Lever arm measurement precision and its impact on exterior orientation parameters in GNSS/IMU integration

2.1 Study area

The aerial mobile mapping dataset was acquired by Lantmäteriet, The Swedish mapping, Cadastral, and land registration authority, on August 31, 2021, in the vicinity of Gävle, Sweden. The data collection process involved 6 strips and 86 aerial photographs captured at a flight height of 2,000 meters. The TerrainMapper 2 system (Leica geosystems 2023), equipped with an IMU (SPAN CNUS5-H, Class 5) operating at a 500 Hz data rate and a GNSS (NovAtel SPAN OEM7) operating at a 10 Hz data rate (but the GNSS data output extracted with 1 Hz), was utilized for data acquisition. The positional root mean square (RMS) error for horizontal measurements is better than 3−5 cm, while for vertical measurements, it is better than 5–7 cm. The attitude RMS errors were within 0.005° for roll and pitch and 0.008° for heading. To establish a reference station, a VRS was employed, which was calculated using the SWEPOS service, which provides virtual RINEX data from the national network of permanent GNSS reference stations (continuously operating reference station) maintained by Lantmäteriet (2023) at a data rate of 1 Hz. To utilize the VRS in the Inertial Explorer software, the data rate needed to be interpolated to 0.5 Hz, and data captured 2 h before and after the flight mission were required for the VRS calculation process. Figure 1 illustrates the aerial mobile mapping trajectory and VRS location in the project area.

Figure 1

Illustration of aerial mobile mapping trajectory and the location of VRS in the project area.

2.2 LA measurements

The determination and computation of the distance from the installed GNSS antenna (Trimble AV39 antenna) on the aircraft’s ceiling to the IMU within the Leica TerrainMapper 2 airborne mapping system are essential for accurate mapping and geospatial analysis. To accomplish this, a total station known as the Leica TS16 and a prism called the Leica GMP111 Miniprism were utilized for the measurement of ten specific points. These points included four on the GNSS antenna, four on the screws surrounding the TerrainMapper 2, and two points on the nose and tail of the aircraft (Figure 2). The measurements were gathered on August 31, 2021, at Borlänge Dala Airport. All the measurements and calculations were conducted with Lantmäteriet.

Figure 2

A schematic illustration of measured points on the GNSS antenna, airborne mobile mapping system, and aircraft (the image is not to scale).

To determine the center position of the GNSS antenna, four reference points were utilized. These encompassed three points on the screws (top left (G1), top right (G2), and bottom left (G3)) and an additional point on the bottom left element of the “m” character within the Trimble mark situated atop the GNSS antenna, denoted as the G-Center point. This precise location was identified as the accurate horizontal midpoint of the antenna. The surveying of three points on the screw positions on the antenna served the purpose of conducting a quality check by comparing the surveyed bolts with the antenna drawing.

The determination of the M-center, the central point of the circular plane in the TerrainMapper 2 system, involved the measurement of four points labeled M1, M2, M3, and M4. The precise coordinates of these points were measured with a total station and utilized to formulate a system of nonlinear equations, taking into account equidistant conditions (see the Supplementary material), which reflects the geometric relationship between the points and the M-center. The system was numerically solved using MATLAB code, resulting in the precise X and Y coordinates of the M-center. The calculated coordinates were then employed to establish the virtual central point in a localized coordinate frame. M5 is also measured on a screw which represents the top of the gimbal, which is utilized in the dZ value calculation of LA in equation (3). This virtual center was further adjusted by a specified value provided by the Leica company to align with the center of the IMU. Additionally, to establish the orientation of the X-axis within a localized coordinate frame where the Y-axis is positioned at a right angle to the X-axis following a right-handed configuration, two points were assessed on both ends of the aircraft (referred to as A-front and A-back points) and Z-axis pointing down. The specific measurements’ coordinates are outlined in Table 1.

Then the offset between the GNSS antenna center and the virtual center of TerrainMapper 2 is calculated as follows.

where h ₁, h ₂, and d are defined as the height of the antenna, the height of the L ₁ center, and the depth of the virtual center, respectively.

The parameter h ₂ is ascertained from the antenna database (DB) for antenna type (AERAT1675_180+NONE) accessible at Antenna Calibrations (n.d.), representing values derived from authoritative geodetic sources to ensure precision in antenna specifications. Furthermore, the parameter d = 0.071m is acquired directly from the Leica Company, either through verbal communication or written correspondence, providing specific details about its measurement or calibration processes.

The final LA between the IMU sensor and GNSS antenna after taking the offset between the virtual center and IMU sensor center is calculated as −0.057, −0.213, and 1.289 m in the X, Y, and Z directions, respectively.

2.3 LA integration and uncertainty analysis in GNSS/IMU processing

The LA measurements, representing the vector distance between the GNSS antenna and the IMU center, are critical in the GNSS/IMU integration process. These measurements are incorporated as an initial condition in the Inertial Explorer software, translating the GNSS antenna position to the IMU center for accurate alignment of the two sensor data streams. This integration is crucial for the precise estimation of EOPs (Montalbano and Humphreys 2018, Stovner and Johansen 2019).

The mathematical relationship between LA uncertainty and position/orientation estimation can be described as follows:

where P GNSS is the position of the GNSS antenna, P IMU is the position of the IMU, R is the rotation matrix representing the orientation of the IMU, and LA is the LA vector.

The rotation matrix R can be expressed as

where ϕ , θ , and ψ represent roll, pitch, and yaw angles, respectively.

The uncertainty in the LA propagates to the position estimation through the rotation matrix R . If the LA uncertainty is denoted as δ LA , the perturbed position δ P GNSS can be written as

This introduces an additional error term δ P , which affects the overall position accuracy.

For orientation estimation, the LA uncertainty affects the angular measurements. The orientation Θ can be influenced by errors in the LA through the following relationship:

where f is a function that relates the LA to the orientation parameters. The function f can be expressed as

where R i j are the elements of the rotation matrix R .

The uncertainty in the LA, δ L A , affects the orientation estimation as

This results in an error term δ Θ , which impacts the orientation accuracy (Fu et al. 2018, Samiei et al. 2024).

To simulate various levels of LA measurement precision, we assign different uncertainty values (2 mm, 1 cm, 5 cm, 10 cm, and 20 cm) to the LA vector components. These uncertainties are introduced as standard deviations in the Inertial Explorer software’s configuration, effectively weighting the LA measurements in the integration algorithm.

By varying the LA uncertainties, we effectively change the weight given to the LA measurements in the GNSS/IMU integration process. Higher uncertainties result in less weight being given to the LA measurements, potentially allowing other measurements to have more influence on the final solution. This approach enables us to systematically evaluate how different levels of LA measurement precision affect the estimation of EOPs, providing insights into the sensitivity of the integration process to LA uncertainties (Possolo et al. 2024, Tate and Plebani 2016).

2.4 Data and EOP processing strategy

In this simulation study, we tested the impact of assigning different LA uncertainties (different weights). We utilized a tightly coupled integration of differential GNSS/IMU to conduct measurements in both forward and reverse directions. Forward processing involves the integration of data from a known starting point, typically the initial position or a previously established position fix, progressing in the direction of motion. This method is commonly employed to estimate the trajectory or path of a moving object based on the accumulated sensor measurements. Conversely, reverse processing entails the integration of data in backward chronological order, starting from the end of the data sequence and working reverse toward the initial point. This approach is often utilized to refine and improve the accuracy of a trajectory by iteratively adjusting the initial conditions or constraints based on observations made toward the end of the data collection period.

The LA was determined by varying the uncertainties at different levels in this study, specifically 2 mm, 1 cm, 5 cm, 10 cm, and 20 cm. However, we also consider applying the LA without weighting in the calculation. Our method involved integrating criteria including a 12° elevation mask, an L1 lock time cutoff of 4 s (if the lock time falls below this threshold, the receiver might disregard the data from that satellite because the signal is considered unreliable), and maximum ranges of 4 km for single frequency and 30 km for dual frequencies GNSS observations. The Inertial Explorer software was utilized to integrate GNSS and IMU data, enabling the smooth merging of information from both sources (Novatel 2023). A VRS served as a reliable GNSS reference station, with both the rover and VRS being capable of capturing signals from GPS and GLONASS satellites. During the integration processing of GNSS/IMU, the SD values used were 1 m for code, 0.1 m for Doppler, and 0.035 m for the GNSS carrier phase observations. Ambiguities were solved with good condition (fixed) for all different LA SD.

2.5 Alignment period determination

In this section, we focus on determining the alignment period in a GNSS/IMU tightly coupled integration for an airborne mobile mapping system. This is a crucial step before studying the impact of LA uncertainties on EOPs. The Allan variance method was employed to determine the optimal alignment period for gyroscopes and other inertial sensors. This method is particularly effective in identifying different types of sensor errors, such as bias instability and random walks. The process involved calculating the sampling frequency derived from GPS time data, establishing distinct time intervals, and computing the mean angular rate over each interval. These calculated means were then compared against the initial interval mean, with differences noted and stored. The variance of these differences was subsequently computed and documented. The Allan variance, denoted by σ 2 ( τ ) , is defined for a given time interval τ as follows (Freescale 2015, Siraya 2020):

where M is the number of non-overlapping intervals of length τ , and X ¯ i is the average of the samples within the i -th interval.

3.1 Alignment period

Allan variance analysis was performed on the angular rate data across the X, Y, and Z axes in both forward and reverse directions. Figure 3 illustrates the Allan variance curves plotted against GPS time for each axis. Unique characteristics are exhibited by each curve over time, offering insights into sensor performance and error contributions at varied averaging times. Notably, the system’s stability, or alignment, is observed at the beginning and end of the data. In the forward direction, the system achieves alignment in 901 s, while in the reverse direction, it takes 225 s for the system to become aligned. The alignment period threshold is indicated by a dotted red vertical line at approximately 201,463 and 203,939 s. These trends and patterns observed in the Allan variance curves are instrumental in determining the optimal alignment period. This alignment period is closely related to the stability of the sensors, thus providing a reliable measure of the performance of the GNSS/IMU tightly coupled integration in the airborne mobile mapping system.

Figure 3

Allan variance analysis of angular rates for optimal alignment period determination.

3.2 Impact of LA uncertainties on EOP estimation

The accuracy of the proposed LA measurements and calculations was assessed by evaluating the SD of the EOPs, which includes both attitude SD and position SD, after assigning different LA uncertainties. The results indicated that the attitude SD remained consistent across all different LA uncertainties.

As illustrated in Figure 4, the SD of attitude parameters (roll, pitch, and heading) is examined under the assumption of an LA without a predetermined uncertainty. Despite testing scenarios with different uncertainties for the LA, the results remained consistent, possibly due to the inherent characteristics of the system and the nature of the measurements. As a result, we decided not to present plots for the other scenarios (different LA uncertainties). The LA, which is the distance between the IMU and the GNSS antenna, is a key factor. The IMU is the primary determinant of the attitude parameters, making them largely unaffected by different LA uncertainties. Interestingly, these attitude parameters are less influenced by changes in the uncertainties of the LA than the position SD. This could potentially be due to the accurate measurement and calculation of the LA (see Section 2.2). Even the influence on position SD was not significant, with changes of about 1 or 2 mm observed in some parts of the data. Thus, it is hypothesized that changes in the uncertainties of an inaccurately measured LA might have the potential to affect the SD of the attitude parameters. However, this remains largely speculative and requires further investigation.

Figure 4

Roll, pitch, and heading SD for LA without assigning SD, demonstrating the independence of system attitude from LA value (the red dash line shows the alignment period).

High sensor noise or poorly calibrated sensors can introduce dominant errors in attitude estimation, overshadowing the impact of LA errors. Consequently, differences in LA uncertainties might not noticeably affect the attitude SD, as the estimation errors are primarily driven by the data quality rather than LA errors. This suggests a masking effect of data quality on the influence of LA uncertainties on attitude SD (Stovner and Johansen 2019).

In addition, the position SD (only for LA without assigning) was evaluated by considering both horizontal SD and vertical SD, alongside the horizontal dilution of precision (HDOP) and vertical dilution of precision (VDOP), respectively (refer to Figures 5 and 6). The horizontal and vertical SDs of the EOPs remained almost consistent (with 1 or 2 mm differences) for all different LA uncertainties. However, it was observed that the horizontal and vertical SD were influenced more by the geometric distribution of satellites in view during the data capturing period (HDOP and VDOP) rather than by the assignment of different LA uncertainties.

Figure 5

Horizontal SD and HDOP for LA without assigning SD, demonstrating the influence of satellite geometry over horizontal SD.

Figure 6

Vertical SD and VDOP for LA without assigning SD, demonstrating the influence of satellite geometry over vertical SD.

The accuracy of the LA significantly influences the precision of the final EOPs. Providing incorrect LA parameters can lead to a degradation in the data outputs for position, velocity, and orientation. Therefore, maintaining the accuracy of the LA is paramount for optimal system performance (Zhang et al. 2020a).

Interestingly, when considering all the EOPs’ (attitudes and position) SD for different LA uncertainties, no differences were observed between the various LA uncertainties (Figures 4–6). This indicates that even if the LA uncertainties vary, the EOPs’ SD remains constant for attitude and almost constant with 2 mm differences in some parts of the data for position. This finding underscores that the LA values are accurately measured and calculated with our previously proposed method in Section 2.2.

3.3 LA accuracy in GNSS/IMU integration: An analysis of attitude and position separations

To investigate the impact of the accuracy of the measured LA on the final EOPs, a series of experiments were conducted using GNSS/IMU integration processing. A measured LA (X = −0.057, Y = −0.213, Z = 1.289 m) was employed with different uncertainties, including 2 mm, 1 cm, 5 cm, 10 cm, 20 cm, and without assigning an SD. The influence of these different LA uncertainties on the accuracy of the EOPs was analyzed through the examination of forward and reverse attitude separation plots (Figures 7–9). If the separation exceeded ±0.1 arcmin (0.0016°), it indicated the possibility of an incorrect LA assignment. Conversely, the forward and reverse position separation plots (Figures 10–12), with a threshold of ±10 cm, showed satisfactory results, indicating an acceptable level of accuracy (Grumer 2021). The thresholds for attitude separation and position separation are the result of experimentation and have been determined through personal communication with Lantmäteriet.

Figure 7

Forward and reverse roll separation plots with different SDs of the LA.

Figure 8

Forward and reverse pitch separation plots with different SDs of the LA.

Figure 9

Forward and reverse heading separation plots with different SDs of the LA.

Figure 10

Forward and reverse east separation plots with different SDs of the LA.

Figure 11

Forward and reverse north separation plots with different SDs of the LA.

Figure 12

Forward and reverse height separation plots with different SDs of the LA vs VDOP, using a 6-degree polynomial fit.

The integration of GNSS and IMU systems poses a significant challenge in determining the alignment period, which corresponds to the time required to solve the initial integration constants. Analysis of attitude and position separation plots revealed alignment periods occurring at the beginning and end of the dataset. Specifically, the alignment period extended from the commencement of data capturing until 201,463 s, and from 203,939 s until the end of the dataset. These alignment periods are visually represented by two vertical dashed dark gray lines in the plots. Furthermore, horizontal dark gray dashed lines were included to denote the ±0.1 arcmin range for attitude forward–reverse separation and the ±10 cm range for position forward-reverse separation. In both attitude and position separation plots, it is crucial for the differences between forward and reverse processing to approach zero, indicating ideal data processing.

The analysis of roll (Figure 7), pitch (Figure 8), and heading (Figure 9) forward–reverse separation plots provided valuable insights into the attitude separation for different uncertainties. Remarkably, all attitude separations remained within the range of the predefined threshold, i.e., ±0.1 arcmin. This finding suggests that the measurement and calculation of the LA before GNSS/IMU post-processing were accurately performed. Even when altering the uncertainties of the LA, ranging from 2 mm to 20 cm, the attitudes consistently remained within the threshold. This demonstrates the robustness and reliability of the LA estimation procedure in maintaining accurate attitude measurements throughout the data processing.

By consistently achieving position separations within the ±10 cm range in our analysis of the East (Figure 10), North (Figure 11), and Height (Figure 12) plots, regardless of the LA uncertainties, we affirm the accuracy and precision of the GNSS/IMU LA measurements. Even deliberate differences in the LA uncertainties, ranging from 2 mm to 20 cm, did not exceed the position separations beyond the acceptable threshold, underscoring the reliability of the LA estimation method.

As we pursued an ideal scenario where position separation plots remain close to zero, an intriguing trend emerged. The increase in LA uncertainties, representing heightened measurement uncertainty, widened the range of change in position separation diagrams. This phenomenon highlights the robustness of conventional surveying methods in providing accurate LA measurements, but it also points to the fact that a higher uncertainty introduces variability, leading to noticeable deviations from the expected zero value (less difference from the reference value, i.e., without SD of LA) in position separations.

3.3.2 Assessing position misclosure and LA accuracy impact

The position misclosure was examined to evaluate the positional difference between the results derived from IMU and the GNSS. A significant disparity greater than the 5 mm threshold (this threshold is the result of experimentation and has been determined through personal communication with Lantmäteriet) in the East and North directions may suggest inaccurate LA values (Grumer 2021).

The position misclosure plot serves as a useful tool for assessing the accuracy of the GNSS/IMU solution and determining the stability of the IMU. Any sudden large jumps or spikes in the plot may indicate an unreliable IMU solution, whereas minimal separations approaching zero confirm the reliability of the GPS solution (NovAtel 2020).

During the data processing, for both the different uncertainties of the LA and particularly without the LA uncertainties, most of the East and North values remained within the range of ±5 mm. However, there were occasional spikes that exceeded this threshold. These results demonstrate that the LA was accurately measured to a sufficient degree for the tightly coupled GNSS/IMU integration. Figure 13 illustrates the position misclosure for GNSS/IMU integration without LA uncertainties. The other scenarios were not elaborated upon in this article due to the similarity of the results.

Figure 13

Position misclosure for GNSS/IMU integration without assigning LA uncertainties.

3.3.3 VRS-rover distance impact on position and attitude separations

It is noteworthy to highlight that the baseline of the base-rover station holds significant importance in assessing the accuracy of the calculated LA. Figure 14 illustrates the correlation between East, North, and height separation diagrams and the horizontal distance between the VRS and rover station (aircraft). When the VRS-rover station horizontal distance is less than 5 km, the position separation diagrams exhibit a stable behavior across different LA uncertainties. For VRS-rover distances below 5 km, an increase in LA uncertainties results in an escalation of East and North separation. However, for VRS-rover distances exceeding 5 km, the east and north separation diagrams display an unstable behavior. In contrast, the behavior of the height separation diagram varies from that of the east and north separation. As discussed in the previous section, the height separation diagrams adhere to the VDOP (Figure 13).

Figure 14

Correlation between position separation diagrams and VRS-rover horizontal distance.

The results presented above indicate that the GNSS/IMU integration algorithm excels in determining a more accurate estimation of the LA when a shorter baseline between the base and rover station is applied.

In parallel with position separation, the diagrams depicting attitude separation, as illustrated in Figure 15, exhibit increased stability when VRS-rover horizontal distances are less than 5 km. Nevertheless, when the VRS-rover horizontal distance exceeds 5 km, the attitude separation diagram displays slight fluctuations. This observation underscores the enhanced precision in LA estimation achieved with shorter baselines between the base and rover stations.

Figure 15

Correlation between attitude separation diagrams and VRS-rover horizontal distance.