1. Introduction
The global navigation satellite system (GNSS) is currently being used in a variety of applications that employ position information. Among GNSS positioning methods, real time kinematic (RTK) is one of the most accurate methods and provides cm-level position estimation in an open sky environment [
1]. In recent years, the development of autonomous vehicles and self-driving support systems has progressed [
2,
3]. RTK is occasionally employed, because automated driving requires highly accurate position information, such as a planar error below 0.3 m. RTK is used not only for automated driving, but also for automated guided vehicle (AGV) and other logistics transport robots, position estimation using light detection and ranging (LiDAR), and simultaneous localization and mapping (SLAM) integration [
4,
5,
6]. It is moreover employed as a reference to evaluate the position results estimated by SLAM [
7,
8].
Currently, RTK can be used easily in several ways. Some GNSS receivers are equipped with RTK, whereas others use the open source software RTKLIB [
1], which is an RTK application that uses raw data from receiver, to estimate RTK. In other cases, RTK is performed by a software receiver using an RF recorder [
9,
10]. RTK uses the carrier phase transmitted from the satellite to estimate the position. The pseudo-range used in single positioning and differential GNSS (DGNSS) is calculated by measuring the propagation time of the satellite signal. Therefore, because the signals are transmitted at the speed of light, a clock error of 1 µs can result in an observation error of approximately 300 m. In contrast, the carrier wave has a wavelength of approximately 0.2 m for an L1 signal, and the resolution of the phase that can be measured is high. However, only the carrier phase can be measured, and the number of waves being transmitted is unknown. This problem is called integer ambiguity, and the solution is referred to as ambiguity resolution (AR). Various methods have been proposed for AR. The most widely used is integer least-squares (ILS). Based on ILS, there are the fast ambiguity resolution approach (FARA) [
11], the least-squares ambiguity decorrelation adjustment (LAMBDA) [
12], and the least-squares ambiguity search technique (LSAST) [
13]. Other AR methods based on the Success Rate Criterion (SRC) have also been proposed [
14,
15]. By accurately solving the integer ambiguity, we can obtain a fix solution, which is a centimeter-class positioning estimation result.
To improve the robustness of GNSS, a method of integrating GNSS and the inertial measurement unit (IMU) has been proposed [
16,
17,
18,
19,
20,
21,
22,
23,
24,
25]. GNSS cannot estimate the position in places where the signal cannot be received, such as tunnels and under elevated buildings. In such cases, the integration of IMU enables position estimation, even in places unreachable by signals. There are two main types of GNSS/IMU integration: loose coupling [
18,
19], which integrates the results of each sensor, and tight coupling [
20,
21], which integrates the raw values of each sensor. A Bayesian filter following a normal distribution, such as the Kalman filter, is commonly used for these integrations. Because the Bayesian filter takes into account the error in the estimation, the accuracy of position estimation can be improved [
22,
23,
24]. Moreover, the integration of RTK and IMU allows for a very accurate position estimation. In this case, the integration is based on trusting the fix solution of RTK. Therefore, when integrating RTK and IMU, the accuracy and reliability of the fix solution is highly important.
GNSS-based position estimation causes accuracy degradation in urban areas. In urban areas, where there are numerous obstructions such as high-rise buildings, the number of observation satellites decreases, and multipath occurs due to reflection and diffraction of satellite signals [
26,
27]. In recent years, multi-GNSS has become more common owing to the increase in the number of satellite systems [
28,
29,
30,
31]. The main multi-GNSS are the global positioning system (GPS) of the United States, BeiDou navigation satellite system (BDS) of China, global navigation satellite system (GLONASS) of Russia, and Galileo of the EU. Multi-GNSS solves the problem of reduced number of observation satellites and improves the utilization rate. In contrast, the signal multipath is directly related to the decrease in the position estimation accuracy. The integration of GNSS/IMU is considered to be effective for this problem [
32,
33]. However, the error due to multipath does not follow a normal distribution. In the Kalman filter, where the error is assumed to be normally distributed, it may diverge due to multipath noise. Understanding the error due to multipath is difficult. Therefore, the integration of GNSS/IMU is not effective unless the amount of error due to multipath is detected or the integration is performed by removing multipath noise [
34]. One solution is to use a GNSS/IMU system; however, its use is limited due to its high cost. Furthermore, multipath degrades the accuracy of the fix solutions of RTK. In urban areas, a fix solution with a non-negligible error may be generated and this is referred to as the missed fix. In such cases, it becomes difficult to use the fix solution as a reliability indicator. Therefore, there is a risk of failure in integration that relies on the fix solution.
Therefore, we propose a novel method to determine the reliability of the fix solution. The goal is to determine whether the position error is within 0.3 m, which is required for applications such as automatic driving. The reliability of the proposed method is assessed not by the horizontal plane, but by the height direction. Two reasons justify our focus on the height direction. The first is that GNSS positioning errors in the height direction tend to be larger than those in the plane due to geometric factors between the satellite and the receiver. The second is because we can accurately estimate the vehicle motion with multipath removed. For these two reasons, we can accurately determine the fix solution even with low-cost GNSS/IMU by focusing on the height direction. If the proposed method is effective, conventional GNSS/IMU integration methods may also operate effectively in urban areas.
The paper is organized as follows. In
Section 2, we explain the generation mechanism of the fix solutions with errors and conventional testing methods.
Section 3 provides a detailed description of the proposed method for determining the reliability. In
Section 4, we discuss the evaluation tests and results of the proposed method.
Section 5 provides the summary and conclusion of the paper.
2. LAMBDA Method and Ratio Test
RTK is a high-precision positioning method using the carrier phase. However, its integer ambiguity is unknown. To address this problem, various methods have been devised. In this study, we describe the LAMBDA method, which is used in RTK algorithm. In RTK algorithm, the state variables are set as follows:
where
x is the state vector,
a is the integer ambiguity,
b is the real value parameter, and
Q is the error covariance. The observation model is defined as follows:
where
y is the observation vector, containing the pseudo-range and carrier phase,
H is the observation matrix, and
v is the observation noise. In RTKLIB, the first step is to solve the problem using the Kalman filter with integer ambiguities as real values. The resulting real-valued ambiguity is called the float ambiguity. Then, the float ambiguity is solved by the ambiguity resolution (AR) as an integer value. A simple AR method is the integer least squares method, which has the following evaluation function.
When the error covariance of the float ambiguity is a true diagonal matrix, rounding off the float solution is used to obtain the optimal solution of the fix solution. However, the actual error variance is not a diagonal matrix. Because there are error correlations for each float ambiguity, estimation by search is required instead of rounding. Consequently, the search cost is high.
Therefore, the LAMBDA method transforms the variables, such that the error variance becomes a diagonal matrix. By applying the transformation and making the errors uncorrelated, it is possible to reduce the search cost for integer solutions [
35,
36]. For this transformation to preserve the integer nature of the ambiguity, the transformation matrix
Z must obey the following conditions:
With
Z satisfying this condition, the float ambiguity is subjected to a variable transformation.
In this situation, the integer least squares method is used to search for the optimal solution that minimizes the evaluation function corresponding to the variable transformation.
Once the search for the optimal solution
z is completed, the inverse z-transform is applied to convert it to the actual position parameters.
In this manner, cm-class position estimation is performed in RTK.
Further, a test is performed to determine whether the fix solution obtained using the evaluation function of the LAMBDA method is plausible. This test is referred to as the ratio-test [
37,
38,
39]. The ratio-test is evaluated by the ratio of the evaluation functions of the first and second solutions obtained by the search of the LAMBDA method and is defined by the following equation.
If the value obtained in Equation (8) is above a certain threshold, the fix solution is considered to be an accurate search solution and is converted into a position result using Equation (7). In contrast, if the solution does not pass the test, the search solution is considered to be inaccurate, and the float solution is converted to the position result. Thus, the ratio-test makes a judgment based on whether the float ambiguity is close in value to the first solution. This is how the fix solution was tested in the past, and the method is also employed in RTKLIB.
However, when multipath noise occurs, the ratio-test may break down. If the ratio-test fails, a fix solution will be generated with an error.
Figure 1 illustrates this situation. The following is a description of the mechanism behind the occurrence of the fixed solution with the error.
Multipath noise due to high obstructions affects the pseudo-range and the carrier phase. Due to multipath noise, the observed noise becomes non-normally distributed. Because the Kalman filter assumes a normal distribution, it may fail to estimate the float ambiguity. Even the failed float ambiguity is searched by the LAMBDA method. Typically, the LAMBDA method fails in the search or is considered unsuccessful by the ratio-test. However, there are cases where the LAMBDA method searches for the first solution near the failed float solution, even though it is far from the true value. In this case, the ratio-test considers it as a pass and outputs it as a fix solution. In reality, it is the fix solution that is far from the true value, and it contains a large error. Under these conditions, the ratio-test fails and outputs the fix solution with a large error. Therefore, we believe that it is necessary to determine the fix solution in a way different from that of the ratio-test.
5. Conclusions
Autonomous driving support systems and self-driving cars require highly accurate and reliable vehicle positions. The fix solution of kinematic positioning is often used as a factor to show the reliability of GNSS positioning results. Most fix solutions are determined by the combination of the LAMBDA method and ratio-test. However, fix solutions with large errors are typically generated in urban areas. Therefore, in this study, we propose another method to determine the reliability in place of the ratio-test. In this study, we consider two aspects. One is that the height variation of the vehicle is small and can be estimated more accurately. The other is that GNSS errors are more likely to occur in the height direction than in the plane direction. Based on these considerations, we proposed a method for detecting a reliable fix solution by using the high trajectory during driving. According to the evaluation tests, the proposed method achieved a decision accuracy of 99.9% on route A, which is an urban area. This determination resulted in a flatness accuracy (RMS) of 0.17 m, achieving the target of 0.3 m or below. In Route B, which is a dense urban area with a harsh environment, the judgment accuracy was 99.7%. Here, the RMS of the plane was 0.22 m, indicating that the confidence level was estimated with high accuracy. Because these results were determined with a higher judgment accuracy than the ratio test, the proposed method is considered to be effective.
However, we have not yet been able to completely assess the reliability. In Route B of the evaluation test, correct assessments could not be made even when the height error was 0.3 m or more. To improve these problems, it is necessary to improve the accuracy of the height trajectory and combine the assessment in horizontal directions. Further, we must solve the multipath problem of GNSS and improve the accuracy of the fix solution.