# Numerical Analysis of GNSS Signal Outage Effect on EOPs Solutions Using Tightly Coupled GNSS/IMU Integration: A Simulated Case Study in Sweden

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Loosely and Tightly Coupled Integration

#### 2.2. Single and Network-Based PPK Solution

#### 2.3. GNSS/IMU Processing Using KF and Smoothing Method

#### 2.4. Data and Analysis

## 3. Results and Discussion

#### 3.1. Comparison of Single and Network-Based PPK Solutions

#### 3.2. Impact of GNSS Signal Outage on EOPs

#### 3.2.1. Smooth and Forward KF Processing Comparison Using 2D Position and Height Uncertainties

#### 3.2.2. Smooth and Forward KF Processing Comparison Using Orientation Uncertainties

#### 3.3. Kalman Filter

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Tightly and Loosely coupled scheme. Note: Primary GNSS receiver is a GNSS receiver mounted on the aircraft (rover antenna). Base GNSS receivers are the receivers mounted on the ground to establish a network of GNSS stations. In a tightly coupled scheme, each of the primary and base GNSS data adds independently to the Kalman filter.

**Figure 2.**Position uncertainties in tightly- and loosely-coupled network-based PPK with smoothing algorithm for 180 s GNSS signal outage.

**Figure 3.**TC GNSS/INS integration data processing strategy, modified after [33]. Note: The subscripts $r$ and $S$ represent the GNSS receiver and satellite number, respectively. ${\lambda}_{i}{\phi}_{i,r}^{s}$ and ${P}_{i,r}^{s}$ are observations of the raw carrier phase and pseudo-range, respectively. ${f}_{ib}^{b}$ is the accelerometer observation of specific force and ${\omega}_{ib}^{b}$ is the gyroscope observation of angular rate (i denotes inertial-frame and b is IMU body-frame [34]). The position, velocity, and orientation that are estimated by INS are shown with ${r}_{ins}$, ${v}_{ins}$, and ${\varphi}_{ins}$, respectively. $\delta r$, $\delta v$, and $\delta \varphi $ are their corresponding correction. ${\lambda}_{i}\nabla \Delta \phi $, $\nabla \Delta {P}_{i}$, and $\nabla \Delta {\dot{\rho}}_{i}$ are the double difference observation of carrier phase, pseudorange, and pseudorange rate, respectively.

**Figure 4.**A schematic illustration of forward KF and smoothing error curve behavior of EOPs during GNSS signal outage.

**Figure 6.**The trajectory of the rover (aircraft) and position of the SWEPOS stations in the study area.

**Figure 7.**North component uncertainties in single-based PPK with forward KF, network-based PPK with forward KF, single-based PPK with smooth processing, and network-based PPK smooth processing modes.

**Figure 8.**The number of satellites, PDOP, bank angle, and baseline length plots during the flight mission. Note: The baseline length is the distance between the rover (aircraft) and the assigned GNSS reference station on the ground (TOUP.0).

**Figure 9.**Mean of EOPs uncertainties in different processing modes, i.e., single-based PPK with forward KF (SF), network-based PPK with forward KF (NF), single-based PPK with smooth processing (SS), and network-based with smooth processing (NS).

**Figure 10.**2D position (north and east) uncertainties using network-based PPK with smooth processing and assuming different GNSS (GPS + GLONASS) signal outages. Note: for better illustration: The inset figure shows the focused zoom of the main plot during the GNSS signal outage. We set a −0.01 m shift for 180 s GLONASS outage and without GNSS outage plots for both inset and whole mission plots to make a better illustration.

**Figure 11.**Height uncertainties using network-based PPK with smooth processing and assuming different GNSS (GPS + GLONASS) signal outages. Note: for better illustration: The inset figure shows the focused zoom of the main plot during the GNSS signal outage. We set a −0.01 m shift for 180 s GLONASS outage and without GNSS outage plots for both inset and whole mission plots to make a better illustration.

**Figure 12.**Uncertainties of the roll angle using network-based PPK with smooth processing and assuming different GNSS (GPS + GLONASS) signal outages. Note: for better illustration: The inset figure shows the focused zoom of the main plot during the GNSS signal outage. We set −0.001 arc minute shift for 180 s GLONASS outage and without GNSS outage plots for both inset and whole mission plots to make a better illustration.

**Figure 13.**Uncertainties of the pitch using network-based PPK with smooth processing and assuming different GNSS (GPS + GLONASS) signal outages. Note: for better illustration: The inset figure shows the focused zoom of the main plot during the GNSS signal outage. We set −0.001 arc minute shift for 180 s GLONASS outage and without GNSS outage plots for both inset and whole mission plots.

**Figure 14.**Uncertainties of heading using network-based PPK with smooth processing and assuming different GNSS (GPS + GLONASS) signal outages. Note: for better illustration: The inset figure shows the focused zoom of the main plot during the GNSS signal outage. We set −0.01 arc minute shift for 180 s GLONASS outage and without GNSS outage plots for both inset and whole mission plots to make a better illustration.

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**MDPI and ACS Style**

Jouybari, A.; Bagherbandi, M.; Nilfouroushan, F.
Numerical Analysis of GNSS Signal Outage Effect on EOPs Solutions Using Tightly Coupled GNSS/IMU Integration: A Simulated Case Study in Sweden. *Sensors* **2023**, *23*, 6361.
https://doi.org/10.3390/s23146361

**AMA Style**

Jouybari A, Bagherbandi M, Nilfouroushan F.
Numerical Analysis of GNSS Signal Outage Effect on EOPs Solutions Using Tightly Coupled GNSS/IMU Integration: A Simulated Case Study in Sweden. *Sensors*. 2023; 23(14):6361.
https://doi.org/10.3390/s23146361

**Chicago/Turabian Style**

Jouybari, Arash, Mohammad Bagherbandi, and Faramarz Nilfouroushan.
2023. "Numerical Analysis of GNSS Signal Outage Effect on EOPs Solutions Using Tightly Coupled GNSS/IMU Integration: A Simulated Case Study in Sweden" *Sensors* 23, no. 14: 6361.
https://doi.org/10.3390/s23146361