Using robotic total station networks for 360° prism deviation reduction
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Gereon Tombrink
,
Lasse Klingbeil
Abstract
The accurate generation of reference trajectories is essential for the empirical evaluation of navigation sensors. Systematic deviations of the reference trajectory should be avoided as much as possible, as they distort the result of the trajectory comparison. While robotic total stations (RTS) already have several advantages over commonly used GNSS/IMU reference trajectories, they are still not free of systematic deviations due to the use of 360° prisms. Depending on the angle of incidence of the RTS signal into the prism, these cause systematic deviations in the position in the range of several millimeters. In this paper, we therefore present a measurement concept that can be used to reduce these deviations. We use a network of multiple RTS that track the same 360° prism. Due to the different RTS-target geometry of each RTS, the resulting deviations also differ and can be reduced when averaged. We support this claim with experiments which show a reduction of horizontal position deviations of up to 38.5 % in a static measurement and up to 22.4 % in a kinematic measurement. However, the vertical deviations cannot be reduced by combining multiple RTS with our measurement setup. We publish the code for controlling and managing RTS networks here: https://github.com/Engineering-Geodesy-Bonn/rts-network.
1 Introduction
In the fields of mobile robotics and mobile sensing, accurate positioning of the used vehicle is crucial. Understanding the quality of this positioning solution is essential to identify its suitable fields of application, to correctly interpret products generated with it, such as a point cloud, to compare it with other solutions, or to improve it through further development. To perform a quality analysis, the positions of a test system are usually empirically compared with those of a reference system that is known to have higher accuracy. Depending on the expected accuracy of the sensor, solutions of high-end GNSS/IMU fusions are often suitable for this purpose [1], 2]. However, if the sensor to be examined is similarly accurate or even more accurate, or the planned application is in GNSS denied environments, then these reference sensors reach their limits. Therefore, robotic total stations (RTS) are being increasingly used to generate a reference trajectory [3], [4], [5], [6]. These geodetic instruments measure the distance, horizontal direction, and vertical angle to a reflector (prism) and can thus determine its position kinematically with millimeter accuracy [7]. Unlike GNSS, there is no requirement for an open view of the sky and the influence of the environment is significantly lower. However, a constant line of sight is required between the RTS and the prism. In order to be able to target the prism from all directions during the measurement process, 360° prisms are commonly used, which ensure this by combining several mirrors. However, these prisms cause systematic deviations of several millimeters in the determined position, depending on the angle of incidence of the RTS signal into the prism [8], 9]. Other systematic deviations of RTS can be detected and corrected in advance by calibration measurements in two faces while random deviations during a kinematic RTS measurement with distances between 2 and 34 m were determined in Tombrink et al. [4] and amounted to σ = 1.25 mm horizontally and σ = 0.68 mm vertically. Thus, 360° prism deviations can be identified as the largest and, above all, systematic source of deviation in kinematic RTS tracking, which so far cannot be reduced or eliminated by suitable measures.
In this paper, we therefore propose a measurement method to reduce these 360° prism deviations using multiple RTS. We track the same 360° prism with spatially distributed RTS that are synchronized using GPS time. Because each RTS has a different angle of incidence into the prism, their resulting deviations also differ and can be reduced when averaged. We support this idea with both static and kinematic experiments. The kinematic measurement is carried out on a closed-loop rail track, which is part of a test environment at the University of Bonn and was also used for the analysis in ref. [4]. With the help of this rail track, we can create a reference trajectory that is free of systematic 360° prism deviations due to the use of a manually oriented round prism. The ability to exactly repeat the trajectory makes it possible to compare this reference solution with the approach presented in this paper. To further increase accuracy, we also take into account any intrinsic synchronization deviations between internal components of the RTS, a well known problem investigated e.g. in refs. [10], 11]. Furthermore, we publish the code that can be used to control, calibrate, and measure with multiple RTS in a user-friendly way via a web interface, to facilitate the reproducibility of our research. With the exception of the RTS itself, we only use low-cost hardware.
The paper is structured as follows. First, we present the state of research regarding the use of several synchronized RTS in Section 2. This is followed by the explanation of our RTS network methodology and software in Section 3. The experiments involving several measurements are described in Section 4, while their results are presented and discussed in Section 5. The paper concludes with a summary in Section 6.
2 Related work
The idea of using multiple robotic total stations (RTS) simultaneously is not new and several studies can be found in the literature. For example, Mao et al. [12] present an RTS network in which several total stations receive their commands from a central server and send their measurements back to it. The focus here is on being able to carry out surveying tasks, e.g. on larger construction sites, efficiently and roughly at the same time. The time systems of the individual RTS are not precisely synchronized. Kerekes & Schwieger [13] propose an approach that ensures continuous tracking of a target, even if not all total stations have a direct line of sight. The authors investigate the improvement of the position accuracy, but due to simple redundancy and not with the focus on 360° prism deviations. The same authors use a similar system in ref. [14] to achieve a higher data rate and increased accuracy in short ranges by using only the angle measurements of the RTS. Thalmann & Neuner [11] propose a wireless network-based approach to synchronize multiple robotic total stations using GPS time while [15] use an Network Time Protocol (NTP) server which is available over the internet. Previous research on RTS networks has thus focused on synchronization and on increasing accuracy by combining measurements. A targeted analysis of what this combination means for the reduction of 360° deviations, as well as a publication of the code, has not yet been carried out, which leads to the novelty of our work.
In addition to synchronizing the total stations with each other, synchronization deviations within each RTS must also be taken into account. This is because within a total station several subsystems operate independently of each other and are responsible for different tasks, such as distance measurement, angle measurement or automatic target recognition (ATR). Multiple studies have shown that there may be synchronization deviations between these subsystems. A delay between the distance and angle measurement is the most critical here, is often referred to as intrinsic delay, and leads to a distorted trajectory [10], 11], 16]. Even though the magnitude of this delay is getting smaller with new product generations (e.g. Leica ATR vs. ATRplus [17]), it is still relevant depending on the RTS used and must be taken into account to achieve the best possible results during kinematic tracking. In Thalmann & Neuner [11], the authors determine this delay by comparing the raw RTS observations with the nominal positions of a robot arm rotating on a circular path. We take a similar approach, but instead of using nominal positions, we exploit the shape of a circular trajectory.
3 RTS network
This section introduces the Robotic Total Station (RTS) network. It provides the basis for the investigations presented in this paper. The overarching goal is to reduce the incidence angle-dependent 360° prism deviations. To achieve this, all RTS must be georeferenced in the same coordinate system and synchronized. The first requirement can be fulfilled by standard procedures during the measurement and is therefore not described further here. We refer to the second requirement as extrinsic synchronization, in reference to [11], and solve it using GPS time. The exact procedure is explained in Section 3.1. Furthermore, synchronization deviations also occur within an RTS, since an RTS combines several sensors that must be synchronized with each other. We approach this intrinsic synchronization problem by means of a calibration method that is described in Section 3.2. Both synchronization problems are shown in Figure 1. This is followed by a description of the open-source code that enables the synchronized operation of multiple RTS within a network in Section 3.3.
Extrinsic and intrinsic synchronization problem in RTS networks.
3.1 Extrinsic synchronization
To control the total stations and to record their measurements, we use multiple external logging devices which are comparable to those used in refs. [11], 18] and consist of two components (see Figure 2). One component is the Raspberry Pi 4, a low-cost single board computer responsible for communicating with the total station and receiving measurements. The communication is serial using the Leica GeoCOM protocol. To ensure precise and absolute time referencing of the incoming GeoCOM messages, the single board computer receives time information from a u-blox LEA-6T GNSS receiver resembling the second component of the logging device. National Marine Electronics Association (NMEA) messages are transferred to the computer via an USB connection providing absolute time information, while a Pulse-Per-Second (PPS) signal is also transmitted by connecting a trigger cable to the GPIO pins of the Raspberry Pi. Through the utilization of chrony, an open-source implementation of the Network Time Protocol (NTP), the system time of the operating system is synchronized with the GPS time. The described setup enables the logging device to timestamp the incoming GeoCOM messages in GPS time. These messages contain the measurements of the total station. Only the receipt of the GeoCOM response from the RTS can be stamped with a GPS time, but not the actual time of measurement in the RTS. This is a limitation of the GeoCOM protocol, which does not allow direct querying of the internal RTS time for synchronization purposes. Therefore, there is still a delay between the GPS time and the RTS time of about 50–100 ms, which depends on the RTS-specific duration between the actual measurement and the arrival of the message at the Raspberry Pi [11], 18], 19]. To keep the offset between multiple RTS as small as possible, it is recommended to use RTS of the same type. Note that there is no communication between the RTS instruments in a network and that synchronization is achieved solely by connecting each RTS to a GPS time-synchronous logging device.
Communication scheme of the logging device involving a low-cost GNSS receiver for a precise time reference and a Raspberry Pi single board computer for sensor communication via the Leica GeoCOM protocol.
3.2 Intrinsic synchronization
During the measurement of an RTS, many different subsystems within the device are involved [20]. The most important of these include the angle measurement, the distance measurement, and the automatic target recognition (ATR). Most subsystems measure independently of each other at different times [11]. The exact time of the measurement cannot be influenced by the user. Figure 3 schematically shows the previously mentioned distance, angle and ATR subsystems as well as their measurement times. The internal time system of Leica RTS, which are used in this paper, is the sensor board time, which is given in milliseconds and starts at 0 when the total station is powered on. As soon as the total station receives a request for a measurement at time
![Figure 3:
Schematic visualization of the angle, distance and ATR subsystem within a Leica RTS. The measurements of other subsystems are interpolated onto the time of the distance measurement. Adapted from ref. [11].](/document/doi/10.1515/jag-2025-0047/asset/graphic/j_jag-2025-0047_fig_003.jpg)
Schematic visualization of the angle, distance and ATR subsystem within a Leica RTS. The measurements of other subsystems are interpolated onto the time of the distance measurement. Adapted from ref. [11].
The problem that occurs in most RTS is that the subsystems are not exactly synchronized and the interpolation is thus based on incorrect time stamps. Depending on the angular speed of the total station, a delay between the distance and angle subsystems can lead to distortions of the measured trajectory in the mm to cm range and must therefore be taken into account [10], 11], 16]. For this, a so-called intrinsic calibration has to be performed. The goal is to estimate the synchronization deviation δ a between the angle and distance subsystem. We use a similar approach to that of Thalmann & Neuner [11]. However, while that work requires exact reference positions, we exploit the geometric shape of a measured circular path, resulting in a simpler and more accessible calibration process. As in previous work, we assume that the delay is the same for the horizontal and vertical angles.
Using a simply constructed rotating arm, we measure the trajectory of a prism attached to the end of it in both directions of rotation (Figure 6). This process is exaggerated in Figure 4 as a simulation that is not to scale, with the different directions of rotation shown in red and green. Due to the opposite direction of rotation, the trajectory is also distorted in opposite directions. Given that the true trajectory lies on a sphere, we estimate the intrinsic delay using an extended sphere estimation problem:
Simulated effect of an intrinsic delay between the angle and the distance subsystem. Changing the direction of rotation also changes the sign of the effect. The effect is greatly exaggerated for visibility. Typical deviations are in the low millimeter range.
In addition to the parameter of interest δ a , the center of the sphere x c , y c and the radius of the sphere r are also estimated. To prevent ambiguity, z c is fixed at the mean of the measured z-components. The observations are the distance measurement d, the horizontal direction h and the vertical angle v. The angular velocities of the horizontal direction ω h and the angular velocities of the vertical angle ω v are computed from these observations. The estimation is done via a least-squares fit using the Gauss-Helmert-Model with variances from manufacturer’s specifications [21], [22], [23].
Software architecture of the RTS network, consisting of four components: dashboard, server, worker and RTS.
3.3 Software architecture
The open source RTS network software[1] is used to easily control multiple RTS in a network and consists of four components: the dashboard, the server, the worker and the RTS itself (see Figure 5). The server stores the measurements of each RTS in a database and also manages the serial and tracking settings, as well as the completed and pending RTS jobs. Possible jobs are, for example, a change of face or the tracking of a target. The workers are essentially the GPS synchronized logging devices described in Section 3.1. In principle, any GNSS receiver and computer combination can be used, as long as it is network-capable, can execute Python and its computer time can be synchronized using NMEA and PPS. Each worker requests pending RTS jobs from the server and executes them. All communication with the RTS is done by the worker via the Leica GeoCOM protocol. The measurements, including a GPS time stamp, are then transmitted to the server via http. The dashboard also communicates with the server via http and is used to display measurement data and to provide easy control and management of the total stations.
Non-scale representation of the intrinsic calibration measurement.
Since only the worker communicates directly with the RTS, it is possible to extend the software to the RTS of other manufacturers without difficulty by implementing a customized worker. Using the provided code, the intrinsic delay can also be determined using the method described in Section 3.2. Furthermore, the measurements of two RTS can be aligned using the methodology presented in ref. [19].
4 Experiments
For the successful combination of the measurements of several RTS, both the extrinsic and the intrinsic synchronization must be ensured and validated. Furthermore, the actual benefit of the RTS combination in terms of reducing 360° prism deviations must be quantified and confirmed.
To achieve these objectives, we performed several experiments, which are explained in this section and can be divided into three subsections. First, the intrinsic calibration measurement that was performed for every involved RTS is explained in Section 4.1. This is followed by the description of a static measurement involving four RTS tracking a static target in Section 4.2. Finally, a kinematic experiment using a closed-loop rail track that is part of a test environment of the University of Bonn is explained in Section 4.3.
4.1 Indoor intrinsic calibration
The measurement for calibrating the intrinsic delay between the angle and distance subsystems of an RTS is shown schematically and not to scale in Figure 6. A 360° prism attached to the end of a rotating arm is kinematically tracked by the RTS to be calibrated. Due to the continuous change of the horizontal direction and thus a non-zero angular velocity, deviations from the circular path occur if the subsystems are not synchronized correctly. The delay can be determined using the method described in Section 3.2. Overall, such measurements were carried out for the Leica TS60, the Leica TS16 family and the Leica MS60 multiple times over a duration of approximately two years. All RTS received regular software updates. The results and their empirical standard deviations are listed in Table 1. They are consistent with the values known from the literature [11]. It is noticeable that the intrinsic delay of the Leica MS60 does not differ significantly from zero, which could be related to its wave form digitizing electronic distance measurement sensor [7]. This finding is consistent with the studies by ref. [24], in which the intrinsic delay of the MS60 was also investigated but not published. Since this is the most advanced RTS, this could indicate that the intrinsic calibration problem will no longer be relevant in future product generations. The raw observations of the TS60 and TS16 are corrected using the corresponding values, the observations of the MS60 are not corrected. However, due to maintenance, the MS60 was not involved in any further experiments.
Intrinsic calibration results.
| RTS | # Calibrations | δ a [ms] | Std. dev. [ms] |
|---|---|---|---|
| Leica TS60 | 6 | 4.98 | 0.48 |
| Leica MS60 | 5 | 0.22 | 0.13 |
| Leica TS16 | 7 | 7.22 | 0.27 |
4.2 Static measurement
As a proof-of-concept, a static measurement was carried out with four RTS tracking the same static target. A Leica TS60 and three Leica TS16s were used, one of them in the TS16i version. These were positioned around the target, as depicted in Figure 7. After measuring three spatially distributed control points with all four RTS in four full sets, we ensured that all RTS are georeferenced in the same local coordinate system by estimating a four parameter similarity transformation (3D translation and z-rotation) between the TS60 positions and each remaining RTS.
Tracking of the same static target using four RTS.
The actual experiment was divided into two parts. First, a Leica GPR121 round prism was installed as a target and measured by all RTS in two faces. The mean of these measurements serves as a reference without 360° prism deviations. Then, the round prism was replaced by a Leica GRZ122 360° prism and tracked by all RTS simultaneously. The 360° prism was rotated by hand as evenly as possible through 360° in approximately 180 s. This measurement allows us to investigate the extent to which the individual RTS are influenced by changes in the horizontal angle of incidence. Due to the measurement setup, no variation of the vertical angle of incidence can be induced and therefore also cannot be analyzed.
4.3 Kinematic rail track measurement
The measurements for the main investigation of this paper were carried out on a rail track, which is part of a test environment at the University of Bonn and is shown in Figure 8. It is approximately 140 m long and allows us to exactly repeat a kinematic trajectory measurement using a rail car.
Top: Bird’s eye view of the measurement setup, showing the rail track, the position of both RTS and the rail car with the prism. Bottom: Roll (blue), pitch and height (green) variations of the rail track.
Similar to the static experiment, two measurements were carried out. First, a Leica GPR121 round prism was mounted on the rail car and kinematically tracked with the Leica TS60 (see Figure 8). During this measurement, which was performed at an average speed of 0.3 m/s, the prism was manually directed towards the total station. In two laps, 4,790 positions were recorded. The measurement serves as a reference, which is free of 360° prism deviations.
For the second measurement, the round prism was replaced by a Leica GRZ122 360° prism. In addition to the TS60, the TS16 also tracked the prism simultaneously. The positions of both RTS can be seen in Figure 8. To ensure that both total stations are georeferenced in the same local coordinate system, three identical points were measured with both instruments in four full sets. By estimating the 3D translation and the z-rotation between both point groups, both RTS could be aligned. The residuals between the transformed coordinates were less than 1 mm. During the kinematic tracking of the 360° prism, the average speed was approximately 0.5 m/s. In a total of five laps, the TS16 measured 4,101 positions and the TS60 measured 6,599 positions. The measurement is suitable for comparing the individual trajectories of both RTS and their combination with the round prism reference. Furthermore, the extrinsic synchronization of both RTS can be verified.
The two round prism laps and the five 360° prism laps were each approximated to a mean trajectory using the methodology of ref. [4] to reduce random deviations. To create the combined TS16 and TS60 trajectory, we first ensured that both trajectories were available at the same points in time using interpolation. The arithmetic mean was then calculated from the position pairs. Finally, the five recorded laps were also approximated to a mean trajectory.
5 Results and discussion
In this section, the results of the experiments are presented and discussed. First, the static measurement is investigated in Section 5.1. After this, the results of the kinematic rail track measurement are discussed in Section 5.2.
5.1 Static measurement
Figure 9 shows the 360° prism deviations with respect to the mean round prism reference as a 2D plot. The directions to each RTS are represented by arrows. The averaged 360° prism positions are shown in black and their mean value is indicated by a cyan marker with a white border. To average all 360° prism measurements, their time series were interpolated to the same points in time and then averaged. The results of the static experiment allow us to analyze the 360° prism deviations as the horizontal angle of incidence changes. The vertical angle of incidence was not varied during the experiment. In future investigations, the vertical angle of incidence could also be varied by using different RTS or prism heights similar to ref. [8]. However, a continuous variation of the vertical angle is difficult from a mechanical point of view, as the prism would have to rotate precisely around its reference point.
2D view on the 360° prism deviations of each RTS with respect to the round prism reference during the static experiment. Directions to each RTS are shown as arrows.
The resolution and precision of the angle and distance measurement is clearly recognizable and both times superior for the angles. This is in line with expectations, as the distance to the target is only a few meters. It is noticeable that the distance measurements to the 360° prism are usually too long compared to the round prism reference, which results in a scattering behind the round prism position along the direction of measurement. Therefore, the averaged positions are also shifted by 0.85 mm with respect to the round prism reference. However, their standard deviation of 0.35 mm is approximately 36 % less than the average standard deviation of a single RTS of 0.56 mm. For each RTS and their average, the RMSE between the round prism reference and the 360° prism measurements is listed in Table 2. At 0.98 mm, the RMSE of the RTS combination is on average 38.5 % lower than that of a single RTS, which validates the approach of reducing 360° prism deviations by averaging several RTS. In terms of both the standard deviation and the RMSE, the measurements of all RTS are within the manufacturer’s specifications for the Leica GRZ122 360° prism of a 3D accuracy of 2.0 mm.
RMSE values of the 360° positions compared to the averaged round prism reference for the static measurement.
| TS60 | TS16 1 | TS16 2 | TS16i | Averaged | |
|---|---|---|---|---|---|
| RMSE [mm] | 1.48 | 1.35 | 1.81 | 1.74 | 0.98 |
Figure 10 shows the deviations between the 360° prism positions and the averaged round prism reference for each RTS for the x, y and z components. Especially in the x and y components, a cyclic pattern is clearly recognizable due to the six prism mirrors. The z-component has significantly smaller deviations and no cyclical behavior. This is consistent with the findings from Lackner & Lienhart [9] for the Leica GRZ122 prism. The result of averaging all RTS is shown as a black time series.
Time series of the 360° prism deviations to the reference during the static experiment.
The experiment illustrates both the potential and the limitations of 360° prism averaging. The effectiveness of averaging to increase accuracy depends strongly on the phase shift of the cyclic patterns. For the x-component this is only minor in three of the four RTS, which means that although the averaged result has smaller deviations from the reference, it still has a clearly visible cyclical pattern. For the y-component, the phase shift is more favorable and averaging has a greater effect on improving the accuracy. In comparison, the y deviations of the TS60 are lower than those of the other RTS. As can be seen in Figure 9, the more precise angle measurement is particularly decisive for the y component due to the position of the TS60. In addition, the angle precision of the TS60 is higher than that of the other RTS. The TS60 also stands out with an offset of approx. −1 mm for the z component, the reasons for which are unknown.
5.2 Kinematic rail track measurement
This section examines and analyzes the results of the kinematic rail track measurement using two RTS. Due to the kinematic nature of the measurement, the extrinsic synchronization of both RTS is essential for the averaging of both RTS trajectories. Therefore, it is verified first. Subsequently, the 360° prism deviations are analyzed by comparing the 360° prism trajectory to the round prism reference, which is averaged from two laps completed on the rail track. The deviations of both laps from their average trajectory have a standard deviation of 0.3 mm in the horizontal cross-track direction and 0.21 mm in the vertical cross-track direction, indicating high reproducibility.
To examine the extrinsic synchronization of both RTS, the time offset between the two trajectories was estimated using the method described in ref. [19]. This approach uses velocities calculated from the positions to estimate a time offset between both trajectories by means of a Gauss-Helmert-Model. Both trajectories were found to have a time offset of 22.2 ms with a standard deviation of 0.1 ms. The remaining time offset can be seen by plotting the along-track deviations between the two trajectories against their velocity (see Figure 11). Before synchronizing the time stamps, it can clearly be seen that the along-track deviations increase with increasing speed. After applying the estimated time offset, this trend is no longer visible. As also mentioned in Section 3.1, one reason for the remaining time offset could be that the TS60 and TS16 each process the observations for different lengths of time before sending them to the logger. For all further investigations, the observations corrected for the delay were used.
Along-track deviations between both 360° prism trajectories tracked by the Leica TS16 and TS60.
Figures 12–15 show the results of our kinematic experiment, first for the horizontal cross-track deviations (Figures 12 and 13) and then for the vertical cross-track deviations (Figures 14 and 15). In Figure 12, the deviations between the three 360° prism trajectories and the round prism reference are shown as histograms. The deviations are divided into horizontal and vertical cross-track directions and computed using the methodology described in ref. [4]. In this and all subsequent figures, A depicts the deviations between the TS16 360° prism trajectory and the round prism reference, B shows the deviations between the TS60 360° and the round prism reference, and C refers to the deviations between the combined trajectory and the reference. The RMSE of the TS16 deviations (A) is 1.47 mm while the TS60 measurements (B) have a RMSE of 1.31 mm. However, histograms A and B show clear systematic deviations between −4 and +4 mm, with two recognizable clusters at approximately −1 mm and +1 mm. These systematic deviations can be attributed to the 360° prism and their magnitude is consistent with the findings of the static experiment (see Section 5.1) and Lackner & Lienhart [9]. When analyzing deviations in kinematic RTS measurements, it should be noted that the electronic distance measurement sensor exhibits systematic cyclic deviations, which could also have an influence on the results shown, but are usually significantly smaller than 360° prism deviations [25]. Histogram C shows the deviations between the combination of TS16 and TS60 and the round prism reference. These are closer to a normal distribution, have a RMSE of 1.14 mm and no longer show any obvious systematic deviations, which supports the idea that these can be reduced by using several RTS (−22.4 % for TS16, −13.0 % for TS60). Since all histograms scatter around zero, their standard deviations do not differ significantly from their RMSE values.
Horizontal cross-track deviations compared to the round prism reference shown as histograms. (A) Leica TS16 360° prism measurement, (B) Leica TS60 360° prism measurement, (C) combined 360° prism measurement.
Horizontal cross-track deviations compared to the round prism reference. (A) Leica TS16 360° prism measurement, (B) Leica TS60 360° prism measurement, (C) combined 360° prism measurement.
Vertical cross-track deviations compared to the round prism reference shown as histograms. (A) Leica TS16 360° prism measurement, (B) Leica TS60 360° prism measurement, (C) combined 360° prism measurement.
Vertical cross-track deviations compared to the round prism reference shown. (A) Leica TS16 360° prism measurement, (B) Leica TS60 360° prism measurement, (C) combined 360° prism measurement.
Figure 13 depicts the same deviations as a 2D plot along the rail track. For all 2D plots, the deviations were smoothed for better visualization with a sliding window of 10 deviations which corresponds to a mean window size of about 0.6 m. In this plot, the systematic deviations can also be seen in A and B, since different deviations occur depending on the targeted mirror of the 360° prism. Due to the different positions of both RTS, the angles of incidence and distances and thus the deviations are also different. Judging by Figure 13, the correlation between the horizontal angle of incidence and the resulting horizontal cross-track deviation appears to be significantly stronger than between the vertical angle of incidence and the resulting horizontal deviation. The deviations of the TS60 (B) turn from positive to negative near the RTS (positioned at x, y = 0), a location where high variations of the vertical angle of incidence occur, but this could also be related to the change in the horizontal angle of incidence. A clean separation of the influences is not possible with our measurement setup. The figure also provides more information about the quality of the combination (C), which remains hidden in the histogram. Even if the overall histogram is closer to a normal distribution after the combination, some systematic deviations remain, albeit in significantly reduced magnitude, similar to the results of the static experiment. It is conceivable that these could further be reduced with additional RTS or a more varied trajectory. However, with kinematic measurements it becomes increasingly difficult to add further RTS that have a significantly different RTS-target geometry and still see the target.
Figure 14 shows the vertical cross-track deviations in histograms. For all three trajectories, clear systematic patterns and a shift of about +2 mm can be observed. Only a few deviations scatter around zero. The trajectory recorded by the TS16 has the smallest deviations with a RMSE of 1.87 mm and a standard deviation of 1.11 mm, while the TS60 trajectory has additional deviations of up to +7 mm, a RMSE of 2.98 mm and a standard deviation of 1.81 mm. These deviations exceed the 3D centering accuracy of 2.0 mm provided by the manufacturer [26]. Overall, the vertical 360° prism deviations can be mitigated by the combination, but not to the same extent as the horizontal cross-track deviations. The resulting trajectory has a RMSE of 2.39 mm and a standard deviation of 1.40 mm and thus larger deviations than the TS16 trajectory alone. As can be seen in the 2D representation of these deviations in Figure 15, they appear to depend primarily on the vertical angle of incidence. The TS16 (A) was further away from the rail car during the measurement and thus there were fewer variations in the vertical angle of incidence. The TS60 (B), on the other hand, was much closer to the track and was therefore exposed to a significantly wider range of vertical angles of incidence. Large deviations occur when the prism is very close to the instrument (positioned at x, y = 0) and the rail car is additionally tilted (see Figure 8). So, the TS16 trajectory is affected by rather constant prism deviations, while these change more frequently with the TS60 trajectory. When combining the two (C), this consequently does not lead to a reduction of the 360° prism deviations. The offset of 2 mm, which can be observed in all trajectories, could be related to the elevated position of both RTS. As a result, the vertical angle of incidence is below 90° (0° at the zenith) most of the time, which could lead to a positive offset. As the vertical angle of incidence approaches values greater than or equal to 90°, this results in 360° prism deviations around zero. However, since this only occurs at a few locations on the rail track (both pitch variations), there are only a few of these deviations. No data is available to make a statement about the deviations at vertical incidence angles significantly greater than 90°. It is noticeable that the horizontal angle of incidence seems to have a significantly lower influence on the resulting vertical 360° prism deviation than the vertical angle of incidence. In areas with similar vertical angles of incidence but changing horizontal angles of incidence, the deviations remain relatively constant around 2 mm.
6 Conclusion and outlook
In this paper, we have investigated the suitability of RTS networks for reducing 360° prism deviations. These deviations depend on the angle of incidence of the RTS signal into the prism. The basic idea is that multiple spatially distributed RTS cause different 360° prism deviations for each RTS, which are then reduced by averaging. To investigate this approach, several experiments were conducted using a static measurement setup and a rail car moving on a rail track. These confirmed that the horizontal 360° prism deviations could indeed be reduced by up to 38.5 % in a static and up to 22.4 % in a kinematic measurement, even though systematic deviations were still present in the averaged position. The systematically too long distance measurement observed during the static experiment suggests that RTS positioned opposite to each other, e.g., in a triangle or a square, might be advantageous for reducing 360° prism deviations. The vertical deviations were unremarkable during the static experiment without variation of the vertical angle of incidence. However, in the kinematic rail track experiment, large vertical deviations of up to 7 mm occurred which could not be reduced by averaging. Here, the TS16, which was further away from the rail track, showed mostly constant 360° prism deviations, while the TS60, which was closer to the rail track, showed greater variations in the vertical angle of incidence and thus also greater vertical cross-track deviations. Therefore, the combination of both did not improve the trajectory. This could change with more or spatially differently distributed RTS. The magnitude of the vertical deviations, from −1 mm to +7 mm, was contrary to expectations for the Leica GRZ122 360° prism. Overall, the horizontal angle of incidence appears to be the determining factor for the horizontal deviations, while the vertical angle of incidence seems to be the decisive criterion for the vertical deviations.
In future studies, the results of the experiments could also be used to parameterize the 360° prism deviations as a function of the rail track length or the angle of incidence and to apply them as a correction in subsequent measurements. Thus, a precise reference trajectory for evaluating navigation sensors could be generated with just one RTS and a 360° prism. However, the transferability to other environments may be challenging. For applications in arbitrary environments, the use of an RTS network is suitable for the reduction of horizontal 360° prism deviations. These improved trajectories could be integrated into a sensor fusion as a GNSS replacement to further increase accuracy.
Funding source: Deutsche Forschungsgemeinschaft
Award Identifier / Grant number: EXC 2070 – 390732324
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Research ethics: Not applicable.
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Informed consent: Not applicable.
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Author contributions: Gereon Tombrink: Conceptualization, Methodology, Software, Validation, Investigation, Writing – Original Draft. Lasse Klingbeil: Supervision, Conceptualization, proof-reading. Heiner Kuhlmann: Supervision, Conceptualization, Project administration. The authors have accepted responsibility for the entire content of this manuscript and approved its submission.
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Use of Large Language Models, AI and Machine Learning Tools: None declared.
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Conflict of interest: The authors state no conflict of interest.
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Research funding: This work has been partially funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC 2070 – 390732324.
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Data availability: The raw data can be obtained on request from the corresponding author.
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