.T. Spacecraft Navigation Using X-ray Pulsars. J. Guid. Control Dyn.
**2006**, 29, 49–63. [Google Scholar] [CrossRef] [Green Version] - Shearer, A.; Golden, A. Implications of the Optical Observations of Isolated Neutron Stars. Astrophys. J.
**2001**, 547, 967. [Google Scholar] [CrossRef] - Hisamoto, C.S.; Sheikh, S.I. Spacecraft Navigation Using Celestial Gamma-Ray Sources. J. Guid. Control Dyn.
**2015**, 38, 1765–1774. [Google Scholar] [CrossRef] [Green Version] - Emadzadeh, A.A.; Speyer, J.L. Navigation in Space by X-ray Pulsars; Springer: New York, NY, USA, 2011. [Google Scholar]
- Emadzadeh, A.A.; Speyer, J.L. On Modeling and Pulse Phase Estimation of X-Ray Pulsars. IEEE Trans. Signal Process.
**2010**, 58, 4484–4495. [Google Scholar] [CrossRef] - Emadzadeh, A.A.; Speyer, J.L. X-ray Pulsar-Based Relative Navigation using Epoch Folding. IEEE Trans. Aerosp. Electron. Syst.
**2011**, 47, 2317–2328. [Google Scholar] [CrossRef] - Golshan, A.R.; Sheikh, S.I. On Pulse Phase Estimation and Tracking of Variable Celestial X-ray Sources. In Proceedings of the 63rd Annual Meeting of The Institute of Navigation (2007), Cambridge, MA, USA, 23–25 April 2007; pp. 413–422. [Google Scholar]
- Winternitz, L.M.B.; Hassouneh, M.A.; Mitchell, J.W.; Valdez, J.E.; Price, S.R.; Semper, S.R.; Yu, W.H.; Ray, P.S.; Wood, K.S.; Arzoumanian, Z.; et al. X-ray pulsar navigation algorithms and testbed for SEXTANT. In Proceedings of the 2015 IEEE Aerospace Conference, Big Sky, MT, USA, 7–14 March 2015; pp. 1–14. [Google Scholar] [CrossRef] [Green Version]
- Sheikh, S.I. The Use of Variable Celestial X-ray Sources for Spacecraft Navigation. Ph.D. Thesis, University of Maryland, College Park, MD, USA, 2005. [Google Scholar]
- Hellings, R.W. Relativistic effects in astronomical timing measurements. Astron. J.
**1986**, 91, 650–659. [Google Scholar] [CrossRef] - Zucca, C.; Tavella, P. The clock model and its relationship with the Allan and related variances. IEEE Trans. Ultrason. Ferroelectr. Freq. Control
**2005**, 52, 289–296. [Google Scholar] [CrossRef] - Simon, J.; Julier, J.K.U. New extension of the Kalman filter to nonlinear systems. In Proceedings of the Signal Processing, Sensor Fusion, and Target Recognition VI, Orlando, FL, USA, 18 July 1997; Volume 3068, p. 3068-12. [Google Scholar] [CrossRef]
- Guo, Y.; Farquhar, R.W. New Horizons mission design. Space Sci. Rev.
**2008**, 140, 49–74. [Google Scholar] [CrossRef] - Golshan, A.R.; Sheikh, S.I.; Pines, D.J. Absolute and relative position determination using Variable Celestial X-ray Sources 2007. In Proceedings of the 30th Annual AAS Guidance and Control Conference, Breckenridge, CO, USA, 3–7 February 2007. [Google Scholar]
- Topputo, F.; Merisio, G.; Franzese, V.; Giordano, C.; Massari, M.; Pilato, G.; Labate, D.; Cervone, A.; Speretta, S.; Menicucci, A.; et al. Meteoroids detection with the LUMIO lunar CubeSat. Icarus
**2023**, 389, 115213. [Google Scholar] [CrossRef] - Cervone, A.; Topputo, F.; Speretta, S.; Menicucci, A.; Turan, E.; Di Lizia, P.; Massari, M.; Franzese, V.; Giordano, C.; Merisio, G.; et al. LUMIO: A CubeSat for observing and characterizing micro-meteoroid impacts on the lunar far side. Acta Astronaut.
**2022**, 195, 309–317. [Google Scholar] [CrossRef]

**Figure 2.**Crab pulsar normalized profile plot with data from [23].

${x}_{0}$ | 1,092,587,085.357 km | ${v}_{x,0}$ | 5.560 km/s |

${y}_{0}$ | −4,206,812,531.514 km | ${v}_{y,0}$ | −12.623 km/s |

${z}_{0}$ | −1,655,791,253.618 km | ${v}_{z,0}$ | −4.892 km/s |

Parameter | Value | Unit | Parameter | Value | Unit |
---|---|---|---|---|---|

${T}_{obs}$ | 3600 | s | ${\sigma}_{pos}$ | ${10}^{2}$ | km |

${T}_{map}$ | 0.5 | d | ${\sigma}_{vel}$ | ${10}^{-4}$ | km/s |

${B}_{x}$ | 0.005 | ph/cm^{2}/s | ${\sigma}_{TDB}$ | ${10}^{-6}$ | km |

${N}_{psr}$ | 4 | ${\sigma}_{bias}$ | ${10}^{-9}$ | s | |

${A}_{det}$ | 1 | m^{2} | ${\sigma}_{bias}$ | ${10}^{-16}$ | 1/s |

$\Delta {t}_{eul}$ | 10 | s | ${a}_{d}$ | ${10}^{-14}$ | s/s |

**Table 3.**Selected pulsars’ parameters. Data are taken from [33] and updated where applicable with more recent data.

Pulsar | ${\mathit{F}}_{\mathit{x}}$ | Galactic Latitude | Galactic Longitude | Period |
---|---|---|---|---|

J1751−305 | 0.180000 ph/cm^{2}/s | −0.0330 deg | 6.27 deg | 2.30 ms |

B0531+21 | 1.540000 ph/cm^{2}/s | −0.1000 deg | 3.22 deg | 33.4 ms |

B1937+21 | 0.000050 ph/cm^{2}/s | −0.0051 deg | 1.00 deg | 1.60 ms |

B1821−24 | 0.000193 ph/cm^{2}/s | −0.0970 deg | 0.14 deg | 3.10 ms |

${\sigma}_{x}$ | 4.532 km | ${\sigma}_{{v}_{x}}$ | 0.8548 mm/s |

${\sigma}_{y}$ | 0.751 km | ${\sigma}_{{v}_{y}}$ | 0.1561 mm/s |

${\sigma}_{z}$ | 2.022 km | ${\sigma}_{{v}_{z}}$ | 0.4056 mm/s |

Start Date | 29 August 2020 | ||
---|---|---|---|

${x}_{0}$ | 211,878.048 km | ${v}_{x,0}$ | 1.187 km/s |

${y}_{0}$ | −345,283.782 km | ${v}_{y,0}$ | 0.691 km/s |

${z}_{0}$ | −136,298.530 km | ${v}_{z,0}$ | 0.221 km/s |

Parameter | Value | Unit | Parameter | Value | Unit |
---|---|---|---|---|---|

${T}_{obs}$ | 300 | s | ${\sigma}_{pos}$ | ${10}^{2}$ | km |

${T}_{map}$ | 1 | d | ${\sigma}_{vel}$ | ${10}^{-4}$ | km/s |

${\lambda}_{back}$ | 0.005 | ph/cm^{2}/s | ${N}_{bin}$ | 2048 | |

${N}_{psr}$ | 1 | ${A}_{det}$ | 0.01 | m^{2} |

${\sigma}_{x}$ | 2.595 km | ${\sigma}_{v,x}$ | 1.484 $\times {10}^{-2}$ m/s |

${\sigma}_{y}$ | 0.804 km | ${\sigma}_{v,y}$ | 6.955$\times {10}^{-3}$ m/s |

${\sigma}_{z}$ | 1.753 km | ${\sigma}_{v,z}$ | 4.196$\times {10}^{-3}$ m/s |

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**MDPI and ACS Style**

Malgarini, A.; Franzese, V.; Topputo, F.
Application of Pulsar-Based Navigation for Deep-Space CubeSats. *Aerospace* **2023**, *10*, 695.
https://doi.org/10.3390/aerospace10080695

**AMA Style**

Malgarini A, Franzese V, Topputo F.
Application of Pulsar-Based Navigation for Deep-Space CubeSats. *Aerospace*. 2023; 10(8):695.
https://doi.org/10.3390/aerospace10080695

**Chicago/Turabian Style**

Malgarini, Andrea, Vittorio Franzese, and Francesco Topputo.
2023. "Application of Pulsar-Based Navigation for Deep-Space CubeSats" *Aerospace* 10, no. 8: 695.
https://doi.org/10.3390/aerospace10080695