# A Sensitivity Study of POD Using Dual-Frequency GPS for CubeSats Data Limitation and Resources

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## Abstract

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## 1. Introduction

## 2. Processing Strategy

- Compute kinematic orbits using single point positioning (SPP) employing the IF combination of the code observations. These kinematic orbits, denoted as vector ${\hat{r}}_{K}$, are discrete and have an accuracy of meters.
- Computation of the code-based reduced-dynamic orbits. The reduced-dynamic orbits are computed with accelerations based on a series of gravitational and nongravitational terms, such as the Earth gravitational terms, the Earth tidal terms, the gravitational attraction from the sun, moon, and other planets, as well as the general relativistic term. Note that mis-modeled effects like the solar radiation pressure and the air drag will be largely absorbed by the estimated dynamic parameters and the stochastic velocity changes or accelerations set up later in the processing [35]. Details of the processing and the dynamic models are given in Table 1. Making use of the kinematic code orbits from the first step, the six Keplerian elements at the initial condition (the semi-major axis of the orbit, the orbital eccentricity, the inclination of the orbital plane, the right ascension of the ascending node, the argument of perigee, and the argument of latitude at the initial condition), and a remaining part of the dynamic models are estimated with a batch least-squares adjustment, which includes at this step nine flight-oriented dynamic parameters. These estimable dynamic parameters contain three constant terms (${a}_{R0}$, ${a}_{S0}$, and ${a}_{W0}$) and six periodic terms (${a}_{RC}$, ${a}_{SC}$, ${a}_{WC}$, ${a}_{RS}$, ${a}_{SS}$, and ${a}_{WS}$) in the radial (R), along-track (S) and cross-track (W) directions. The total acceleration $a$ can then be distributed into the term ${a}_{0}$, which is assumed known by applying the models given in Table 1, and an additional dynamic term ${a}_{dyn}$ that is to be adjusted:$$a={a}_{0}+{a}_{dyn},$$$${a}_{dyn}={a}_{R}{e}_{R}+{a}_{S}{e}_{S}+{a}_{W}{e}_{W},$$$${a}_{R}={a}_{R0}+{a}_{RC}\mathrm{cos}\left(U\right)+{a}_{RS}\mathrm{sin}\left(U\right),$$$${a}_{S}={a}_{S0}+{a}_{SC}\mathrm{cos}\left(U\right)+{a}_{SS}\mathrm{sin}\left(U\right),$$$${a}_{W}={a}_{W0}+{a}_{WC}\mathrm{cos}\left(U\right)+{a}_{WS}\mathrm{sin}\left(U\right),$$$$\mathrm{E}\left({\hat{r}}_{K}-{\hat{r}}_{0}\right)=\left[{A}_{rk},\text{}{A}_{rd}\right]{\left[{x}_{k},\text{}{x}_{d}\right]}^{T},$$

- 3.
- Phase preprocessing and orbit improvements. This step preprocesses the raw phase observations to detect cycle slips and mark bad observations. The preprocessing goes through several iterations to improve the LEO orbit quality. The orbit improvement is realized through estimation of stochastic velocity changes [38] in addition to the 15 parameters mentioned in the second step, and is performed in a least-squares adjustment making use of the IF combination of the phase observations. One set of the stochastic velocity changes is considered in each predefined time interval, e.g., every 15 min. The linearized phase observation equation at the epoch ${t}_{i}$ can be expressed as:$$\mathrm{E}\left(\mathsf{\Delta}{\phi}_{IF}\right)=\left[{A}_{lk},{A}_{ld},{A}_{lv}\right]{\left[{x}_{k},{x}_{d},{x}_{v}\right]}^{T}+c\times \mathsf{\Delta}{t}_{r}+{\lambda}_{IF}{N}_{IF},$$$${\lambda}_{IF}=\frac{c}{{f}_{1}+{f}_{2}},$$$${N}_{IF}=\frac{{f}_{1}^{2}{\lambda}_{1}{N}_{1}-{f}_{2}^{2}{\lambda}_{2}{N}_{2}}{c\left({f}_{1}-{f}_{2}\right)},$$
_{IF}represents the observed-minus-computed (O-C) term of the IF phase observations. c and Δt_{r}denote the speed of light and the receiver clock error, respectively. λ_{j}, f_{j}, and N_{j}represent the wavelength, the frequency, and the ambiguity on frequency j (j = 1, 2), respectively. Note that N_{IF}is not an integer. The receiver clock error is estimated epoch-wise independently, and the ambiguity is assumed constant before the detection of a cycle slip. Note that new ambiguities are setup for estimation at the beginning of each round of duty cycling. x_{v}stands for the vector containing all stochastic velocity changes in the RSW directions from the first to the current epoch and note that x_{v}is constrained to zero with a predefined a priori standard deviation. The design matrices A_{lk}, A_{ld}, and A_{lv}contain the partial derivatives of the O-C terms with respect to the x_{k}, x_{d}, and x_{v}, respectively. To be estimated are the vector [x_{k},x_{d},x_{v}]^{T}, the receiver clock offset Δt_{r}, and the term N_{IF}. Note that very little code observations are used to avoid the problem of matrix singularity between the receiver clock offset and the ambiguity terms. Note that the ambiguities on L1 and L2 are not attempted to be fixed in this study.

- 4.
- Generation of final orbits. With the preprocessed phase observations, the six Keplerian elements, the three constant dynamic parameters (${a}_{R0}$, ${a}_{S0}$, and ${a}_{W0}$) are estimated together with stochastic accelerations in the RSW directions. The accelerations are set up in shorter time intervals compared to those in Step 3. The linearized phase observation equation is thus formulated as:$$\mathrm{E}\left(\mathsf{\Delta}{\phi}_{IF}\right)=\left[{A}_{lk},{A}_{ld0},{A}_{la}\right]{\left[{x}_{k},{x}_{d0},{x}_{a}\right]}^{T}+c\times \mathsf{\Delta}{t}_{r}+{\lambda}_{IF}{N}_{IF},$$

## 3. Orbit Determination under Different Scenarios

#### 3.1. Duty Cycling and Satellite Numbers

#### 3.2. Sampling Rate of the Observations

#### 3.3. Latency Applying Different GPS Products

#### 3.4. Antenna Attitude

#### 3.5. Length of Arc

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Distribution of the tracked satellite numbers per epoch and (

**b**) the time history of the satellite numbers having both the L1/L2 observations and at least the L1 observations. The data was collected from the CubeSat STF-1 from 15:39:37 in GPS time (GPST) on 27 February 2019 to 3:55:10 on 28 February 2019.

**Figure 2.**Flow-diagram of the processing procedure and the estimated parameters. $n$ and $m$ are the numbers of the stochastic parameter vectors considered during the estimation in Step 3 and 4, respectively.

**Figure 3.**3D orbital errors of the code-based kinematic (red) and reduced-dynamic daily arc (blue) with a sampling interval of 10 s. The IGS final products were used for the GPS satellites.

**Figure 4.**3D orbital errors for different duty-cycles (from 20% to 80%) of the phase-based reduced-dynamic orbits using data of GRACE FO-1 with a sampling interval of 10 s. The elevation mask was set to 5 degrees with a mean satellite number of about 9.

**Figure 5.**(

**a**) Distribution of the satellite numbers per epoch (observing at least P1 and P2 observations) applying different elevation mask angles and (

**b**) the corresponding 3D orbital errors of the phase-based reduced-dynamic orbits with a duty-cycle of 100% and mean satellite numbers of 7 and 6.

**Figure 6.**3D RMSE of the reduced-dynamic orbits under different duty-cycles, observation sampling intervals, and mean satellite numbers using the IGS final products. The cases having 9, 7, and 6 satellites (on average) are given between each pair of the dashed lines.

**Figure 7.**Differences in the between-satellite clocks for G02 and G32 between the IGS final and rapid clocks.

**Figure 8.**3D orbital errors of the phase-based reduced-dynamic orbits using data of GRACE FO-1 with a 100% duty-cycle, a mean satellite number of 9, and a sampling interval of 10 s (having the IGS final as the reference). Different IGS products were applied.

**Figure 9.**3D RMSE of the reduced-dynamic orbits under different duty-cycles, observation sampling intervals and mean satellite numbers using (

**a**) the IGR and (

**b**) the IGC products. The cases having 9, 7, and 6 satellites (on average) are given between each pair of the dashed lines.

**Figure 10.**The antenna reference frame (ARF, shown in red), the satellite reference frame (SRF, shown in yellow), and the radial, along-track, and cross-track (RSW, shown in green) system. ${t}_{1}$ and ${t}_{2}$ represent two conservative time epochs for the same satellite. The figure is scaled for a better presentation.

**Figure 11.**Angle differences between the east, north, and up directions in the antenna reference frame (ARF) and the opposite along-track direction, the cross-track direction, and the radial direction. The data for GRACE FO-1 was used for the plot. Note that the blue line is almost overwritten by the green dashed line.

**Figure 12.**3D orbital errors not applying the antenna attitude information. The IGS final products were used for observations with a duty-cycle of 100%, a sampling interval of 10 s, and a mean satellite number of 9.

**Figure 13.**3D orbital errors computed with different arc lengths using (

**a**) the IGS final products and (

**b**) the IGS rapid products. A duty-cycle of 100%, an observation interval of 10 s, and a mean satellite number of 9 were assumed for the processing. The arc length experiment here refers to the processing using e.g., $k$ hours of data in each processing round, and in each hour of these $k$ hours, data are tracked in $X$% of the time.

**Figure 14.**3D RMSE of the reduced-dynamic orbits when processing with the arc lengths of 24 h, 12 h, and 6 h applying (

**a**) the IGS final products, (

**b**) the IGS rapid (IGR) products, and (

**c**) the IGS real-time (IGC) products. Between each pair of the dashed lines, the cases are sorted from the highest duty-cycle of 100% to the lowest duty-cycle of 20%, first for the mean satellite number of 9, and then similarly for the mean satellite numbers of 7 and 6, respectively.

Measurement Model | GPS code P1 + P2, phase L1 + L2 |

IF linear combination | |

Sampling interval: 10 s, 20 s, 30 s, 60 s, 120 s | |

Elevation mask: 5°, 15°, 25° (for different mean satellite numbers) | |

Arc length: 6 h, 12 h, 24 h | |

GPS orbits and clocks: IGS final, rapid, real-time products | |

Dynamic Model | Earth gravity: EGM2008 [39], Earth potential degree: 120 |

N-body gravity: JPL DE405 [40] (Planetary ephemeris) | |

Solid Earth tides: IERS Conventions 2010 | |

Pole tides: IERS Conventions 2010 [41] | |

Ocean tides: FES2004 [42] | |

General relativistic term | |

Reference Frame | IGS14, J2000.0 (Julian epoch) |

Coordinate Transformation | Nutation and precession: IAU2000R06 [34] |

Sub-daily pole variations: IERS Conventions 2010 [41] | |

Earth rotation parameters: IGS final, rapid, ultra-rapid products |

**Table 2.**Mean number of satellites used in SPP and percentiles of valid SPP solutions applying different elevation masks. Note that the mean satellite numbers are given as rounded values. The real values of the mean satellite numbers amount to about 8.5, 7.4, and 5.7 for an elevation mask of 5, 15, and 25 degrees, respectively.

Elevation Mask [Degree] | Mean Integer Number of Satellites | Percentile of Valid SPP Solutions |
---|---|---|

5 | 9 | 99.9% |

15 | 7 | 99.1% |

25 | 6 | 82.9% |

**Table 3.**3D RMSE (in cm) of the phase-based reduced-dynamic orbits under different duty-cycles and mean satellite numbers.

Duty-Cycle/Mean # Satellite | 9 Satellites | 7 Satellites | 6 Satellites |
---|---|---|---|

100% | -- | 1.1 | 2.1 |

80% | 1.6 | 1.9 | 2.4 |

60% | 2.0 | 2.1 | 2.7 |

40% | 2.6 | 2.9 | 3.5 |

20% | 3.5 | 3.9 | 4.8 |

Orbits/Clocks | Identifier | Latency | Accuracy [cm] Orbit/Clock | Satellite Clock Sampling Interval [s] | 3D RMSE [cm] |
---|---|---|---|---|---|

IGS Final | IGS | 12–18 days | 2.5/2.25 | 30 | -- |

IGS rapid | IGR | 17–41 h | 2.5/2.25 | 300 | 4.8 |

IGS RTS | IGC | (Near)-real-time | 2–5/3–5 | 30 | 3.2 |

**Table 5.**3D RMSE of the phase-based reduced-dynamic orbits with different arc lengths and using different IGS products. 10 s observation data with a duty-cycle of 100% and a mean satellite number of 9 were used for the processing.

Arc Length/Products | IGS Final [cm] | IGR [cm] | IGC [cm] |
---|---|---|---|

24 h | -- | 4.8 | 3.2 |

12 h | 0.6 | 5.1 | 3.1 |

6 h | 1.0 | 5.6 | 3.5 |

**Table 6.**Increase in the 3D RMSE (in cm) of phase-based reduced-dynamic orbits when shortening the arc length from 24 h to 12 h. The IGS final products were used.

Duty-Cycle/Mean # Satellite Sampling Interval [s] | 8 Satellites 10/20/30/60/120 | 7 Satellites 10/20/30/60/120 | 5 Satellites 10/20/30/60/120 |
---|---|---|---|

100% | 0.6/0.2/0.1/0.2/−0.1 | 0.1/0.1/0.2/0.2/0.2 | 0.1/0.0/0.1/0.1/0.2 |

80% | 0.2/0.1/0.5/0.4/−0.4 | 0.1/0.1/0.2/0.2/0.5 | 0.2/0.2/0.2/0.3/0.4 |

60% | 0.1/0.0/0.3/1.0/−0.4 | 0.1/0.1/0.2/0.2/0.5 | 0.0/0.1/0.1/0.1/1.0 |

40% | 0.2/0.3/0.3/0.6/0.8 | 0.2/0.0/0.2/0.7/1.7 | 0.2/0.3/0.5/2.0/3.5 |

20% | 2.0/2.2/2.8/3.3/-- | 1.6/2.2/3.0/3.6/-- | 2.9/2.0/3.4/4.8/-- |

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**MDPI and ACS Style**

Wang, K.; Allahvirdi-Zadeh, A.; El-Mowafy, A.; Gross, J.N. A Sensitivity Study of POD Using Dual-Frequency GPS for CubeSats Data Limitation and Resources. *Remote Sens.* **2020**, *12*, 2107.
https://doi.org/10.3390/rs12132107

**AMA Style**

Wang K, Allahvirdi-Zadeh A, El-Mowafy A, Gross JN. A Sensitivity Study of POD Using Dual-Frequency GPS for CubeSats Data Limitation and Resources. *Remote Sensing*. 2020; 12(13):2107.
https://doi.org/10.3390/rs12132107

**Chicago/Turabian Style**

Wang, Kan, Amir Allahvirdi-Zadeh, Ahmed El-Mowafy, and Jason N. Gross. 2020. "A Sensitivity Study of POD Using Dual-Frequency GPS for CubeSats Data Limitation and Resources" *Remote Sensing* 12, no. 13: 2107.
https://doi.org/10.3390/rs12132107