Influences of Discontinuous Attitudes on GNSS/LEO Integrated Precise Orbit Determination Based on Sparse or Regional Networks

Abstract

A uniformly distributed global ground network is essential for the accurate determination of GNSS orbit and clock parameters. However, achieving an ideal ground network is often difficult. When limited to a sparse or regional network of ground stations, the integration of LEO satellites can substantially enhance the accuracy of GNSS Precise Orbit Determination (POD). In practical processing, discontinuities with complicated gaps can occur in LEO attitude quaternions, particularly when working with a restricted observation network. This hampers the accuracy of determining GNSS/LEO integrated orbits. To address this, an investigation was conducted using data from seven LEO satellites, including those from Sentinel-3, GRACE-FO, and Swarm, to evaluate integrated POD performance under sparse or regional station conditions. Particular focus was placed on addressing attitude discontinuities. Four scenarios were analyzed, encompassing both continuous data availability and one-, two-, and three-hour interruptions after one hour of continuous data availability. The results showed that the proposed quaternion rotation matrix interpolation method is reliable for the integrated POD of GNSSs and LEOs with strict attitude control.

1. Introduction

A uniformly distributed global network is essential for accurate orbit and clock determination in GNSS POD processing. Currently, with a network of approximately 100 global stations, the orbit accuracy of GPS, GLONASS, Galileo, and BDS MEO satellites can achieve centimeter-level precision [1,2,3]. However, it is often impractical to maintain an ideal global ground network. When the number of ground observation stations is limited, resulting in a sparse or regional network, integrating LEO satellites can significantly enhance the accuracy of GNSS orbit determination [4].

The development of LEO satellites for navigation has advanced significantly, starting with the first-generation navigation system, the TRANSIT project, which was launched by the United States in 1957 [5]. This system, consisting of six LEO satellites in polar circular orbits, was eventually discontinued due to its limited coverage. With the recent rise of numerous LEO satellite communication constellation initiatives, LEO-based navigation enhancement systems and independent navigation systems have become prominent research topics [6,7,8].

The enhancing effect of LEO satellites on navigation systems is primarily attributed to their proximity to Earth and high-speed movement. When integrated into the global observation network as dynamic “rovers”, LEO satellites significantly improve the network’s geometric configuration, particularly benefiting geostationary (GEO) satellites. Numerous studies using real observation data have demonstrated the critical role of LEO satellites in enhancing GNSS Precise Orbit Determination (POD). Examples include missions such as GRACE [9,10,11], GOCE [12], Swarm [4], Sentinel [13], FY-3C/3D [14,15], and HY-2A [16].

In actual processing, it has been observed that due to the short transit time, the time window for LEO data downlinking is often rather limited. Consequently, when processing data from a sparse observation network and LEO satellites, essential information, such as the attitude quaternions of LEO satellites, may be incomplete or missing. The accuracy of dynamic LEO POD heavily depends on the precision of the dynamic models used, and deficiencies in these models can introduce systematic errors in POD. These errors may accumulate with increasing arc length. For LEO satellite POD, these deficiencies are primarily related to non-gravitational forces, including atmospheric drag and solar radiation pressure (SRP). Accurate modeling of atmospheric drag requires precise information on the upper atmospheric density, a detailed understanding of the interaction between atmospheric particles and the satellite surface, and the satellite’s attitude in space. Therefore, missing attitude information can lead to mis-modeling of atmospheric drag and SRP. Although stochastic accelerations or velocity pulses can partially absorb certain mis-modeled forces in reduced-dynamic POD, some studies have shown that employing more accurate models is preferable to using simplified models with loosely constrained empirical accelerations [17,18]. Regarding the observational model, the antenna sensor offsets and Phase Center Offsets (PCOs) of LEO satellites are often provided in the satellite body-fixed frame. To accurately transform these hardware-related offsets from the body-fixed frame to the Earth-Centered Inertial (ECI) system, precise attitude information of the corresponding LEO satellites is essential.

Some studies have been performed on the attitude influence on LEO satellite POD. For example, Jason-1/2/3 satellite orbits using the nominal attitude and quaternions at an overall time interval of 25 years were generated, and the results showed that the use of attitude observations for POD enhances the orbit accuracy compared with orbits based on Satellite Laser Ranging (SLR) observations [19]. An attitude model based on the actual yaw angle was developed for the Haiyang 2C satellite and an orbit uncertainty of less than 2 cm was obtained through SLR validation [20]. Nominal attitude, a combined attitude model, and real quaternions were tested as three scenarios for the Jason-3 satellite, and the results showed that the orbital accuracies with quaternions are the best and most stable among the three modes [21]. The results also showed that the combination of measured attitude data and a modeled Phase Center Variation (PCV) map can provide a better orbit solution for Jason-3 [22]. The effect of attitude data on kinematic and reduced-dynamic orbits indicates that nominal attitude data can reliably replace measured attitude data in GRACE-FO orbit determination through piecewise constant empirical accelerations that have a time resolution of 6 min and are set up in the radial, along-track, and cross-track directions [23]. Research also showed that real quaternions are the best and most stable option in different attitude settings [19].

Based on the above research, it is necessary to investigate the impact of discontinuous attitude quaternions on integrated POD and develop solutions under the condition of a non-ideal ground observation network. In this paper, seven LEO satellites, including those from Sentinel-3, GRACE-FO, and Swarm, were selected to evaluate the performance of integrated POD under sparse and regional station conditions, with a particular focus on attitude discontinuities. This study is organized as follows: In Section 2, the integrated processing approach is briefly introduced, and the attitude algorithm based on discontinuous quaternions is explained. Section 3 presents the selected LEO satellites and the ground network, followed by the data processing strategy and experimental scenario. In Section 4, the results of the integrated POD under different scenarios of attitude quaternions, based on sparse and regional station networks, are compared. Finally, the discussion and conclusions are provided in Section 5 and Section 6, respectively, based on the aforementioned results and analysis.

2. Algorithm

2.1. Integrated POD Algorithm

In actual processing, the observation equations of ground stations and LEO satellites can be expressed as follows:

$$P_{g,f}^s={\rho }_g^s+c\delta t_g-c\delta t^s+I_{g,f}^s+T_g^s{+\delta }_{g,rel}^s\left(t\right)+B_{g,f}-{B_f}^s+{\epsilon }_{P_{g,f}^s}$$

(1)

$$P_{l,f}^s={\rho }_l^s+c\delta t_l-c\delta t^s+I_{l,f}^s{+\delta }_{l,rel}^s\left(t\right)+B_{l,f}-{B_f}^s+{\epsilon }_{P_{l,f}^s}$$

(2)

$$L_{g,f}^s={\rho }_g^s+c\delta t_g-c\delta t^s-I_{g,f}^s+T_g^s{+\delta }_{g,rel}^s\left(t\right)+{\lambda }_f(B_{g,f}-{B_f}^s)+{\lambda }_f{N}_{g,f}^s+{\epsilon }_{L_{g,f}^s}$$

(3)

$$L_{l,f}^s={\rho }_l^s+c\delta t_l-c\delta t^s-I_{l,f}^s{+\delta }_{l,rel}^s\left(t\right)+{\lambda }_f{(B}_{l,f}-{B_f}^s)+{\lambda }_f{N}_{l,f}^s+{\epsilon }_{L_{l,f}^s}$$

(4)

where $g$, $s$, and $l$ represent the ground station, GNSS satellite, and LEO satellite, respectively. $f$ represents frequency. $c$ represents the speed of light. $P_{g,f}^s$, $P_{l,f}^s$, $L_{g,f}^s$, and $L_{l,f}^s$ are the pseudo-range and carrier phase observations received by the ground station and LEO satellite. ${\rho }_g^s$ denotes the geometric distance of ground station and GNSS satellite, which is in meters between the satellite antenna at the signal transmission time and the receiver antenna at the signal reception time. ${\rho }_l^s$ denotes the geometric distance of LEO satellites and GNSS satellite, which is in meters between the satellite antenna at the signal transmission time and the receiver antenna at the signal reception time. $\delta t_g$, $\delta t^s$, and $\delta t_l$ represent the clock (in seconds) of the ground station receiver, GNSS satellite, and LEO satellite, respectively. $T_g^s$ represents the tropospheric delay (in meters) at the ground station. $I_{g,f}^s$ and $I_{l,f}^s$ represent the ionospheric delays (in meters) at frequency $f$ for the ground station and LEO satellite, respectively. ${\delta }_{g,rel}^s\left(t\right)$ and ${\delta }_{l,rel}^s\left(t\right)$ represent the relativistic effects (in meters) on signal propagation for the ground station and LEO satellite, respectively. ${\lambda }_f$ denotes the wavelength (in meters) corresponding to frequency. $N_{g,f}^s$ and $N_{l,f}^s$ represent the integer phase ambiguities (in cycles). $B_{g,f}$, $B_{l,f}$, and ${B_f}^s$ represent the hardware delay of the ground station, LEO satellite, GNSS satellite, respectively. ${\epsilon }_{P_{g,f}^s}$, ${\epsilon }_{P_{l,f}^s}$, ${\epsilon }_{L_{g,f}^s}$, and ${\epsilon }_{L_{l,f}^s}$ represent the observation noise (in meters) for the code and phase measurements, respectively.

Then, based on the principle of statistical orbit determination, the ground- and LEO-based observation equations at epoch $t_i$ can be expressed as:

$$Y_{g,i}^s=F_g^s(x_{g,i},x^{s,i},P_{g,i},{P_i}^s,t_i)+{\epsilon }_g^s$$

(5)

$$Y_{leo,i}^s=F_{leo}^s(x_{leo,i},x^{s,i},{P_i}^s,P_{leo,i},t_i)+{pco}_{leo}+{\epsilon }_{leo}^s$$

(6)

where $Y_{g,i}^s$ and $Y_{leo,i}^s$ are GNSS measurements tracked by ground stations and LEO satellites, respectively. $x_{g,i}$, $x^{s,i}$, and $x_{leo,i}$ represent the position of ground stations, GPS satellites, and LEO satellites, respectively, at $t_i$. ${pco}_{leo}$ is the PCO of LEOs. Since the attitude of LEO satellites is related to the PCO, it is listed separately here. $P_{g,i}$, ${P_i}^s$, and $P_{leo,i}$ are the estimated parameters at $t_i$, such as satellite clocks, receiver clocks, LEO satellite clocks, and dynamic parameters of GPS and LEO satellites, such as SRP for GPS satellites, atmospheric drag and empirical force model parameters for LEO Satellites. In practical processing, the parameters to be estimated for GPS and LEO are different, as detailed in Table 3. ${\epsilon }_g^s$ and ${\epsilon }_{leo}^s$ contain unmodeled effects and measurement errors.

The positions of GNSS and LEO satellites at any time can be obtained from the state transition matrix and initial orbits according, which can be expressed as follows:

$$x^{s,i}={\mathsf{\Psi }}_{GNSS}(t,t_0)x^{s,0}$$

(7)

$$x_{leo,i}={\mathsf{\Psi }}_{LEO}(t,t_0)x_{leo,0}$$

(8)

where $x^{s,0}$ and $x_{leo,0}$ represent the initial positions of GPS and LEO satellites. ${\mathsf{\Psi }}_{GNSS}(t,t_0)$ and ${\mathsf{\Psi }}_{LEO}(t,t_0)$ denotes the state transition matrix, which can be obtained by numerical integration. $x^{s,i}$ and $x_{leo,i}$ are the positions of the GPS and LEO satellites at the required time.

The process of integrated POD involves iteratively adjusting and solving the observation Equations (5) and (6) to obtain the initial values of the satellite’s orbit by using the least squares estimation. In this process, the satellite position at any time in the error equation can be calculated using Equations (7) and (8).

2.2. LEOs’ Attitude Algorithm

During the movement of LEO satellites, there are typically three attitude control modes. The Acquisition and Safe Mode (ASM) is designed to ensure the spacecraft’s power and thermal survival. The Normal Mode (NOM) is the standard operational mode, providing precise three-axis attitude control based on a defined reference. The Orbit Control Mode (OCM) is used for necessary orbit changes and maintenance maneuvers during the mission [24]. This paper focuses exclusively on attitude data in NOM.

With continuous and complete attitude quaternions, which the LEO satellite Task Force provides for the aforementioned satellites Under normal circumstances, primarily forming the transformation matrix from the Earth-Centered Inertial (ECI) frame to the body-fixed frame. However, it is important to note that the quaternions in Swarm L1B products define the transformation matrix from the Earth-Centered Earth-Fixed (ECEF) frame to the body-fixed frame.

The transformation matrix based on the real attitude quaternions can be expressed as follows [25]:

$$R_{real}=\left[\begin{array}{ccc}{q}_0^2+q_1^2-q_2^2-q_3^2& 2\left(q_1{q}_2+q_0{q}_3\right)& 2\left(q_1{q}_3-q_0{q}_2\right)\\ 2\left(q_1{q}_2-q_0{q}_3\right)& q_0^2-q_1^2+q_2^2-q_3^2& 2\left(q_2{q}_3+q_0{q}_1\right)\\ 2\left(q_1{q}_3+q_0{q}_2\right)& 2\left(q_2{q}_3+q_0{q}_1\right)& q_0^2-q_1^2-q_2^2+q_3^2\end{array}\right]$$

(9)

where $q_0$, $q_1$, $q_2$, and $q_3$ represent one real unit and three imaginary units. $R_{real}$ represents the rotation matrix from the ECI frame to the body-fixed frame.

Meanwhile, without the attitude quaternions, some LEO satellites with strict attitude control requirements will use the transformation matrix from the ECI frame to the body-fixed frame under nominal attitude. The body-fixed frame under a nominal attitude definition varies according to each satellite’s specifications. For most LEO satellites, in the satellite orbital frame, the x-axis points towards the along-track direction (T), the y-axis points in the opposite direction to the cross-track direction (-A), and the z-axis points in the nadir direction (-R), where R, T, and A represent the radial, along-track, and cross-track directions, respectively. For convenience, the body-fixed frame under nominal attitude will hereafter be referred to as the RTA frame. It is calculated based on the position ($\overrightarrow{r}$) and velocity vectors ($\dot{\overrightarrow{r}}$) from the ECI frame, which can be expressed as follows:

$$R_{nominal}(3,i)=-{\displaystyle \frac{\overrightarrow{r}}{\left|\overrightarrow{r}\right|}}$$

$$R_{nominal}(2,i)=-{\displaystyle \frac{\overrightarrow{r}\times \dot{\overrightarrow{r}}}{\left|\overrightarrow{r}\times \dot{\overrightarrow{r}}\right|}}$$

(10)

$$R_{nominal}(1,i)=R_{nominal}(2,i)\times R_{nominal}(3,i)$$

where $i$ = 1, 2, 3 corresponds to the three components of each row vector in the transformation matrix $R_{nominal}$. $R_{nominal}(2,i)$ and $R_{nominal}(3,i)$ are the row vectors of the second and third rows of the matrix $R_{nominal}$. $R_{nominal}$ represents the rotation matrix from the ECI frame to the RTA frame.

As outlined in the introduction, incomplete quaternion data may be encountered during the processing of data from a limited observation network and LEO satellites. However, when the interruption in satellite attitude quaternions exceeds a certain threshold, it is not feasible to directly use interpolation to generate new quaternions. Moreover, constructing the C-matrix based on quaternions (see Equation (9)) and applying rotation according to the default definition (see Equation (10)) at epochs with missing quaternions would be unreliable. This is because there exists a certain angle between the actual attitude and the nominal attitude. If mixed use occurs, it is as if the satellite attitude rotates a certain angle every hour, constantly adjusting the attitude, leading to anomalous orbit determination. On the other hand, whether using continuous actual attitude or nominal attitude, it is equivalent to the satellite attitude being stable. In this condition, the accelerations from the experience forces can absorb the errors well, and the orbit determination precision reaches a certain level.

During normal attitude periods for certain satellites, the rotation angle between the RTA frame and the body-fixed frame remains stable. Consequently, in integrated POD processing, when quaternions are discontinuous, the actual quaternion rotation matrix at disrupted epochs can be represented as follows:

$$R_{real,i}=R_{real,0}{R_{nominal,0}}^T{R}_{nominal,i}$$

(11)

where $R_{real,0}$ is the rotation matrix based on real quaternions of the last available epoch before the interruption can be achieved. $R_{nominal,0}$ is the rotation matrix from the inertial frame to the RTA frame of the last available epoch before the interruption. $R_{nominal,i}$ is the rotation matrix from the inertial frame to the RTA frame in the missing epoch. $R_{real,i}$ is the rotation matrix that is to be calculated, which is close to the real quaternions [26].

2.3. LEOs’ Attitude Stability

The rotation matrix R represents the relationship between the real quaternion rotation matrix $R_{real}$ and the nominal rotation matrix $R_{nominal}$. The former defines the transformation from the inertial frame to the body-fixed frame, and the latter defines the transformation from the inertial frame to the RTA frame. R can be described with the rotation by angle ${\alpha }_x$ around the X-axis of RTA frame, followed by the rotation by angle ${\alpha }_y$ around the Y-axis of RTA frame, followed by the rotation by angle ${\alpha }_z$ around the Z-axis of RTA frame. So, matrix R can be expressed as follows:

$$R_{real}=R·R_{nominal}$$

(12)

$$R=R_x\left({\alpha }_x\right)R_y\left({\alpha }_y\right)R_z\left({\alpha }_z\right)$$

(13)

Then, the rotation angle between the RTA frame and body-fixed frame is calculated with the following:

$${\alpha }_x=arctan\left({\displaystyle \frac{R\left(\mathrm{2,3}\right)}{R\left(\mathrm{3,3}\right)}}\right)$$

$${\alpha }_y=arctan\left({\displaystyle \frac{R\left(\mathrm{1,3}\right)}{\sqrt{R^2\left(\mathrm{1,1}\right)+R^2\left(\mathrm{1,2}\right)}}}\right)$$

(14)

$${\alpha }_z=arctan({\displaystyle \frac{R\left(\mathrm{1,2}\right)}{R\left(\mathrm{1,1}\right)}})$$

Before processing the integrated POD, the rotation matrix angles between the RTA frame and body-fixed frame of the GRACE-FO and Sentinel-3 satellites were calculated using the data from Day 130 of 2021. The rotation angle results are shown in Figure 1 and Figure 2. As depicted in

Table 1. RMS and STD of the rotation angles of the transformed body-fixed frame and the RTA frame for LEO satellites (unit: degrees).

		GRACE-C	GRACE-D	Sentinel-3A	Sentinel-3B
${\mathit{\alpha }}_{\mathit{x}}$	STD	0.04	0.04	0.04	0.04
	RMS	0.04	0.04	0.04	0.04
${\mathit{\alpha }}_{\mathit{y}}$	STD	0.03	0.04	0.11	0.11
	RMS	0.86	0.81	0.11	0.11
${\mathit{\alpha }}_{\mathit{z}}$	STD	0.08	0.08	0.01	0.01
	RMS	0.08	0.08	0.01	0.01

, the STDs of the rotation angles of the GRACE-FO and Sentinel-3 satellites are at around or below 0.1 degrees. For the Swarm satellite, they are less than 1 degree. Note that the document on Swarm L1B processor algorithms provides transformation of S/C to ACC sensor frames, which are also RTA frames [27].

3. Integrated POD Strategy

The integrated POD is conducted by simultaneously using the LEO onboard and ground network GPS data. In this section, the LEO satellites and ground network used for the experiments are introduced. Then, the data processing strategy is presented, including the observation model, the orbit dynamic model, and the parametrization.

3.1. LEO Configuration and Ground Network

For nominal attitude control, the accuracy of GRACE-FO attitude is better than 30 mrad at the 3σ confidence level [24]. According to the previous studies, the RMS of the rotation angles between real quaternions and the nominal attitudes of Sentinel-3B and Sentinel-6A are generally at the same level, around or below 0.1 degree [26].

This means that in POD processing, there are only slight differences when using the real quaternions or nominal attitude [26]. The discrepancy in single-satellite orbit determination due to different attitudes is generally in millimeters level. However, for the centimeter-level precision required in the integrated POD, the experimental objective is to harness the actual attitude quaternions available to achieve an even higher degree of orbital accuracy. Based on the above research, these LEO satellites showed a stable attitude in the NOM model. Therefore, GRACE-FO, SWARM [28], and Sentinel-3 satellites were selected for integrated POD experiments. The information on these LEO satellites is shown in

Table 2. LEO satellite information.

Satellite	Launch Time	Height	Inclination
GRACE-FO	22 May 2018	490 km	89°
SWARM-A/C	22 Nov. 2013	450 km	87.4°
SWARM-B	22 Nov. 2013	530 km	87.4°
Sentinel-3A	16 Feb. 2016	814 km	98.65°
Sentinel-3B	15 Apr. 2018	814 km	98.65°

. The quaternions in the 1B data provided by the GRACE-FO and Sentinel series satellites are supposed to form the rotation matrix from the ECI frame to the body-fixed frame, while the SWARM satellite is different from these two different satellite series. The quaternions in the 1B data of SWARM satellites are used to form a rotation matrix from the International Terrestrial Reference Frame (ITRF) to the body-fixed frame.

About 22 global stations were selected to form a sparse network, and among them, 7 stations were selected to form the regional network to be used for the experiments. The distribution of these stations is shown in Figure 3 and Figure 4.

3.2. Strategy

In the integrated processing of GPS and LEO satellite orbits, the Positioning and Navigation Data Analyst (PANDA) was employed, which offers advanced capabilities for multi-technique processing, including GNSS, Very-Long-Baseline Interferometry (VLBI), and Satellite Laser Ranging (SLR) [29,30]. The integrated processing strategy of LEO-onboard and ground network GPS data is shown in Table 3. And a one-step approach was used in integrated POD processing. Seven satellites, including Sentinel-3, GRACE-FO, and Swarm, were selected to investigate the integrated POD under sparse and regional station conditions, with a focus on attitude discontinuities. The attitude quaternion discontinuity cases were divided into four scenarios:

Case 1: Continuous and complete attitude quaternions for all LEO satellites.

Case 2: Attitude quaternions are interrupted for one hour every hour.

Case 3: Attitude quaternions are interrupted for two hours every hour.

Case 4: Attitude quaternions are interrupted for three hours every hour.

When processing LEO and ground station data in real time, only the predicted Earth Rotation Parameters (ERPs) are available. Given the strong correlation between the ERPs and the orbits, we also studied whether ERPs should be added to the estimations under sparse/regional stations and the influence of ERPs on the integrated POD in this case. The following trials are based on the results from Days 126 to 133 in 2021, for a total of 8 days.

Table 3. The integrated POD strategy.

	GPS	LEO
Observational model
Signal	GPS:L1/L2	L1/L2
Arc length	24 h
Sampling rate	30 s
Troposphere	GPS: A prior model [31] is applied to remove the dry delay; zenith wet delay is estimated with the GMF mapping function [32]. LEO: no
Reference frame	ITRF2014
Station displacement	Solid Earth tide, pole tide, and ocean loading tide [33]
Dynamic Models
Solid Earth, ocean, pole tide	IERS Conventions 2010 [33]	IERS Conventions 2010
N-body perturbation	JPL DE405 [34]	JPL DE405
Relativity	IERS Conventions 2010 [33]	IERS Conventions 2010
Earth gravity field	EGM 12 × 12 [35]	EGM 150 × 150
Atmospheric drag	NO	DTM94 [36]
Solar radiation pressure	GPS: ECOM1 + Boxwing [37,38]	PAN_GEN
Attitude	Yaw attitude	Based on different test settings
Estimated Parameter
Station coordinates	Tightly Constrained
Orbit parameter	6 orbit elements and ECOM1	6 orbit elements and 6 piecewise empirical parameters: 90 min
Atmosphere drag	No	Yes and for every 90 min
Clock offsets	Epoch-wise
Phase ambiguities	Float
Rotation parameter	As introduced

Table 3. The integrated POD strategy.

	GPS	LEO
Observational model
Signal	GPS:L1/L2	L1/L2
Arc length	24 h
Sampling rate	30 s
Troposphere	GPS: A prior model [31] is applied to remove the dry delay; zenith wet delay is estimated with the GMF mapping function [32]. LEO: no
Reference frame	ITRF2014
Station displacement	Solid Earth tide, pole tide, and ocean loading tide [33]
Dynamic Models
Solid Earth, ocean, pole tide	IERS Conventions 2010 [33]	IERS Conventions 2010
N-body perturbation	JPL DE405 [34]	JPL DE405
Relativity	IERS Conventions 2010 [33]	IERS Conventions 2010
Earth gravity field	EGM 12 × 12 [35]	EGM 150 × 150
Atmospheric drag	NO	DTM94 [36]
Solar radiation pressure	GPS: ECOM1 + Boxwing [37,38]	PAN_GEN
Attitude	Yaw attitude	Based on different test settings
Estimated Parameter
Station coordinates	Tightly Constrained
Orbit parameter	6 orbit elements and ECOM1	6 orbit elements and 6 piecewise empirical parameters: 90 min
Atmosphere drag	No	Yes and for every 90 min
Clock offsets	Epoch-wise
Phase ambiguities	Float
Rotation parameter	As introduced

4. Test Results

4.1. Orbital Dilution of Precision

During integrated POD, the LEO satellites and the ground stations are regarded as tracking stations for GNSS satellites. The average, maximum, and minimum values of the observations that can simultaneously track each GPS satellite are recorded to reflect the satellite geometry status for the integrated POD. The LEO satellite’s real-time navigation augmentation capabilities can be obtained by calculating the Orbital Dilution of Precision ($ODOP$) as follows:

$$H=\left[\begin{array}{ccc}{\displaystyle \frac{x_1-x^s}{\left|\stackrel{⃑}{r_1}\right|}}& {\displaystyle \frac{y_1-y^s}{\left|\stackrel{⃑}{r_1}\right|}}& \begin{array}{cc}{\displaystyle \frac{z_1-z^s}{\left|\stackrel{⃑}{r_1}\right|}}& 1\end{array}\\ {\displaystyle \frac{x_2-x^s}{\left|\stackrel{⃑}{r_2}\right|}}& {\displaystyle \frac{y_2-y^s}{\left|\stackrel{⃑}{r_2}\right|}}& \begin{array}{cc}{\displaystyle \frac{z_2-z^s}{\left|\stackrel{⃑}{r_2}\right|}}& 1\end{array}\\ \begin{array}{c}⋮\\ {\displaystyle \frac{x_n-x^s}{\left|\stackrel{⃑}{r_n}\right|}}\end{array}& \begin{array}{c}⋮\\ {\displaystyle \frac{y_n-y^s}{\left|\stackrel{⃑}{r_n}\right|}}\end{array}& \begin{array}{cc}\begin{array}{c}⋮\\ {\displaystyle \frac{z_n-z^s}{\left|\stackrel{⃑}{r_n}\right|}}\end{array}& \begin{array}{c}⋮\\ 1\end{array}\end{array}\end{array}\right]$$

(15)

$$Q={\left(H^{T}H\right)}^{-1}$$

(16)

$$ODOP=\sqrt{Q\left(\mathrm{1,1}\right)+Q\left(\mathrm{2,2}\right)+Q\left(\mathrm{3,3}\right)}$$

(17)

According to the analysis of observational data on DOY 130 of 2021, the average observation number simultaneously tracking a GPS satellite for the seven LEO satellites and sparse/regional networks is as shown in

Table 4. The statistics of simultaneously tracked observations for each GPS satellite for the sparse/regional networks and 7 LEO satellites.

Obs.	Satellites	Average Sta/LEOs	Max Sta/LEOs	Min Sta/LEOs
7LEOs + 7Sta	GPS	3.87	13	0
7LEOs + 22Sta	GPS	9.48	20	1

. The results show that the average number of observations per GPS satellite is close to four when using the regional network, which implies it is likely that in some cases the integrated POD result could be very undesirable. However, for the sparse network, the average number of observations is about nine. Although from Figure 5, it can be observed that some observations are not of good quality, should have little influence on the integrated POD, and could have greater influences on the clock determination. The ODOPs for the two types of networks are also illustrated in Figure 5 and Figure 6. For the regional network, it would be difficult to achieve good orbital results in the integrated POD. In other words, the results would be very sensitive to the quality of the observations and the GPS satellite’s initial orbits. For the sparse network, the ODOPs for every satellite are under 40 in most epochs. The better satellite geometry gives better promise for the integrated POD compared to the regional network.

4.2. Earth Rotation Parameters

Currently, ITRF is established using VLBI, SLR, DORIS, and GNSS observations. Some studies have demonstrated that incorporating LEO satellites in the estimation process can improve the determination of the geocenter location and ERPs to varying degrees [4,12]. Numerous scholars have confirmed that the inclusion of LEO data enhances the accuracy of GNSS orbit determination [10,12]. From the perspective of global network solutions, the addition of LEO satellites positively impacts both the ERPs and the geocenter location estimation [39]. However, these studies are primarily based on global network configurations. Based on the above analysis, the orbit determination results for regional stations are highly sensitive to the involvement of seven satellites. Therefore, whether or not to estimate ERPs and geocenter location is beyond the scope of this discussion. Instead, we focus on analyzing the impact of ERP and geocenter location estimation under sparse station configurations. The ERPs here refer to the pole motions and the Length of Day (LOD).

Finally, the cumulative distribution of orbit errors with and without ERPs and geocenter estimation is shown in Figure 7. The horizontal axis of each subplot represents the orbital accuracy in the radial, along-track, cross-track, and 1D directions compared with the International GNSS Service (IGS) final product in millimeters, while the vertical axis indicates the percentage distribution. When ERPs and geocenter are estimated in the integrated POD, the GPS orbit accuracy amounts to 16.5 mm, 28.3 mm, 24.24 mm, and 24.1 mm in the radial, along-track, cross-track, and 1D directions, respectively, as shown in Figure 7. The situation improves without estimating the ERPs. The final statistical results indicate that the integrated POD accuracy, when estimating the ERPs and the geocenter, degrades by 3.25%, 14.6%, and 13.4% in the radial, along-track, and cross-track directions, respectively, compared to the strategy that does not estimate the ERPs. Since ERP is strongly correlated with the orbit, when the ground network is sparse/regional, which is not an ideal situation for high-precision GNSS orbit determination, estimating it during the adjustment process is likely to introduce more inaccurate parameters. That leads to a reduction in accuracy.

The inclusion of LEO satellites in integrated POD has also shown a degradation in accuracy as shown in Figure 8.

Table 5. The RMS of LEO satellite orbital errors compared with the corresponding LEO satellite final orbit products in the radial, along-track, cross-track, and 1D directions with or without estimation of the ERPs and geocenter based on the sparse network.

		Radial (mm)	Along-Track (mm)	Cross-Track (mm)	1D (mm)
GRACE-C	Without ERP and CoE	11.5	8.1	20.5	14.4
	With ERP and CoE	11.6	10.4	20.3	14.9
GRACE-D	Without ERP and CoE	11.7	8.4	20.3	14.4
	With ERP and CoE	11.9	10.1	20.2	14.9
Swarm-A	Without ERP and CoE	15.8	28.2	23.2	23.1
	With ERP and CoE	17.5	32.2	23.7	25.1
Swarm-B	Without ERP and CoE	16.6	27.4	19.4	21.7
	With ERP and CoE	17.7	29.7	21.1	23.4
Swarm-C	Without ERP and CoE	15.8	28.8	23.4	23.3
	With ERP and CoE	16.9	32.9	23.9	25.5
Sentinel-3A	Without ERP and CoE	16.1	33.1	21.05	24.6
	With ERP and CoE	18.1	36.3	20.1	26.3
Sentinel-3B	Without ERP and CoE	15.1	30.5	21.2	23.2
	With ERP and CoE	17.2	33.4	20.9	24.9

shows the RMS of LEO satellite orbital errors compared with the corresponding LEO satellite final orbit products in the radial, along-track, cross-track, and 1D directions with or without estimation of the ERPs and the geocenter. For GRACE-C and GRACE-D, the 1D error increased slightly from 14.4 mm to 14.9 mm after considering ERPs and the geocenter location, with minor changes observed in the radial, along-track, and cross-track directions. For the Swarm satellites, the 1D error increased from 23.1 mm to 25.1 mm for Swarm-A, from 21.7 mm to 23.4 mm for Swarm-B, and from 23.3 mm to 25.5 mm for Swarm-C, demonstrating the influence of ERP estimation on the orbital accuracy. For Sentinel-3A and Sentinel-3B, the 1D errors increased from 24.6 mm to 26.3 mm and from 23.2 mm to 24.9 mm, respectively, with the along-track direction showing a more significant increase. Overall, the inclusion of ERP and geocenter location degrades the accuracy of the LEO satellite orbit determination, though the impact varies across different satellite types.

4.3. Integrated POD Result with Attitude Algorithm

4.3.1. Sparse Network

The International GNSS Service (IGS) regularly publishes GPS and GLONASS orbit and clock offset products, with accuracies of approximately 2 cm and 75 ps, respectively [40]. These final products are derived from a combination of the orbits and clock offsets determined by the IGS Analysis Centers (ACs) as input. As Figure 9 showed, based on the statistical analysis of data from DOYs 126-133/2021, it was found that when the attitude data remained continuous, the GPS orbital results obtained from integrated POD showed RMS differences of 16 mm, 25 mm, and 21 mm in the radial, along-track, and cross-track directions, respectively, compared with the IGS final products as. However, when quaternion data were interrupted for one hour at regular intervals and interpolated using the previously mentioned algorithm (see Section 2), the 1D RMS orbital difference for GPS satellites increased to 24 mm, which is comparable to the results observed in case 3. When the interruptions were further extended to three hours, the accuracy degraded by approximately 29%. Furthermore,

Table 6. The RMS of LEO satellite orbital errors in 1D under 4 attitude situations. The unit is millimeters.

Satellite	Case 1	Case 2	Case 3	Case4
GRACE-C	14.37	14.12	14.28	14.30
GRACE-D	14.38	14.16	14.35	14.38
Swarm-A	23.05	25.42	24.40	26.68
Swarm-B	21.65	24.30	23.28	25.62
Swarm-C	23.28	25.60	24.60	26.93
Sentinel-3A	24.58	27.26	36.50	29.13
Sentinel-3B	23.22	27.04	25.20	28.88

shows the RMS of the LEO satellite orbital errors in 1D under four attitude situations. The distributions of orbital errors of the 7 LEO satellites are shown in Figure 10, Figure 11 and Figure 12, the vertical axis represents the 1D deviations in millimeters, while the horizontal axis shows four test cases. The red dots represent outliers, and the blue box indicates the interquartile range, with the red line marking the median value. The results from different tests did not exhibit significant differences for GRACE-C and GRACE-D (The 1D mean time series in the orbital errors as Figure 13 shown). For Swarm-A, Swarm-B, and Swarm-C, the orbital errors exhibited larger differences compared with the corresponding LEO satellite final orbital products in the 1D direction, i.e., with values varying between approximately 21.65 mm and 26.93 mm, reflecting slightly higher variability. The 1D mean time series of Swarm-B in the orbital errors as Figure 14 shown exhibiting the same phenomenon. The 1D mean time series of Sentinel-3B in the orbital errors is as Figure 15 shown. Sentinel-3A and Sentinel-3B show the most significant variations, particularly for Sentinel-3A in case 3, where the deviation reaches 36.50 mm, indicating potential challenges in specific cases. Overall, the results highlight that GRACE satellites achieve better orbit determination accuracy compared to the Swarm and Sentinel satellites, with noticeable differences influenced by different attitude situations.

4.3.2. Regional Network

Based on the regional network, it was found that when the attitude data remained continuous, the GPS orbital results obtained from the integrated POD showed RMS differences of 35 mm, 63 mm, and 48 mm in the radial, along-track, and cross-track directions, respectively, when compared with the IGS final products, as shown in Figure 16. When quaternion data were interrupted for one hour at regular intervals and interpolated using the previously mentioned algorithm, the 1D RMS orbital differences for the GPS satellites increased to 59 mm. When the interruptions were further extended to three hours, the accuracy degraded slightly, by approximately 11%. However, the results obtained were not as bad as for the sparse network when the interruptions were for two hours. This is likely due to the insufficient sample size and the exclusion of too many outliers during the 3σ rule-based statistical process.

Table 7. The RMS of LEO satellite orbital errors in 1D under 4 attitude situations (unit: millimeters).

Satellite	Case 1	Case 2	Case 3	Case 4
GRACE-C	58.17	67.37	54.83	53.55
GRACE-D	65.70	77.47	53.40	76.93
Swarm-A	54.07	52.57	41.57	55.77
Swarm-B	54.47	52.60	41.43	56.00
Swarm-C	54.40	52.90	42.40	56.03
Sentinel-3A	56.97	53.50	46.70	52.60
Sentinel-3B	61.23	63.07	46.67	61.80

shows the RMS of LEO satellite orbital errors in 1D under four attitude situations. The distributions of orbital errors of the seven LEO satellites based on different attitude situations are shown in Figure 17, Figure 18 and Figure 19. As mentioned in the previous ODOP analysis, when combining seven ground stations and seven LEO satellites for integrated POD, the observational data are insufficient to meet the calculation conditions during certain epochs. This implies that during parameter estimation and integration, some correct data might be excluded, or outliers may not be removed. Therefore, the statistical analysis of the orbit determination results only serves as a reference within a certain precision range. The results showed that for LEO satellites, the orbital errors are around 5–7 cm for every LEO satellite in different attitude situations.

5. Discussion

This paper addresses the practical issue of discontinuous attitude quaternions in LEO satellites and explores LEO/GNSS integrated orbit determination using limited attitude quaternion data of LEO satellites. Seven LEO satellites, including those from GRACE-FO, Swarm, and Sentinel-3, were analyzed. The results showed that the use of 22 globally distributed stations and 7 LEO satellites allowed integrated POD to achieve a GPS orbit accuracy of 2–3 cm and LEO satellite orbit accuracy of 1–3 cm under four scenarios, continuous quaternions and quaternion interruptions of 1, 2, and 3 h after one hour of continuous data, employing interpolation between actual and nominal attitude rotation matrices. The results in complete and continuous quaternions are close to other integrated POD studies based on PANDA software [4,41]. When using a regional network of seven stations, the achieved GPS satellite’s accuracy was 5–6 cm, and for LEO satellites, the orbital accuracy was 5–7 cm under the same scenarios. The results demonstrate that the proposed interpolation method is effective for joint orbit determination in cases of sparse or interrupted quaternions for satellites with strict attitude control.

6. Conclusions

When conducting integrated POD using LEO satellites and sparse or regional networks, handling sparse or interrupted quaternion data poses significant challenges. Discarding quaternion data in such cases is regrettable, yet long-term interruptions make it impractical to rely solely on conventional interpolation methods such as Slerp or spline interpolation. For satellites with strict attitude control, the nominal and actual attitudes typically deviate by a small angle, which is generally less than 1°. In this study, quaternions are interpolated during epochs with interruptions based on real quaternions and the nominal attitudes of the LEO satellite. The analysis focuses on orbit determination accuracy under sparse and regional network scenarios. The key findings are as follows:

• The ODOP statistical results indicate that the precision of joint orbit determination using regional stations and seven LEO satellites is relatively poor, and the data volume is insufficient to meet future real-time calculation demands.
• For joint orbit determination under sparse station conditions with 22 stations, not estimating the ERPs is slightly superior to estimating them.
• Based on 22 sparse stations, an analysis was conducted for cases where quaternion data were continuous and complete, and were interrupted for 1 h, 2 h, and 3 h every hour, in a total of four scenarios. The results showed that the accuracies of 1D orbit determination using quaternion rotation matrix interpolation are 21 mm, 24 mm, 24 mm, and 28 mm in the four scenarios, respectively.
• For seven regional stations and seven LEO satellites performing joint orbit determination, the accuracy of GPS orbit determination is between 5 and 6 cm, and that of LEO orbit determination is between 5 and 7 cm across the four scenarios.

These results demonstrate that the quaternion rotation matrix interpolation method is reliable for the integrated POD of GNSSs and LEOs with strict attitude control. Additionally, this method offers valuable insights for joint orbit determination during the deployment of LEO satellite constellations.