Highly Efficient Real-Time Kinematic-Based Precise Relative Navigation for Autonomous Rendezvous CubeSat

2.1 System Configuration

The GNSS Laboratory at Seoul National University initiated the SNUGLITE-III CubeSat project, comprising two CubeSats, A and B, adhering to the 3-unit (3U) standard. The primary mission involves collecting GPS radio occultation (RO) data for Earth remote sensing, culminating in a demonstration of autonomous formation flying and rendezvous docking technology. The SNUGLITE-III A/B CubeSats are designed to combine, meeting the 6-unit (6U) size standard, and will be positioned on the 6U P-POD. Figure 1(a) illustrates the configuration, showing the combined state of the SNUGLITE-III A/B CubeSats. The SNUGLITE-III A/B CubeSats, while identical in configuration, differ based on whether they face the front or rear direction (X-axis) during the docking process, as depicted in Figure 1(b). Specifically, SNUGLITE-III A is equipped with a docking device in the rear, whereas SNUGLITE-III B has a docking device at the front, resulting in distinct body coordinate systems for each. Tables 1 and 2 present information about the system configuration and hardware used in the SNUGLITE-III relative navigation system.

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FIGURE 1

SNUGLITE-III CubeSats: (a) joint 6U configuration; (b) separated SNUGLITE-III A/B CubeSats, each in an identical 3U configuration

Both SNUGLITE-III CubeSats A and B are equipped with three GPS receivers and antennas dedicated to the relative navigation system. The GPS receivers are specifically customized for the SNUGLITE-III CubeSat project, drawing from the experience gained with space-proven receivers in earlier SNUGLITE CubeSat projects. For relative navigation, both CubeSats use GPS antennas oriented in the same direction. This approach maximizes the number of common visible GPS satellites between the two CubeSats, to minimize common GPS error factors. To achieve this, the collected GPS measurements are transmitted in real time between satellites, using state-of-the-art long-range (LoRa) communication to apply low-power, high-efficiency technology to CubeSats (Devalal & Karthikeyan, 2018).

The SNUGLITE-III CubeSats are scheduled for launch in 2025 aboard the Korean satellite launch vehicle II (KSLV-II, Nuri). The mission scenario, depicted in Figure 2, involves SNUGLITE-III A/B CubeSats initially functioning as a jointed 6U platform immediately after launch. During this phase, the SNUGLITE-III B CubeSat executes attitude control to maintain a nadir-pointing orientation in the 6U joint state. After a two-month commissioning phase to assess CubeSat conditions, the CubeSats will separate. Upon separation, each CubeSat will perform three-axis active attitude control. SNUGLITE-III A (target) will transmit GPS measurements to SNUGLITE-III B (chaser) at intervals of 1–5 s through an ISL. Initially, the relative velocity caused by the separation mechanism will gradually move the two CubeSats apart via orbital perturbations, resulting in an increased relative distance. This divergence initiates formation flying. The SNUGLITE-III A and B satellites use orbit control through atmospheric drag in low orbit without thrusters, operating at very low speeds and using relative air resistance available in the delicate low-orbit environment. After successful formation flying, the CubeSats will proceed to a rendezvous and docking mission. When within a close distance of approximately 1 m, they will employ an electromagnetic device for docking. Equipped cameras will capture the docking scene, and opposing forces from electromagnets will facilitate undocking.

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FIGURE 2

Operational scenario of the SNUGLITE-III CubeSats with precise relative navigation in formation flying, rendezvous, and docking phases

The necessary GPS relative navigation for the formation flying and rendezvous docking missions is performed in two operational scenarios. The first is the coarse relative navigation mode, which provides decimeter-level accuracy based on pseudorange measurements. In this mode, GPS measurements are transmitted through the ISL at 10-s intervals, providing relative navigation solutions at 1-s intervals. This mode verifies power stability and technical soundness during the initial CubeSat operation, using orbit propagation modules between ISL communication intervals to facilitate decimeter-level formation flying. The second scenario is the RTK relative navigation mode, which provides centimeter-level accuracy. This mode, designed explicitly for rendezvous and docking missions, can be initiated through ground station commands. GPS measurements are transmitted through the ISL at 1-s intervals in this mode.

This study presents a practical methodology for implementing centimeter-level RTK relative navigation for CubeSats. As highlighted in Table 2, the SNUGLITE-III CubeSat hardware has performance limitations. The processor designated for relative navigation operates at 66 MHz, with 91 Dhrystone millions of instructions per second (DMIPS), which is in stark contrast to desktop computers (Intel i5-11600k, 346350 DMIPS at 4.92 GHz) and processors designed for larger satellites (European Space Agency, LEON 4, 2550 DMIPS at 1.5 GHz). This difference underscores the inherent computing power constraints of CubeSat processors. Additionally, the OBC processor, which is responsible for relative navigation and multiple subsystems, imposes computational speed limitations on real-time CubeSat missions.

In the context of GPS relative navigation for CubeSats, the ISL communication from the target to the chaser CubeSat significantly influences the overall mission. To address power operation constraints, the chosen LoRa module theoretically facilitates communication up to a maximum distance of 55 km (Table 2). However, as a communication module, it introduces latency due to the ISL communication speed, which directly impacts real-time systems and is associated with securing the OBC computation time. Therefore, it is crucial to use algorithms that minimize data usage while optimizing computational efficiency and minimizing transmission delays. Moreover, hardware performance limitations, including those of GPS patch antennas, must be thoroughly addressed. Considering these challenges, this study introduces an efficient single-frequency GPS relative navigation approach tailored for CubeSat platforms. This approach aims to diversify GPS applications and make a significant contribution to the advancement of modern CubeSat technology.

2.2 Space GPS Receiver for CubeSats

The GPS receiver embedded in the SNUGLITE-III CubeSats is an L1/L2C GPS receiver for LEO collaboratively developed by Seoul National University and Danam Systems. This receiver is an evolution of the CubeSat GPS receiver integrated into the SNUGLITE-I CubeSat in 2018, primarily for meter-level performance verification. Subsequent enhancements focused on hardware miniaturization and firmware improvements. The receiver’s performance was validated in two CubeSat missions (SNUGLITE-I and SNUGLITE-II) via orbit determination techniques (Park et al., 2023; Yu et al., 2020). The results demonstrated navigation solutions with errors within 10 m, meeting the navigation error requirements for LEO satellites equipped with single-frequency GPS receivers (Sandau et al., 2008).

Dual-frequency GPS measurements are essential in GPS RO atmospheric observation (Kursinski et al., 1997) for SNUGLITE-III CubeSat missions. The comparison of L1 and L2C measurements serves as the foundation for distinguishing atmospheric and ionospheric contributions. However, the focus of this paper is on relative navigation utilizing single-frequency GPS measurements transmitted through ISL communication. The aim is to demonstrate the feasibility of precise GPS relative navigation in orbit while minimizing data volume to alleviate communication delays and facilitate efficient computations.

The configuration and specifications of the GPS receiver are shown in Figure 3 and Table 3, respectively. Three GPS receivers are integrated following the PC/104 standard to make them compatible with CubeSat applications, with each connected to its respective GPS antenna. As outlined in Table 3, the power consumption of one GPS receiver is designed to be approximately 0.45 W, which is roughly one third of the power consumption of commercial GPS receivers commonly used in CubeSats. The hardware adheres to NASA STD-8739.1B specifications and features a parylene C chemical vapor deposition coating and additional shielding to enhance the radio frequency signal quality.

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FIGURE 3

Seoul National University CubeSat L1/L2C GPS receiver consisting of three GPS receivers on one board, before and after can shielding

3.1 Overall Block Diagram

The comprehensive configuration of the proposed precise relative navigation system is shown in Figure 4. GPS measurements from the SNUGLITE-III A CubeSat are transmitted to the SNUGLITE-III B CubeSat through the ISL. After preprocessing, including memory allocation and time synchronization, the transmitted GPS measurements are directed to the relative navigation module. The data to be transmitted through the ISL are summarized in Table 4. For example, with 10 visible satellites, the data transmitted is 275 bytes. With a transmission rate of 23 kbps (as described in Table 1), the theoretical transmission delay time is estimated to be at least 94 ms. GPS measurements are updated every second, to provide a relative navigation solution within 1 s, considering both communication and computation time within the module. Minimizing the communication delay reduces power consumption and ensures timely computation due to the limited processing speed of the OBC.

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FIGURE 4

Overall block diagram of the relative navigation system

The precise relative navigation module primarily comprises a DGPS module, an RTK relative navigation module, and a position-domain module for handling outlier cases. The DGPS module computes pseudorange-based relative positions to minimize integer ambiguity candidates. It continually provides a relative navigation solution every second and features an optimal Hatch filter designed for fast maneuverability in the LEO environment using standard GPS satellites for the SNUGLITE-III A/B CubeSats.

The RTK solution from the DGPS-calculated relative navigation is merged with double-difference carrier-phase (DDCP) measurements to compute the primary integer ambiguity covariance. Although similar to ground-user RTK schemes, this implementation excludes a Kalman filter for efficient computation. The enhanced DGPS position solution minimizes integer ambiguity candidates and estimates the covariance. Because of rapid changes in the GPS constellation in space, the reference satellites used for DDCP measurements also frequently change. Therefore, continuous consideration of integer ambiguities and variations in the DDCP measurements is essential for applying RTK in space.

The space environment lacks tropospheric and multipath errors typically present in ground environments, providing a higher number of visible GPS satellites and significant advantages in terms of ambiguity dilution of precision (ADOP). This study leverages this characteristic to resolve integer ambiguities for RTK from single-frequency DGPS relative navigation. Because dual-frequency measurements are not used, typical linear carrier measurement combinations are impossible. The key idea for performing RTK in this study is to use a Hatch filter to enhance the accuracy of DGPS relative navigation. Although the Hatch filter is typically affected by ionospheric delays, the mission’s baseline of 1 km for autonomous formation flying and rendezvous docking minimizes the impact of errors. Thus, it is possible to enhance the DGPS performance when using a single frequency for subsequent RTK applications.

Subsequently, the least-squares ambiguity decorrelation adjustment (LAMBDA) technique is used to estimate integer ambiguities. For cases in which the exact integer ambiguity cannot be fixed, the relative position and velocity based on DGPS are provided. To prevent divergence and improve computing efficiency, the number of iterations for the LAMBDA calculations is limited. If the ISL communication fails to execute the relative navigation module, a time update is initiated using an orbital propagation algorithm. The orbit propagation algorithm used is based on the Hill–Clohessy–Wiltshire (HCW) equation, which describes circular relative motion. Once the integer ambiguity is resolved by LAMBDA, achieving centimeter-level precision in relative navigation becomes feasible. Thus, this study achieves a diversity of GPS applications through the implementation of these methods.

3.2 Range-Domain DGPS with a Hatch Filter

Range-domain DGPS enhances relative navigation by mitigating errors through shared GPS satellites between the two CubeSats. This approach aims to address issues in relative positioning performance caused by non-removable ionospheric errors and other factors present in position-domain DGPS, representing the positional difference between stand-alone GPS solutions. However, as a pseudorange-based algorithm, range-domain DGPS is susceptible to relatively high noise levels associated with pseudorange, which can affect the positional accuracy.

To address this challenge, this study introduces the use of a Hatch filter to refine the noise level of pseudorange measurements, as described in previous research (Shim et al., 2023). Hatch filtering is a crucial algorithm for enhancing relative navigation performance when using single-frequency GPS measurements. Falling within the category of smoothing filters, this filter targets an improvement in the noise level of pseudorange measurements. It effectively combines carrier-phase measurements with millimeter-level noise and pseudorange measurements characterized by meter-level noise (Kaplan & Hegarty, 2017). By integrating carrier-phase measurements with pseudorange, this filter estimates corrected pseudorange measurements, thus improving noise levels without calculating integer ambiguity. The pseudorange smoothing measurements achieved by using a Hatch filter for the k-th epoch can be expressed as shown in Equation (1):

1

2

where ρ(k) denotes the pseudorange measurement in the current epoch, Δϕ(k) denotes the change in the carrier-phase measurement, and denote the corrected pseudorange in the current and previous epochs, respectively, and σ_ρ(k) represent the corrected pseudorange and pseudorange-assumed noise, respectively, and N_h is the Hatch filter smoothing coefficient.

The Hatch filter relies exclusively on previously smoothed pseudorange correction measurements and current measurements. Theoretically, a larger value for the smoothing coefficient, which determines the filter’s performance, leads to a lower estimated noise level as more samples are used for estimation. However, filter divergence may occur in the presence of ionospheric effects, where pseudorange and carrier-phase measurements have different signs. To prevent divergence, the smoothing coefficient should be increased in line with the number of measured values and subsequently kept constant after a certain interval is reached.

To facilitate range-domain DGPS-based relative navigation, single-frequency pseudorange measurements from the j-th GPS satellite, collected from both the target (ρ_r) and chaser (.ρ_u.), can be mathematically represented by Equation (3):

3

Here, d, I, B, and b are the distance between the user and GPS satellite, ionospheric error, receiver clock error, and GPS satellite clock error, respectively, and the subscripts r and u denote the target (reference) and chaser (user), respectively.

The single differential equation between the CubeSats and the j-th satellite is given in Equation (4). Here, , , , and ê represent the GPS satellite position, user position, DGPS relative position, and line-of-sight vector, respectively. Because the SNUGLITE-III CubeSats perform formation flying within a maximum range of 1 km, allowing for very short baselines (ΔI ≈ 0, ) between the two satellites, the equation for conducting relative navigation can be simplified as shown in Equation (5):

4

5

Subsequently, we used the least-squares method and Equation (6) to derive a relative position based on range-domain DGPS, as expressed in Equation (7). In this context, the + superscript denotes a pseudo-inverse matrix:

6

Here, the noise level of the pseudorange varies based on the Hatch filter coefficient (N_h) for each GPS satellite. Consequently, weights corresponding to the coefficient of each satellite can be established using the weighted least-squares method:

7

3.3 Single-Frequency RTK Approach

The RTK method enables users to calculate position information with exceptional accuracy down to the centimeter level; in this study, the RTK method is applied to determine relative positions between CubeSats. Unlike regular ground GPS users who require absolute positions, the emphasis here is solely on relative positions, allowing for autonomous rendezvous missions even as the absolute position of the reference CubeSat changes.

The RTK method uses carrier-phase measurements known for millimeter-level noise, in contrast to the meter-level noise associated with the pseudorange commonly used in traditional GPS positioning methods. This approach achieves centimeter-level relative navigation primarily through a technique known as DDCP measurement. Employing DDCP measurements effectively eliminates errors associated with satellite and receiver clocks. However, despite these efforts, DDCP measurements retain GPS errors, and resolving the integer ambiguity of carrier-phase measurements is necessary for accurate position determination. Traditional RTK methods solve this challenge by linearly combining double-frequency carrier-phase measurements, such as wide-lane, ionosphere-free, and Melbourne–Wuebbena combinations, to remove residual errors and address integer ambiguity.

For RTK relative navigation in the SNUGLITE-III CubeSats, we provide a relative position using only single-frequency carrier-phase measurements rather than applying traditional multi-frequency-based RTK methods This approach is possible because the space environment for performing relative navigation in SNUGLITE-III maintains a very short baseline between the two CubeSats, minimizing common residual errors. In addition, approaching RTK relative navigation is achieved by reducing the candidate pool for integer ambiguity through the DGPS module. The small size of CubeSats minimizes multipath errors, and the number of visible satellites in LEO is highly advantageous for the essential integer ambiguity solution required for RTK. The mathematical representation of single-frequency carrier-phase measurements from the j-th GPS satellite, collected from both the target (ϕ_r) and chaser (ϕ_u) CubeSats, can be expressed by Equation (8). Here, N refers to the integer ambiguity, and λ refers to the L1 frequency wavelength:

8

For the j-th reference satellite, the DDCP measurements for the k-th satellite are defined as follows:

9

Working with a very short baseline (▽ΔI ≈ 0, ) between the reference station and the user, Equation (10) specifies the definition of the DDCP measurement. Thus, the observation matrix, state variable, and measured value vector are defined as shown in Equation (11):

10

11

Consequently, by reorganizing Equation (10) with Equation (11), we obtain the following structure:

12

As evident from Equation (11), the DDCP has the advantage of eliminating GPS error factors, but achieving precise positioning requires resolving the integer ambiguity. This study approaches the resolution of integer ambiguity using the measurement equation derived from Equation (12). Therefore, the float solution of the integer ambiguity is obtained using the DGPS solution calculated in Section 3.2, allowing for the covariance computation. The primary covariance can be defined as in Equation (13) to resolve the integer ambiguity as follows:

13

14

Here, δ represents the errors obtained by subtracting the mean of each component from the defined Equation (12) to define the covariance. In Equation (14), P₁ represents the covariance of the carrier measurement noise, P₂ represents the covariance of the DGPS position, and P₃ and P₄ are the correlations between the Hatch filter used in the DGPS and DDCP measurements.

The covariance P₁ is defined as the noise of the DDCP measurements, and P₂ is defined from the weighted least squares of the DGPS. Given that the carrier measurement values have noise at the millimeter level, it can be anticipated that the dominant factor is variance P₂, which is determined from pseudorange measurements at the meter level. The covariances P₁ and P₂ can be expressed as shown in Equations (15) and (16):

15

16

P₃ and P₄ represent the covariances arising from the correlation between the DDCP measurements and the carrier measurements using the Hatch filter in DGPS. For the m-th visible GPS satellite, P₃ can be expressed as shown in Equation (17):

17

Here, regarding the corrected pseudorange from the Hatch filter, as indicated in Equation (1), we are interested in the carrier-phase correlation at the current epoch time (k) and assumed that there is no time correlation, which can be expressed as in Equation (18):

18

Rewriting the corrected pseudorange, as defined in Equation (17), in terms of the carrier-phase measurements, P₃ can be expanded as a function of the Hatch filter coefficient (N_h):

19

To transform the DDCP into a defined equation in terms of single-difference carrier-phase (SDCP) measurements, we introduce a linear transformation matrix that aligns the dimensions of the covariance matrix, which serves to link the SDCP and DDCP measurements used in the DGPS framework. Equation (20) shows the relationship for the defined linear transformation matrix:

20

By introducing this matrix into Equation (19), P₃ can be calculated as shown in Equation (21):

21

Similarly, P₄ is defined as given in Equation (22):

22

The covariance matrix for the double-difference integer ambiguity can be expressed as follows:

23

Here, an approximation is made for the expression involving the Hatch filter coefficient for computational convenience.

Now, using the float solution derived from Equation (12) and the covariance matrix of the integer ambiguity calculated from Equation (23), the LAMBDA method is applied to determine the integer ambiguity easily. If the determined integer ambiguity passes the F-ratio test, it provides precise relative positioning. Additionally, it adapts to the continuously changing satellite constellation by using linear combinations of integer ambiguity. In this study, the LAMBDA method is applied by using the following strategies to enhance the success rate of integer ambiguity determination, addressing the limitations of single-frequency GPS measurements in LEO environments.

• To control the orbit of CubeSats for formation flying, it is important to maintain the common GPS visible satellites even if there are changes in attitude control. This approach is achieved using GPS antennas attached to the cross-track (Y-axis in the body frame). Both CubeSats need to maintain attitude control to ensure that the GPS antennas face the same direction. At distances closer than 1 m, RTK relative navigation is performed using a radial antenna (Z-axis) with electromagnetic forces.

• After the Hatch filter coefficients converge, a technique is applied to reduce the candidate pool of integer ambiguity from the DGPS module.

• Considering the space environment, noise modeling for GPS measurements assumes an uncorrelated normal distribution without using an elevation-dependent model (Bischoff et al., 2005; Rothacher et al., 1998).

• The GPS satellite with the highest elevation angle based on the antenna frame is selected as the reference satellite, as it is considered the best visible satellite.

• The LAMBDA method is applied in segments for which the ADOP is less than 0.12 to satisfy the probability of integer ambiguity determination exceeding 99.9% when there are more than nine visible satellites (Odijk & Teunissen, 2008; Teunissen & Odijk, 1997).

• An F-ratio test of 3 is applied in the LAMBDA method to conservatively enhance the success-fixed rate of integer ambiguity.

• To simplify the calculations, the LAMBDA method is used in each epoch, and the integer least-square method is limited to 1000 iterations to avoid infinite loops. If the maximum iteration is reached, it is considered that the integer ambiguity determination has failed, and the DGPS solution is provided. The integer ambiguity determination will be attempted again in the next epoch.

• If a cycle slip is detected for a certain GPS pseudorandom noise (PRN), the integer ambiguity is recalculated anew, treated as if it were a newly risen GPS PRN.

3.4 Cycle-Slip Detection Using Dynamic Characteristics

In an RTK relative navigation system, achieving precise centimeter-level positioning relies on carrier-phase measurements. This precision is contingent on an accurate determination of the integer ambiguity. However, temporary signal losses in the receiver can lead to a discontinuity in the carrier-phase measurement, known as a cycle slip. While GPS cycle slips are more common on the ground in urban environments or areas with obstacles such as trees, in the LEO environment, they can result from signal tracking loss caused by the rapid mobility of satellites. When cycle slips occur, there is a sudden increase in position error, which poses a significant challenge to the satellite’s mission. Therefore, detecting cycle slips is crucial for maintaining centimeter-level precision using RTK relative navigation.

The proposed relative navigation system in this study, which uses single-frequency GPS measurements, faces limitations in detecting cycle slips due to the restricted set of measurements. Moreover, because no additional sensors are used, there are constraints in correction capabilities. Acknowledging these limitations, this study presents a simple yet practical cycle-slip detection method suitable for the SNUGLITE-III CubeSat mission. To define the equation for relative dynamics between the two satellites, assuming they are very close (within 1 km), with a measurement update rate of 1 s and orbiting in a circular path, we can establish the well-known HCW equation in the local-vertical–local-horizontal (LVLH) coordinate system based on the chaser CubeSat, as expressed in Equation (24) (Sidi, 2014; Vallado & McClain, 2007). Here, n denotes the mean motion:

24

To analyze the dynamic characteristics of carrier-phase measurements between the two CubeSats, we defined the distance from GPS satellites to the chaser and target CubeSats using Equation (25). With the chaser CubeSat as the reference, the expression for the target can be formulated. The equation for the relative position can then be written in a form similar to the single-difference measurement:

25

The equation describing the acceleration for the relative distance between the two CubeSats from Equation (25) takes the same form as the HCW equation in Equation (24):

26

Based on the equation above, assuming a worst-case scenario for SNUGLITE-III with a maximum relative distance between satellites along the in-track direction of 1 km and a maximum relative velocity of 10 cm/s, the maximum acceleration for the relative distance between satellites is calculated to be on the order of 10⁻³ m/s². This implies that the acceleration term for the distance of carrier-phase measurements is negligible, given the very close distance between satellites. In other words, performing a second time differentiation of the carrier-phase measurements allows us to ignore the bias term. Thus, this study proposes a technique for detecting cycle slips by performing a second time differentiation of the DDCP measurements used in RTK, as defined in Equation (9). The resulting double-time double-difference carrier-phase (DTDDCP) measurements are expressed by Equation (27):

27

28

Here, refers to the difference in time between two epochs, whereas ε_ϕ represents the noise in the carrier-phase measurements. If no cycle slip occurs, the values in Equation (28) should remain below a certain threshold (T_λ). This threshold can be estimated based on the noise level present in the carrier-phase measurements. If we examine the time difference for the noise in the carrier phase at the k-th epoch, the time difference can be expressed as follows:

29

Assuming that the defined noise follows a normal distribution, is independent over time, and has no correlation, the noise for the double-time-difference carrier-phase and DTDDCP measurements can be defined as shown in Equation (30):

30

If we assume a standard deviation of 2 mm for the noise in the carrier-phase measurements, the noise in the DTDDCP measurements calculated from Equation (30) will be . Therefore, the cycle-slip algorithm proposed in this study sets the threshold at , which is significantly larger than the 3-sigma noise level of the DTDDCP measurements from Equation (28). These measurements allow the algorithm to detect half-cycle slips, considering the proximity of the SNUGLITE-III CubeSat orbits. Notably, this signifies that even with single-frequency GPS measurements, the algorithm is capable of precise cycle-slip detection.

3.5 Simulation Results Using a Software GPS Simulator

A software GPS simulator was used to evaluate the performance of the proposed method. The simulation analyzed the performance using 6 h of data, assuming a cross-track separation of SNUGLITE-III A/B, maintained by a spring force of 0.1 N, resulting in a relative distance of 0–100 m. This scenario was designed to verify the centimeter-level performance of the RTK relative navigation for the final rendezvous and docking mission. GPS measurements were obtained at 1-s intervals, assuming pseudorange and carrier noise of 1 m and 2 mm, respectively, following a normal distribution. The GPS antenna elevation mask was set to 5°, pointing in the cross-track direction (Y-axis).

The average number of common GPS satellites affecting the performance of RTK relative navigation was 9.2. For SNUGLITE-III CubeSat body coordinates, a skyplot for the GPS antenna attached to the Y-axis and the PRN of visible satellites over time are illustrated in Figure 5. The skyplot was referenced to the CubeSat’s body frame coordinate system, assuming that the CubeSat’s attitude control aligns with the LVLH frame. In this frame, the Z, X, and Y axes represent the radial, in-track, and cross-track directions, respectively. Therefore, visible satellites in the Y-direction (cross-track) of the GPS antenna showed trajectories rotating around the maximum elevation angle. Because of signal obstruction from the Earth, predictions were made concerning the quality of the DOP in the X-axis (in-track), Z-axis (radial), and Y-axis (cross-track) order. This pattern resembles the challenging DOP in the northern direction when ground users in mid-latitudes use GPS for positioning. The DGPS relative navigation module, which determines the range of potential candidates for integer ambiguity in CubeSat relative navigation, uses a Hatch filter coefficient of 100.

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FIGURE 5

LEO GPS simulation results for 6 h: (a) skyplot in the body frame when a GPS antenna is used in the +Y direction; (b) visible GPS PRNs

Graphs illustrating the error in relative positions over time are presented in Figures 6–7. First, Figure 6 compares the performance of DGPS and RTK relative navigation. Figure 6(a) displays the relative position errors along each axis in the LVLH frame, whereas Figure 6(b) shows the three-dimensional (3D) root mean square error (RMSE) for the DOP. From the figures, it is evident that RTK, using carrier-phase measurements, exhibits significantly superior performance compared with range-domain-based DGPS relative navigation. In RTK relative navigation, ambiguity resolution begins at approximately 236 s, when the number of satellites reaches nine. Additionally, by examining the relative errors along each axis in Figure 6(a), it can be observed that the performance in the cross-track direction is the least favorable. This approach is attributed to the GPS antenna being attached to the Y-axis in the body frame and the Z-axis in the LVLH frame, which is interpreted as a cross-track direction.

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FIGURE 6

Relative position error during LEO simulations in LVLH coordinates (a) for each axis and (b) with 3D normalization and DOP

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FIGURE 7

Results of enlarging the relative position error during LEO simulations in LVLH coordinates (a) for each axis and (b) with 3D normalization and the number of visible satellites (the red line represents the reference PRN)

While interpreted as vertical DOP (VDOP) in the LVLH frame, this DOP is better than horizontal DOP (HDOP) for the GPS antenna attached to the side of the CubeSat, as indicated in Figure 5. Furthermore, Figure 7 shows the results for examining the performance of RTK relative navigation. In Figure 7(a), RTK maintains centimeter-level accuracy along each axis, and in Figure 7(b), the RTK adapts to changes in the reference satellite very quickly. Therefore, if the integer ambiguity is fixed early, it poses no issues in providing precise relative positions. The relative positioning performance of DGPS and RTK resulted in 3D RMSE values of 0.21 m and 0.69 cm, respectively. Importantly, the process of resolving integer ambiguity resulted in a time to first fix (TTFF) of 1 s and a success-fix rate of 100%. This indicates that the performance was favorable because of the higher number of visible GPS satellites available, which led to a better availability of ADOP compared with ground-based users.

To assess the effectiveness of the proposed cycle-slip detection method, Figure 8 illustrates the threshold set for detecting half-cycle slips using DTDDCP measurements over time. As depicted in the figure, the proximity of the two satellites ensures that the influence of distance changes over time does not result in bias. The RMSE value, measured at 9.8 mm, aligns with the theoretically calculated value from Equation (30), confirming the feasibility of cycle-slip detection using single-frequency measurements. Here, measurements exceeding the threshold are considered as cycle-slip detections for the respective PRNs, leading to a recalculation of the integer ambiguities. While cycle slips were not accounted for in the software simulator, the standard deviation of the carrier-phase measurements, which were assumed to have a normal distribution, was set at 2 mm. This resulted in noise exceeding the threshold predicted by Equation (30) with a 6-sigma probability (0.000000197%).

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FIGURE 8

DTDDCP measurements distributed within the threshold set by a half-cycle slip

4.1 Ground Plane Design for the Patch Antenna

Unlike the high-cost GPS antennas commonly used for RTK on the ground, GPS antennas deployed on CubeSats are low-cost, compact patch antennas. Recent advancements in microstrip antenna design technology have significantly progressed, leading to the development of size-reduction techniques for patch antennas. These advancements enhance microstrip antenna characteristics, including gain, bandwidth, and radiation pattern, thereby benefiting from various compact antenna structures (Bhattacharyya, 1990).

Consequently, the patch antenna, especially in a 3U platform such as the SNUGLITE-III CubeSat, emerges as the most suitable option to overcome spatial limitations. The selected GPS antenna is a COTS patch antenna designed for gain, bandwidth, and radiation pattern. When integrating an antenna into a CubeSat, the system engineer must consider the antenna’s ground plane. As is well known, the ground plane design significantly influences patch antenna performance (Khan et al., 2015).

For the antenna used in the SNUGLITE-III CubeSat, as outlined in Table 2, a 10-cm-diameter circular ground plane under optimal conditions is recommended by the manufacturer. However, the size limitation of the 3U CubeSat platform, with a maximum diameter of 82.6 mm for the solar panel to which the antenna is attached, makes it impractical to adopt the recommended ground plane size. To prevent degradation of the GPS patch antenna performance during relative navigation, the ground plane must be optimally positioned without affecting the internal component layout. Additionally, because the ground plane is not designed according to the manufacturer’s recommendations, its performance must be verified through residual error measurements.

In this study, we fabricated a ground plane for the SNUGLITE-III CubeSat platform using a printed circuit board (PCB). To address the impact of the ground plane design, a ground plane with the same dimensions as the solar panels of the 3U CubeSat was fabricated using a PCB and covered with ground copper (2 ounce). The PCB consists of four layers. The top and bottom layers are used for attaching solar cells and connecting connectors, while the two inner layers are composed of ground copper. The antenna mounted on the fabricated PCB is shown in Figure 9. For the flight model, the ground plane is the same as that in the test model, except that solar cells are attached in the space and holes are added for connectors and a star tracker.

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FIGURE 9

Design of the PCB-based ground plane of the GPS patch antenna: engineering model (above) and flight model (below)

The antenna pattern was analyzed in advance in an anechoic chamber to confirm the impact of the proposed PCB-based ground plane. The anechoic chamber is a space shielded from external signals and noise, allowing for accurate antenna performance measurements without environmental interference. Notably, the proposed PCB-based ground plane features an elongated rectangular model rather than a uniform shape centered around the antenna, necessitating an evaluation of its directivity. Antenna patterns are compared before and after the ground plane installation for the GPS L1 signal (1575.42 MHz). The antenna mounted in the anechoic chamber is depicted in Figure 10(a), and the defined frame is presented in Figure 10(b). Measurements were taken at 5° intervals for elevation (θ) and azimuth (ϕ). Because this is an active antenna, the measured antenna pattern includes the gain provided by the low-noise amplifier caused by the external power supply. Unlike typical passive antennas, this analysis focuses solely on changes in the pattern shape for the COTS antenna.

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FIGURE 10

Anechoic chamber setup for measuring the antenna pattern with a PCB-based ground plane: (a) antenna mounted in a chamber; (b) axis definition for the antenna pattern, where the Z-direction represents the main lobe

Figures 11 and 12 display the measured antenna patterns for right-handed circular polarization (RHCP) and left-handed circular polarization (LHCP) in the main and back lobes, respectively, before and after the ground plane installation, in 3D. The measured average gains are presented in Table 5. Initially, in Figure 11, the RHCP antenna pattern appears distorted because of the nonuniform shape of the ground plane. Yet, there is an overall improvement of 6.2% in the gain of the main lobe. The back lobe exhibits no significant improvement. The peak gain was 23.58 dBic without the ground plane at an elevation of 100° and azimuth of 80°. With the ground plane, the peak gain improved to 24.95 dBic at an elevation of 95° and azimuth of 75°, indicating an enhanced directivity towards the zenith.

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FIGURE 11

Comparison of 3D antenna patterns with and without the ground plane for the main and back lobes (RHCP)

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FIGURE 12

Comparison of 3D antenna patterns with and without the ground plane for the main and back lobes (LHCP)

However, Figure 12 demonstrates a substantial decrease of 23.4% in gain in the back lobe for LHCP, indicating a reduced impact from LHCP signals reflected into the back lobe from GPS signals. This result suggests that installing the ground plane enhances performance, particularly against multipath effects reflected as LHCP. Notably, research in the field of GNSS reflectometry has shown that GPS signals reflected from soil are LHCP (Edokossi et al., 2020; Hong et al., 2022; Jia & Pei, 2018; Katzberg et al., 2006; Rodriguez-Alvarez et al., 2023; X. Wu et al., 2021). Given this, the results imply that installing the ground plane can reduce multipath effects in open-sky environments, particularly in the context of ground experiments. Although the directivity and uniformity of the pattern indicate significant degradation compared with high-end antennas (Rykała et al., 2023; Sun et al., 2023; Zhang & Schwieger, 2018), such as those with choke rings, these issues are considered inherent limitations of low-cost patch antennas and are attributed to the rectangular shape of the ground plane.

In addition, based on the antenna pattern, if patch antennas are utilized to validate the performance of the proposed relative navigation system in a ground environment, it can be predicted that the performance of residual error measurements will degrade owing to signals reflected from the back, namely multipath effects. This is particularly true because multipath significantly affects pseudorange, inevitably leading to a degradation in DGPS performance. This issue inevitably arises when patch antennas are utilized, as the maximum diameter of a 3U CubeSat is limited to 10 cm. However, in space environments, unlike on the ground, the signals that are reflected and enter originate from the satellite body itself. Yet, because the entire satellite body is shielded as ground, better performance can be expected compared with the multipath effects of ground environments.

4.2 Experimental Environment and Setup

Considering the limitations that arise in equipping CubeSats with redundancy, the recommendation for enhancing mission success rates centers around rigorous testing (Bouwmeester et al., 2022). It is crucial to validate the performance of the developed RTK-based relative navigation algorithm using actual measurements for seamless integration into the SNUGLITE-III CubeSat. The algorithm was developed by using an LEO GPS simulator, which makes it difficult to replicate the same conditions on the ground. Additionally, the simulations generated idealized measurements assuming a normal distribution, making it challenging to reflect issues arising from actual hardware.

When comparing ground and LEO environments, the ground environment simulates harsher conditions due to error factors such as tropospheric, ionospheric, and multipath errors. This condition is compounded by fewer visible GPS satellites due to terrain and obstacles, especially for a single-frequency DGPS algorithm in which pseudorange measurements are used. To minimize these error factors and maximize satellite visibility in the ground environment, the proposed approach involved very short baseline data collection between the target and chaser CubeSats in an open-sky setting.

The relative navigation algorithm for CubeSats was validated in the proposed environment, and data were collected for 2 h. The experimental setup is depicted in Figure 13, and a detailed description of the experimental environment, along with photographs, is presented in Figure 14 and Table 6. In this experiment, we restricted the baselines to less than 1 m to minimize the relative errors caused by the troposphere and ionosphere. This restriction was imposed because of the short formation flying distance of the SNUGLITE-III CubeSat (maximum of 1 km) and the assumption that the impact of the ionosphere was negligible for baselines within 10 km in actual LEO environments (Montenbruck et al., 2011). A short baseline was selected to minimize the impact of tropospheric errors, which do not occur in space, in the evaluation of the algorithm’s performance. Furthermore, in comparing between the skyplot generated by the LEO GPS simulator in Figure 5(a) and the ground environment illustrated in Figure 14(c), attention was focused on regions in which satellite visibility is hindered by the Earth’s orientation and the reduced number of satellites in the northern direction on the ground, influenced by the arrangement of the GPS constellation.

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FIGURE 13

Equipment settings for data collection

The same GPS signal can be received by connecting the high-cost receiver to the GPS patch antenna and the CubeSat receiver with a splitter.

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FIGURE 14

Experimental setup at the time of data collection: (a) GPS receivers and monitor PC; (b) GPS patch antennas and reference station settings; (c) skyplot in the body frame for 2 h in the ground environment

Data collection was conducted in three phases within the same environment. The first phase assessed the impact of the ground plane, specifically examining the influence of the presence or absence of the ground plane on signal strength. Upon evaluating the ground plane’s impact, we performed a repeat experiment in the same location. The experiments used reliable commercial high-cost GPS receivers and antennas to compare the hardware performance limitations associated with the CubeSat GPS receiver’s patch antennas. In the second phase, a splitter connected high-cost and CubeSat GPS receivers to the GPS patch antenna. The performance of the GPS receivers was evaluated by examining the residual error measurements. Because of the uneven distribution of the ground plane for the antenna, the impact of a nonuniform ground plane was assessed by rotating the PCB 180°, as shown in Figure 13. The verified performance of the algorithm served as a reference for comparing the performance of CubeSat GPS receivers and patch antennas installed on the SNUGLITE-III CubeSat. Finally, based on the collected measurements, the third phase evaluated the performance of relative navigation. This study further explores the performance degradation caused by GPS patch antennas mounted on a 3U CubeSat using both commercial high-cost and CubeSat GPS receivers. The hardware characteristics of the GPS receivers and patch antennas were analyzed by comparing the performance of high-cost and actual space GPS receivers for the same antennas. Additionally, the RTK relative navigation results were compared. All true trajectories were generated using commercial software, including compensations for the antenna’s phase center variation and offset.

4.3 Analysis of Experimental Results

4.3.1 Effect of the Ground Plane

Experiments on signal strength were conducted with actual measurements to validate the impact of the PCB-based ground plane, which was predicted from the antenna pattern. This testing was conducted to assess how the presence or absence of the ground plane affects a high-cost GPS receiver, as depicted in Figure 15(a). In the figure, the red line presents the signal received without a ground plane, the blue line illustrates the signal strength obtained when using the PCB-based ground plane, and the yellow line shows results from the reference for comparison.

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FIGURE 15

Performance evaluation results of GPS measurements with a GPS patch antenna (yellow presents results for a high-cost GPS receiver and antenna as a reference): (a) carrier-to-noise ratio (C/N0) for satellite elevation in the presence or absence of a PCB-based ground plane; (b) DDPR residual error; (c) DDCP residual error

The figure shows that the signal strength is notably unstable when there is no ground plane. In contrast, when the ground plane is attached, an improvement in signal strength is evident. For the case in which a ground plane is included, the results are similar to the reference results near low elevations. For high elevations, the signal strength is maintained at or above 40 dB-Hz, indicating that the low-elevation region with lower signal strength is a more critical factor for improving signal quality. This instability in signal strength due to multipath effects in ground environments is confirmed by the signal gain observed in the antenna pattern, as discussed in Section 4.1.

Subsequently, the measurement residual errors for GPS receiver–antenna pairs over a very short baseline were examined, as depicted in Figures 15(b) and 15(c). The red line presents results for the CubeSat GPS receivers, and the blue line presents results for the high-cost GPS receivers, with both receivers connected to the patch antennas with a ground plane. The yellow line presents data obtained from the high-cost commercial GPS receivers and antennas for comparison. An initial examination of the double-difference pseudorange (DDPR) residual errors in Figure 15(b) revealed that the performance degraded in the order of the CubeSat GPS receiver, high-cost GPS receiver, and reference. The precision of the CubeSat GPS receiver’s pseudorange is influenced by the hardware of the receiver clock, as it was equipped with a temperature-compensated crystal oscillator. In contrast, the high-cost GPS receiver and reference, which support clock steering functionality, showed improved results. Because an elevation-dependent model was not used for relative navigation and all GPS measurements were assumed to follow the same normal distribution, the residual error statistics for the entire set of DDPR measurements showed RMSE values of 1.32, 0.67, and 0.41 m for the CubeSat, high-cost, and reference GPS receivers, respectively, indicating that even when a patch antenna is used, the high-cost receiver demonstrated satisfactory quality, while the CubeSat receiver exhibited a noise level higher than that of the general reference.

In Figure 15(c), which depicts the results for DDCP measurements, it is evident that the impact of the patch antenna is more significant than that of receivers for carrier-phase measurements. Similarly, assuming that all GPS measurements follow a normal distribution, the overall residual error statistics for DDCP measurements showed RMSE levels of 8.67, 8.38, and 4.13 mm for the CubeSat, high-cost, and reference receivers, respectively. This signifies that the patch antenna has degraded the quality of the measurements, as evidenced by the noise level in the DDCP measurements for the reference GPS receiver.

4.3.2 Results of Single-Frequency GPS Relative Navigation

The single-frequency relative navigation proposed in this study was validated and analyzed for its performance based on the measurements discussed in Section 4.3.1. The experimental results for the DGPS module are presented in Figure 16, and the performance metrics are shown in Table 7. In the figure, the red line presents results for the CubeSat GPS receiver, and the blue line presents results for the high-cost receiver, both of which use the GPS patch antenna. The performance of the DGPS module yields 3D RMSE levels of 1.80 and 1.10 m, respectively. The better performance of the high-cost receiver can be attributed to hardware differences in the receivers, as indicated by the residual errors in the pseudorange measurements of the CubeSat GPS receiver, as discussed in Section 4.3.1.

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FIGURE 16

DGPS relative position error during the experiment in LVLH coordinates (a) for each axis and (b) with 3D normalization and DOP

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TABLE 7

RMSE for DGPS Relative Position in LVLH Coordinates

The next step involves analyzing the performance of the RTK relative navigation module based on the DGPS module, as illustrated in Figure 17, with the performance metrics presented in Table 8. Using the same approach as in Section 4.3.1, which involved analyzing residual errors in DDCP measurements, we found that the trend of carrier-phase noise was consistent between the two receivers. This result indicates that the relative positions derived from the RTK relative navigation module for the CubeSat and high-cost receivers are similar, with values of 1.55 cm and 1.49 cm, respectively. Both the DGPS and RTK relative navigation modules exhibit an increase in errors at approximately 1.2 h, which is attributed to a decrease in the number of satellites (increased position DOP [PDOP]), with only six visible satellites. As shown by the LEO simulation in Figure 7(b), a minimum of seven visible satellites is expected in space conditions, potentially resulting in improved relative navigation performance compared with the ground environment.

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FIGURE 17

RTK relative position error during the experiment in LVLH coordinates (a) for each axis and (b) with 3D normalization and the number of visible satellites (red and blue lines present the number of satellites; yellow and green lines present the reference PRN)

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TABLE 8

RMSE for RTK Relative Position in LVLH Coordinates

The hardware characteristics influenced by the algorithm proposed in this study, specifically the pseudorange and carrier-phase residual errors caused by the CubeSat GPS receiver and the GPS patch antenna, increased. However, the required performance of 2 cm for RTK relative navigation was achieved. Nevertheless, the performance degradation of DGPS due to the influence of pseudorange highlights the limitations of the single-frequency RTK technique, particularly in reducing the candidate pool for integer ambiguity resolution. This limitation was evident in the simulation phase, where considering the special conditions of space environments and enhancing ADOP conditions by securing more than nine visible satellites proved sufficient to overcome this limitation. The performance of the undifferentiated integer ambiguity resolution in ground environments under these conditions is summarized in Table 9. Even when actual measurements were used, a 100% success-fix rate was confirmed. However, there is a substantial time difference in the TTFF, at approximately 20 s for CubeSat GPS receivers and 1.9 s for high-cost receivers, which can be attributed to the high F-ratio test of 3 in LAMBDA and the impact of reduced precision in float solutions due to degraded DGPS performance. Despite these challenges, the algorithm proposed in this study demonstrated an efficient execution of centimeter-level precision relative navigation using only single-frequency measurements without additional sensors or filters for integer ambiguity resolution. Moreover, the algorithm is applicable to the CubeSat platform, even in challenging scenarios with hardware limitations.

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TABLE 9

Performance of Integer Ambiguity Determination

The results of cycle-slip detection for each receiver, using the algorithm proposed in Section 3.4, are shown in Figure 18. As shown in the figure, the CubeSat GPS receiver experienced cycle slips, which were successfully detected and used to exclude the affected satellites. In contrast, the high-cost receiver showed stable operation without significant cycle slips. The CubeSat receiver exhibited relatively frequent cycle slips, which were attributed to the rapid mobility required for Doppler tracking in the GPS satellite tracking algorithm. Even when the algorithm proposed in this study was used with actual measurements, the cycle-slip detection was successful. Additionally, the residual errors of the DTDDCP measurements were approximately 7.2 and 5.7 mm for each receiver. Based on Equation (30), the estimated noise levels for carrier-phase measurements for each receiver were approximately 1.5 and 1.2 mm, which aligns with the typical range of noise in GPS carrier-phase measurements (Cai et al., 2016; Nistor & Buda, 2016).

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FIGURE 18

DTDDCP measurements for cycle-slip detection for (a) the CubeSat GPS receiver and (b) the high-cost GPS receiver