Integrity Monitoring for BDS/INS Real-Time Kinematic Positioning Between Two Moving Platforms

Abstract

In recent years, the rapid development of moving platforms, especially unmanned aerial vehicles (UAVs) and unmanned ground vehicles (UGVs), has promoted their widespread applications in various fields such as precision agriculture and formation flight. In these applications, for accurate real-time kinematic positioning between two moving platforms, receiver autonomous integrity monitoring (RAIM) is necessary to assure the reliability of the obtained relative positioning. However, the existing carrier phase-based RAIM (CRAIM) algorithms are mainly a direct extension of pseudorange-based RAIM (PRAIM), whose availability is also a major challenge in signal-harsh environments. Learning from the integrated system between Global Navigation Satellite System (GNSS) and INS and based on a multiple hypothesis solution separation (MHSS) algorithm, we have developed an improved CRAIM algorithm, which combines Beidou Navigation Satellite System (BDS) and INS to offer integrity information for real-time kinematic relative positioning between two moving platforms in challenging environments. To achieve more robust and efficient fault detection and exclusion (FDE) results, an algorithm of observation-domain outlier detection combined with MHSS (OOD-MHSS) is also proposed. In this algorithm, the kinematic relative positioning method with INS addition is performed first, then, based on double-difference (DD) phase observations with known integer ambiguities and the OOD-MHSS method, the integrity monitoring information can be provided for the kinematic relative positioning between two moving platforms. To assess the performance of the OOD-MHSS and the improved CRAIM algorithm, a series of kinematic experiments between different platforms was analyzed and discussed. The results show that the improved CRAIM algorithm can perform effective FDE and provide reliable integrity information, which offers centimeter-level relative position solutions with decimeter-level protection levels (PLs) (integrity budget: ${1\times 10}^{-5}$/h). Both observation outlier detection and INS improve the continuity and availability of kinematic relative positioning and the PLs in horizontal and vertical directions. The PL values have been improved by up to 24.3%, and availability has reached 96.67% in harsh urban areas. This is of great significance for applications requiring higher precision and integrity in kinematic relative positioning.

1. Introduction

Real-time, precise, and reliable kinematic positioning between two moving platforms plays a critical role in a wide range of applications, including vehicle-to-everything (V2X), formation flying, and autonomous navigation. This level of precise kinematic positioning can be achieved by the Global Navigation Satellite System (GNSS), using the BeiDou Navigation Satellite System (BDS) as an example, or other navigation sensors such as an inertial navigation system (INS) [1,2,3]. In practice, however, various factors such as multipath, signal blockage, and sensor errors can introduce biases and degrade the accuracy and reliability. A wrong kinematic positioning solution between two moving platforms can lead to dangerous accidents, such as collisions and crashes. Therefore, the reliability of kinematic positioning is crucial, which is typically described by integrity monitoring that reflects the confidence level of positioning solutions and detects any potential failures in real time.

Existing real-time GNSS precise kinematic positioning methods mainly include real-time precise point positioning (RT-PPP), real-time kinematic (RTK) positioning, and PPP-RTK [4,5]. RT-PPP only requires a single receiver to offer high accuracy and global centimeter positioning, but its long convergence time and dependence on external precise correction sources limit its applications in real-time scenarios [6]. Conventional “static-to-kinematic” RTK method requires a static base station with a known position, but in some scenarios, no known static station can be provided for relative positioning service. PPP-RTK performance relies on the augmentation correction provided by the server system and requires a network of static base stations to be deployed [7]. Interruptions or lack of coverage will degrade the real-time performance, and the positioning service cannot be available in scenarios without base stations or outside the service coverage areas. By contrast, the “kinematic-to-kinematic” RTK method is performed between one selected moving base station and other moving platforms. Once the integer ambiguity is correctly fixed, each moving platform can obtain centimeter-level positioning results relative to the moving base station [8]. This mode is more flexible and suitable for real-time, precise, and reliable positioning scenarios between two moving platforms, where only the precise relative location is of interest.

Beyond positioning accuracy, the integrity of the results is also an important indicator for evaluating the performance of kinematic positioning. Based on the assumption that residual errors are normally distributed, receiver autonomous integrity monitoring (RAIM) can identify and eliminate outliers using redundant observations or other auxiliary information, and provide users with integrity monitoring and fast alarm services [9,10]. Advanced RAIM (ARAIM) extends traditional RAIM by leveraging multi-constellation GNSS (GPS, Galileo, BDS, GLONASS) and advanced error bounding to support precision approaches (LPV-200) without ground infrastructure. The airborne algorithm still performs a self-consistency check of the GNSS measurements, but now under a multiple-hypothesis, solution–separation framework. Integrity is guaranteed by computing protection levels (PLs) that bound the horizontal and vertical position errors with a probability derived from the target hazard level (10⁻⁷ for LPV-200). A prior fault probability model is broadcast to the aircraft via the Integrity Support Message (ISM) that contains per-satellite and per-constellation failure rates, plus the nominal SIS error statistics. The main algorithms for GNSS observations can be divided into Pseudorange-based RAIM (PRAIM) and carrier phase-based RAIM (CRAIM). PRAIM algorithms have been studied for decades and are widely used in the aviation approach phase [11,12]. However, for higher precision positioning applications, CRAIM algorithms are necessary. Many contributions have extensively investigated the CRAIM algorithm and its performance [13,14,15]. Assuming knowledge of integer ambiguities, Pervan et al. [16] directly extended the PRAIM concept to CRAIM. However, this assumption is not always reliable. Chang et al. [17] proposed a single difference-based CRAIM approach, but this approach is also impractical due to the same problem of float ambiguities above. In 2009, Feng et al. [18] presented a new RTK-CRAIM algorithm with an extended Kalman filter (EKF). The algorithm obtained the integer ambiguity by using the least-squares ambiguity decorrelation adjustment (LAMBDA) method and detected satellite failures by using chi-squared (${\mathsf{\chi }}^2$) test, but it cannot identify and eliminate failures. Feng et al. [19] also designed a PPP-CRAIM algorithm monitoring the ambiguity resolution and positioning, and verified the power and efficiency of the algorithm using simulation and real data. Zhang et al. [20] proposed an integer ambiguity scheme for multi-frequency multi-constellation uncombined PPP-RTK, applying the advanced RAIM (ARAIM) theory by Blanch et al. [21], and the results showed that the protection levels (PLs) estimated by ambiguity-fixed solutions could be significantly reduced, and fast convergence in accuracy and the position error bounds in horizontal position components of PPP-RTK could be achieved.

To achieve precise and reliable positioning in harsh signal environments, the integration of various navigation systems is often employed. One such example is the integration of GNSS and INS, in which mode centimeter-level real-time kinematic positioning solutions between two moving platforms can be offered in various signal-harsh environments [22,23]. Moreover, it is also crucial to ensure the integrity of the positioning solution by monitoring and detecting the security risk. El-Mowafy and Kubo [24] showed that the integrity availability of RTK integrated with INS could be superior to 99%, where the PL value was smaller than an alert limit of 1 m. However, detailed methods, especially for multiple faults and the relationship between different fault detection and exclusion (FDE) methods, were not provided. Gunning et al. [25] showed that INS could improve the positioning performance and continuity of PPP with a solution separation algorithm. ARAIM has been extended to integrated navigation systems (e.g., IMU/Lidar/camera) to constrain the hypothesis space and improve integrity monitoring performance through deep learning techniques. Nevertheless, the CRAIM algorithm designed for the RTK positioning between two moving platforms with INS addition has not been given sufficient consideration in current research.

MHSS is designed to handle multiple simultaneous faulty measurements and compute protection levels under the assumption that no more than K measurements are simultaneously faulty. Multiple hypothesis solution separation (MHSS) is one of the most used ARAIM algorithms that is based on the direct evaluation comparison between the position results obtained by a full-set filter (using all satellites) and those obtained by each one of the subset filters (using part of the satellites) [26,27]. The ARAIM baseline algorithm includes MHSS as its core algorithm, incorporating two key procedures: real-time fault detection and exclusion (FDE) and PL computation. Noticed that the PL computation performs only for those satellites that have passed the FDE successfully. In practice, MHSS compares the position estimate obtained using all satellites (all-in-view solution) with estimates from subset solutions (obtained by excluding some hypothetical faults) to detect faults. However, the current MHSS algorithm has some deficiencies, such as the weak robustness to observation errors, excessive subset filters, and high computational burden in real time [28,29]. To address computational complexity and improve robustness, recent works have combined MHSS with techniques like least square residual (LSR) for spoofing detection, gross error detection, or deep learning to reduce the hypothesis space. Since the integrity criteria of kinematic positioning between two moving platforms are more critical than those of aviation applications, hazard alerts must be generated within a very short time. Therefore, the robustness and efficiency of the MHSS ARAIM algorithm in real time are crucial issues to be improved [13].

Based on MHSS ARAIM, we propose an improved BDS RTK/INS CRAIM algorithm between two moving platforms to provide precise position solutions with integrity information in signal-harsh environments. To ensure the effectiveness of monitored observations, only the carrier phase observations with the fixed integer ambiguity are used. A kinematic relative positioning mode augmented with INS is introduced to keep CRAIM available in signal-harsh environments. Additionally, an FDE method combining outlier detection from original observations and the MHSS ARAIM algorithm is proposed to improve the robustness and efficiency. The CRAIM algorithm is aimed at detecting and mitigating any potential failures in real time that may lead to hazardous situations, thereby ensuring the accuracy and reliability of the kinematic relative positioning solution in various scenes.

2. Materials and Methods

As described in Li et al. [30], each moving platform performs the TDCP/INS method on its own to achieve autonomous navigation and relative positioning, thereby remaining unaffected by the unexpected problems, such as instrument failure or interruption of the communication link, that occur in other platforms. Thus, based on the TDCP/INS method, the real-time BDS/INS kinematic relative positioning is first introduced to provide a baseline solution and ambiguity resolution in this section. Then, using DD phase observations with known integer ambiguity resolution, together with original observation-domain outlier detection (OOD) and MHSS ARAIM methods, the BDS/INS CRAIM algorithm for baseline solution between two moving platforms is presented finally.

2.1. BDS/INS Kinematic Relative Positioning Method Between Two Moving Platforms

Although the high-rate position datum by the TDCP/INS method has a systematic bias from the true position caused by the initial position, this bias does not affect the relative positioning results and integrity, especially for short baselines (especially less than 5 km). Note that the bias can be further decreased by other technologies, such as the initial position directly obtained by RTK or RT-PPP.

Based on the position datum from the base platform by the TDCP/INS method, the DD pseudo-range and carrier phase observation equation for short baseline can eliminate almost all common errors, which can be expressed as follows [31]:

$$\left\{\begin{array}{c}\nabla ∆P_{AB}^{pq}={\nabla ∆\rho }_{AB}^{pq}+\nabla ∆{\epsilon }_P\\ \nabla ∆{\Phi }_{AB}^{pq}={\nabla ∆\rho }_{AB}^{pq}+\lambda ·{\nabla ∆\mathit{N}}_{AB}^{pq}+\nabla ∆{\epsilon }_{\Phi }\end{array}\right.$$

(1)

where ∇∆ refers to the DD pseudo-range and carrier phase observation. A and B indicate the base and rover platform, respectively. p and q represent the satellites observed by platforms A and B at the same time. P and $\Phi$ denote pseudo-range and carrier observations. ρ is the geometric distance between the satellite and the platforms. ${\mathit{N}}_{AB}^{pq}$ and $\lambda$ represent the ambiguities and their wavelength, respectively. ${\epsilon }_P$ and ${\epsilon }_{\Phi }$ indicate residual errors of pseudo-range and carrier observations.

Further, the linearized equations of (1) can be rewritten as follows:

$$\left\{\begin{array}{c}(\nabla ∆p_{AB}^{pq}-{\nabla ∆\rho }_{AB}^{pq})=-\left({\mathit{l}}_B^q-{\mathit{l}}_B^p\right)\mathit{b}+\nabla ∆{\epsilon }_P\\ \left(\nabla ∆{\Phi }_{AB}^{pq}-{\nabla ∆\rho }_{AB}^{pq}\right)=-\left({\mathit{l}}_B^q-{\mathit{l}}_B^p\right)\mathit{b}+\lambda ·\nabla ∆N_{AB}^{pq}+\nabla ∆{\epsilon }_{\Phi }\end{array}\right.$$

(2)

where l represents the direction matrix with a unit sight vector between the satellites and the platforms, and b = (dx, dy, dz) refers to the baseline state vector.

Therefore, the measurement model for n + 1 satellites can be written in matrix form as follows:

$$\left[\begin{array}{c}\nabla ∆\mathit{P}-\nabla ∆{\mathit{\rho }}_0\\ \nabla ∆\mathit{\Phi }-\nabla ∆{\mathit{\rho }}_0\end{array}\right]=\left[\begin{array}{cc}\mathit{H}& {\mathbf{0}}_{n\times n}\\ \mathit{H}& \lambda {\mathit{I}}_{n\times n}\end{array}\right]\left[\begin{array}{c}\mathit{b}\\ \nabla ∆{\mathit{N}}_{n\times 1}\end{array}\right]+\left[\begin{array}{c}{\nabla ∆\epsilon }_P\\ {\nabla ∆\epsilon }_{\Phi }\end{array}\right]$$

(3)

where $\nabla ∆{\mathit{\rho }}_{\mathbf{0}}$ denotes the predicted geometric range computed with the aid of the INS-derived position. H is the geometry matrix with a dimension of n × 3. I indicates the identity matrix with the dimension presented in the subscripts.

Based on the above equations, the float ambiguity resolution can be obtained and fixed using the LAMBDA method afterwards [32,33,34]. The partial ambiguity resolution (PAR) method was used to fix the integer ambiguities as much as possible in a short period [35]. Due to the influence of observation noise, atmospheric residual, and multipath in the harsh environment, the ambiguities may be incorrectly fixed. Hence, only when the integer ambiguity resolution set could simultaneously pass the validation of the ratio and success rate tests, this resolution would be accepted and considered as fixed [36]. It should be noted that the uncertainty of the integer estimated ambiguities has not been taken into account here, as the relevant validation methods have already been well established in [35,36]. In this procedure, we can obtain a real-time baseline solution and ambiguity resolution.

2.2. CRAIM Algorithm for BDS/INS Kinematic Relative Positioning

Once the integer ambiguity resolution is obtained, the carrier phase observations are equivalent to the high-precision distances that determine the baseline solution, instead of pseudorange observations. Therefore, only the carrier phase observations with the known integer ambiguity resolution are used for integrity monitoring. In addition, only the baseline solution with integer ambiguity resolution is monitored, and the float ambiguity resolution is directly issued an unavailable alarm. Thus, the linearized DD carrier phase observation in (2) can be simply expressed as follows:

$$\mathit{y}=\mathit{G}\mathit{x}+\mathit{v}$$

(4)

where y is the vector of DD phase measurements minus the expected range. $\mathit{G}$ and $\mathit{v}$ denote the geometry matrix and the zero-mean error with Gaussian distribution, respectively. $\mathit{x}$ is the baseline state. The simultaneous multiple faults detection with the MHSS ARAIM algorithm based on (4) is implemented. Thus, the fault-tolerant baseline solution and the standard deviations are given as follows [37]:

$$\left\{\begin{array}{c} {\widehat{\mathit{x}}}^{\left(k\right)}={\mathit{S}}^{\left(k\right)}\mathit{y}\\ {\mathit{S}}^{\left(k\right)}=({\mathit{G}}^T{\mathit{W}}^{\left(\mathit{k}\right)}\mathit{G}{)}^{-1}{\mathit{G}}^T{\mathit{W}}^{\left(\mathit{k}\right)}\\ {\mathit{\sigma }}^{\left(k\right)}=({\mathit{G}}^T{\mathit{W}}^{\left(\mathit{k}\right)}\mathit{G}{)}^{-1}\end{array}\right.$$

(5)

where k is the index of the monitored fault mode and 0 indicates the all-in-view solution (fault-free mode).$k\in [1,N_{fault,max}]$.$N_{fault,max}$ indicates the maximum number of simultaneous faults; the detailed calculation can be referred to in [21]. ${\widehat{\mathit{x}}}^{\left(k\right)}$ and ${\mathsf{\sigma }}^{\left(k\right)}$ denote the fault-tolerant baseline solution and the corresponding standard deviations, respectively. ${\widehat{\mathit{x}}}^{\left(0\right)}$ and ${\mathsf{\sigma }}^{\left(0\right)}$ are the all-in view solution and that of standard deviation. ${\mathit{W}}^{\left(\mathit{k}\right)}$ is the weighting matrix, where that of the satellite fault is set to 0.

In the CRAIM framework for moving platform users, there are many possible fault subsets, yet the probability of most subsets is very low. Too many computational subsets lead to a large amount of computation in real time. Therefore, if the probability of simultaneous fault subsets is less than the integrity risk threshold, then the resulting risk will not exceed the integrity risk budget. Therefore, these subsets of faults do not need to be monitored, and monitoring only a small part of these is enough. To limit computational burden, when the probability of $N_{fault,max}$ or more simultaneous faults is lower than the threshold for the integrity risk coming from unmonitored faults, the fault subsets of faults number greater than $N_{fault,max}$ do not require monitoring. In other words, the fault subset with $N_{fault,max}$ or fewer observations that need to be monitored.

According to (5), we can see that the MHSS ARAIM algorithm does not detect an outlier on the DD carrier phase observation in (4). Meanwhile, it can also be seen that the fault-tolerant baseline solution needs to compute multiple subsets in the case of multiple faults. The maximum number and monitored fault subset can be determined according to the integrity risk threshold [37], but the fault subset number monitored by the MHSS ARAIM algorithm and the computation amount are still large in real time. If the number of DD observations is n and the number of faults is 1, the MHSS ARAIM algorithm needs to calculate $C_n^1$ subsets; if the number of faults is 2, $C_n^2$ needs to be calculated. Consequently, the greater the number of faults, the more subsets that need to be computed, and the more substantial the computational requirements become.

Considering that the fault detection of the MHSS ARAIM algorithm is performed in the position domain and has limited observation robustness, the quality of the phase observations really determines the performance of the baseline solution. Therefore, we propose an improved FDE method combining OOD and MHSS ARAIM algorithm. The proposed method is presented in a sequential manner as follows:

(1)

Initial outlier detection: Use the OOD method to identify and eliminate outliers in the DD carrier phase observations.

(2)

Subset processing: Apply the MHSS ARAIM algorithm to the remaining observation subsets.

(3)

Efficiency gain: If the MHSS ARAIM algorithm confirms the outlier again, further identification of remaining subsets is avoided. This reduces the computational load by avoiding unnecessary subset computations.

(4)

PL computation: After the satellite faults are identified and excluded, the PLs are computed based on the integrity requirement.

(5)

Alarm decision: If PL values exceed the alert limit (AL), a hazardous alarm is issued, indicating that the kinematic relative positioning solution is not reliable.

In summary, the process is given in Figure 1 of the CRAIM algorithm for BDS/INS real-time precise kinematic relative positioning between two moving platforms. The main process detail of this method is given as follows. Therefore, the advantages of the proposed method can be summarized:

(1)

Enhanced robustness: The initial OOD step improves overall FDE robustness by identifying and eliminating outliers before the MHSS ARAIM algorithm is applied.

(2)

Reduced computational load: By confirming outliers early and avoiding unnecessary subset computations, the computational complexity is significantly lower compared to the traditional MHSS ARAIM algorithm.

(3)

Improved positioning accuracy: The combination of OOD and MHSS ARAIM ensures higher quality observations, yielding more accurate positioning results.

2.3. OOD-MHSS Method

OOD-MHSS method mainly consists of OOD using χ² and w-test (w-) methods, and MHSS ARAIM algorithm, the specific details of which are as follows.

• OOD using χ² and w-test (w-) methods

Firstly, we assume that the ith phase observation contains outliers; hence, the equation of fault detection is as follows:

$$\mathit{y}\left(i\right)=\mathit{G}\left(i\right)\mathit{x}+\mathit{e}\left(i\right)\nabla b+\mathit{v}\left(i\right),i=1\dots n$$

(6)

where $\mathit{e}\left(i\right)$ is a vector whose the ith element is 1 and the other are 0. $\nabla b$ denotes the outlier value. Therefore, the ${\mathsf{\chi }}^2$ statistic and threshold are constructed for global outlier detection, which can be expressed as follows:

$$\left\{\begin{array}{c}T\left(0\right)={\mathit{v}}^T{\mathit{D}}_{nn}^{-1}\mathit{v}\\ T\left({\mathsf{\chi }}^2\right)={\mathit{F}}^{-1}(1-\alpha |n-3)\end{array}\right.$$

(7)

where $T\left(0\right)$ and $T\left({\mathsf{\chi }}^2\right)$ denote the chi-square test statistic and its threshold, respectively. $\alpha$ is the misleading information (MI) probability. n and $\mathit{v}$ indicate the number and residual vector of phase observations. ${\mathit{D}}_{nn}$ is the diagonal covariance matrix of phase observations. The operator ${\mathit{F}}^{-1}\left(p\right)$ is the cumulative distribution function of a chi-square distribution with $n-3$ degrees of freedom. If the $T\left(0\right)$ statistic follows the centralized distribution, the phase observations do not contain outliers, and the corresponding MHSS ARAIM algorithm without fault is performed. Once the $T\left(0\right)>T\left({\mathsf{\chi }}^2\right)$, the phase observations contain outliers, and then, Baarda’s w-test (w-) method is used to identify the outliers [38]. Hence, based on the normal residuals, the w statistic and threshold can be given as follows:

$$w(i)={\displaystyle \frac{{\mathit{e}\left(\mathit{i}\right)}^{\mathit{T}}{\mathit{D}}_{\mathit{v}\mathit{v}}^{-1}\mathit{v}}{\sqrt{{\mathit{e}\left(\mathit{i}\right)}^{\mathit{T}}{\mathit{D}}_{\mathit{v}\mathit{v}}^{-1}{\mathit{D}}_{\mathit{v}\mathit{v}}{\mathit{D}}_{\mathit{v}\mathit{v}}^{-1}\mathit{e}\left(\mathit{i}\right)}}}$$

(8)

where ${\mathit{D}}_{\mathit{v}\mathit{v}}$ is the covariance matrix of the residual. If the $w\left(i\right)$ follows a centralized distribution, this phase observation does not contain an outlier. Once the $w\left(i\right)>N(1-\alpha /2)$, this phase observation contains outliers.

According to the value of $N_{fault,max}$, the outlier in the phase observations exists in the following case.

(1)

$N_{fault,max}=0$, all the phase observations are deemed fault-free. MHSS ARAIM algorithm for the fault-free mode is performed, and the PLs are computed directly.

(2)

$N_{fault,max}=1$, the largest of $w\left(i\right)$ corresponding to phase observation is marked as an outlier. MHSS ARAIM algorithm for a single fault is performed to identify and exclude the satellite. Once the outlier is re-confirmed, the remaining $C_n^1-1$ subsets are skipped, and the PLs are then computed.

(3)

$N_{fault,max}=2$, the two largest values of $w\left(i\right)$ corresponding to phase observations are marked as outliers. MHSS ARAIM algorithm for two faults is performed to identify and exclude the fault observations. Once the two outliers are confirmed again, the remaining $C_n^2-1$ subsets will not be identified, and the PLs are then computed.

(4)

$N_{fault,max}>2$, the phase observations that contained outliers can be determined and marked in a similar way. MHSS ARAIM algorithm for more than two faults is performed to identify and exclude the fault satellites. The results of this step are that the fault satellites index in the phase observations.

The MHSS ARAIM algorithm combined with the ODO method compensates for missed phase-observations FDE, and reduces the computation of the observation subset in (5) under fault conditions.

• MHSS ARAIM algorithm

In the real-time data process, the faulted satellites are first excluded, and the remaining subsets are used to estimate the state vector and its error covariance of the fault-tolerant solution. Then, the fault identification and exclusion are performed by checking the consistency of the difference between the fault-tolerant solution and the all-in-view solution [21]. The indices q = 1, 2, and 3 designate the east (E), north (N), and up (U) components of the baseline solution, respectively. For each k from 1 to $N_{fault modes}$, the statistic to determine whether a fault is as follows [37]:

$$\left\{\begin{array}{c}{t}_{k,q} = {\displaystyle \frac{|{\widehat{\mathit{x}}}_q^{\left(k\right)}-{\widehat{\mathit{x}}}_q^{\left(0\right)}|}{∆{\sigma }_{ss,q}^{\left(k\right)}}} (k=\mathrm{1,2},\dots N_{fault})\\ ∆{\mathsf{\sigma }}_{ss,q}^{\left(k\right)}={\mathit{e}}_q^T∆{\mathit{\sigma }}_{ss}^{\left(k\right)}{\mathit{e}}_q\end{array}\right.$$

(9)

where $N_{fault}$ is the number of monitored fault subsets. $∆{\mathsf{\sigma }}_{ss,q}^{\left(k\right)}$ is the $∆{\mathit{\sigma }}_{ss}^{\left(k\right)}$ of solution separation in the qth direction. ${\mathit{e}}_q$ denotes a vector whose qth entry is 1 and all others are 0. For each fault subset, there are three solution separation threshold tests in each baseline component. The threshold $T_{k,q}$, indexed by the fault index k and the baseline components index q, can be computed in [21] only if all k and q pass the following test:

$$t_{k,q}={\displaystyle \frac{|{\widehat{x}}_q^{\left(k\right)}-{\widehat{x}}_q^{\left(0\right)}|}{T_{k,q}}}\le 1$$

(10)

This satellite is confirmed as the true fault and must be excluded. Then this subset is declared as fault-free, and PLs can be computed. Should any of the k or q tests fail, this excluded satellite in the subset is not a fault, and the outlier detection method for phase observations is repeated to find and index the fault satellite. Therefore, the FDE flowchart of the FDE method combining ODO in original observations and the MHSS ARAIM algorithm is shown in Figure 2.

After the satellite faults are identified and excluded, the PLs are computed by the integrity requirement, which are the error upper bounds under a given integrity risk. For example, the vertical protection level (${VPL}_k$) and horizontal protection level (${HPL}_k$) for the fault index k can be determined by the integrity risk allocated to the vertical and horizontal errors. The final VPL and HPL values are taken as the maximum in all subsets. If the $VPL>VAL$ or $HPL>HAL$, a hazardous alarm is issued to the platform’s user, indicating that the kinematic relative positioning solution is not reliable.

3. Results

In order to analyze and assess the performance of OOD-MHSS and BDS/INS CRAIM algorithm for real-time kinematic relative positioning, different kinematic experiments were carried out between two moving platforms in the urban environment of Wuhan, China. The experiment was carried out under different scenarios and platforms, primarily comprising vehicle and unmanned aerial vehicle (UAV). Both datasets came from the same receiver and INS setup mounted on the platforms. The first experiment involved a vehicle and a UAV, both equipped with integrated BDS and INS systems. The second experiment was conducted in a more challenging urban environment to evaluate the performance of the BDS/INS CRAIM algorithm. Two integrated BDS and INS devices were installed in the same manned vehicle. Satellite availability of GPS and BDS, and the position dilution of precision (PDOP) of the moving device during the observation are analyzed to reflect the integrity monitoring environment. Then, a quantitative analysis was performed, and the numerical integrity results obtained from OOD-MHSS and BDS/INS CRAIM are evaluated.

3.1. Data Collection

A K708 GNSS original equipment manufacturer (OEM) board manufactured by ComNav Technology Ltd. (Shanghai, China) in China and consumer-grade ADIS16470 MEMS manufactured by Analog Devices Inc. (Wilmington, MA, USA) were used synchronously for BDS and INS data collections. Moreover, the OEM and MEMS modules were integrated into a device that can be installed on the moving platforms, as shown in Figure 3a. BDS pseudorange and carrier phase measurements were used, while the MEMS operated at 100 Hz. Meanwhile, the specifications and configuration of each module are shown in

Table 1. Specification and configuration of the K708 GNSS OEM board.

Features	BDS-2	BDS-3
used signals	B1I, B2I, B3I	B1I, B2a, B3I
code noise	B1I: 10 cm; B2I: 10 cm; B3I: 5 cm	B1I: 10 cm; B2a: 10 cm; B3I: 5 cm
phase noise	B1I: 0.5 mm; B2I: 0.5 mm; B3I: 0.5 mm	B1I: 0.5 mm; B2a: 0.5 mm; B3I: 0.5 mm
sampling rate	1 Hz

and

Table 2. Specification and configuration of the ADIS16470 MEMS module.

Features	Gyroscope	Accelerometer
in run stability	8°/h	13 μg
random walk	$0.34°/\sqrt{\mathrm{h}}$	$0.037 \mathrm{m}/\mathrm{s}/\sqrt{\mathrm{h}}$
output noise	0.17°/s	2.3 mg
sampling rate	100 Hz

, respectively. It should be noted that the three frequency signals of BDS-2 and BDS-3 processed in this paper were B1I, B2I, B3I, and B1I, B2a, B3I, respectively. Owing to the low dynamicity and poor geometric distribution of the BDS geostationary earth orbit (GEO) satellite, the five satellites (C01–C05) were excluded in this paper. In our data processing, the critical ratio value and a priori success rate were set to 3.0 and 99.9%, respectively.

The GNSS OEM board was connected to an antenna via the cables, and then it was integrated with the MEMS module. Two sets of the integrated devices were placed on the roof rack of a sport utility vehicle.

Table 3. Strategies applied in the data processing of kinematic relative positioning between two moving platforms.

Parameter	Model	Constraint
observation	BDS-2 and BDS-3	0.02 cycles
observation weight	elevation weighted	/
cutoff elevation	15 degree	/
phase center pattern	Igs20.atx	/
ionospheric delay	DD	/
tropospheric delay	DD	/
ephemerids	broadcast	/
receiver clock drift	DD	/
EOP	fixed to IERS	/
Ambiguity resolution	Estimated+LAMBDA

gives the strategies applied in the data processing of kinematic relative positioning between two moving platforms. The inputs and constants for the CRAIM algorithm are listed in

Table 4. Models and strategies applied in the data processing of CRAIM.

Name	Description	Value (/1 h)
$P_{sat}$	Prior probability of fault in satellite	$1\times {10}^{-4}$
$P_{HMI}$	Total integrity budget	$1\times {10}^{-5}$
$P_{HMI\_NAV}$	Integrity budget allocated to satellite faults	$5\times {10}^{-6}$
$P_{HMI\_VERT}$	Integrity budget for the vertical component	$9\times {10}^{-7}$
$P_{HMI\_HOR}$	Integrity budget for the horizontal component	$1\times {10}^{-7}$
$P_{HMI\_UN}$	Threshold for the integrity risk coming from unmonitored faults	$4\times {10}^{-6}$
$P_{CONT}$	Total continuity budget	$1\times {10}^{-5}$
$P_{FA\_NAV}$	Continuity budget allocated to satellite faults	$5\times {10}^{-6}$
$P_{FA\_VERT}$	Continuity budget allocated to the vertical mode	$4\times {10}^{-6}$
$P_{FA\_HOR}$	Continuity budget allocated to the horizontal mode	$1\times {10}^{-6}$

, which refers to civil aviation [37], can preliminarily represent the case in moving platforms tasks.

3.2. OOD-MHSS Method Performance Between UAV and Vehicle

Firstly, a kinematic experiment between UAV and vehicle was carried out to assess the performance of the OOD-MHSS algorithm with respect to the conventional MHSS method in suburban Wuhan on 23 June 2021. In this experiment, the environment was generally benign and close to open-sky conditions; thus, only BDS data was used for assessment. The used vehicle and UAV are shown in Figure 3b. The vehicle was equipped with one integrated device marked as T013, and one device in an UAV was marked as T003. In this experiment, the vehicle first ran around a pond, and at the same time, the UAV circled overhead at a height of 50 m for about 25 min, then both returned to the start point. It should be noted that during the experience, there were periods when the vehicle’s travel was obstructed by trees, while the UAV flight was not affected by any obstructions.

A static base station was set up in the Science Park of Wuhan University, and the GINS RTK/INS software 1.9.0 developed by Wuhan MAP Space Time Navigation Technology Ltd. was applied, and the post-processing GPS/BDS RTK tightly combined results were used as the reference for the position error (PE) between the vehicle and UAV. In order to give a quantitative analysis and comparison between the proposed method and the conventional method, the conventional method with only the MHSS algorithm is also analyzed herein.

The entire trajectories for the vehicle and the UAV are also displayed in Figure 4. As can be seen, the vehicle follows a relatively smooth path along the predefined route, while the UAV’s trajectory is more complex and dynamic. The velocity of the vehicle and UAV throughout the data collection is displayed in Figure 5. They were generally slow and remained mostly in the 5 to 10 m/s range. The satellite availability and PDOP value are two important indicators that affect the final CRAIM algorithm performance. Figure 6 also shows the tracked BDS satellites and the corresponding PDOP values of T013 on the vehicle and T003 on the UAV during the test (15° elevation cutoff angle). The average number of satellites for the vehicle and UAV is 14.5 and 14.6, and the average PDOP values are about 2.01 and 2.02, respectively. Therefore, the satellite availability and the PDOP value satisfy the requirements of the OOD-MHSS algorithm.

Figure 6 shows the results of different methods, including position error, protection levels, and integrity information obtained in the suburban scenario. In the ARAIM subgraph, there are three kinds of integrity results: 0 indicates no fault is detected; 2 indicates that the PL value exceeds the alarm threshold; and 1 indicates that the faults are detected and excluded, and the PL value is below the thresholds. It should be noted that the blue lines separately represent horizontal AL (HAL) and vertical AL (VAL) thresholds. The protection level, driven by the precise phase observations, converges to the 5 cm range in the horizontal directions and to approximately 10 cm in the vertical direction within the first 10 s. These protection levels bound the position error quite well in Figure 6a. However, sometimes the PL values jumped and re-converged since the partial BDS observations had outages as the vehicle passed under tree cover. Once the horizontal position error (HPE) or vertical position error (VPE) value exceeds the alarm threshold, an alarm could be generated accordingly.

It can be seen from Figure 6b that the inclusion of the OOD method aiding the MHSS ARAIM algorithm enables outlier detection in measurements and adjustment of the protection levels. The OOD method mitigates the adverse influence of the outlier, resulting in smoother protection levels. The number of outlier-induced jumps in the protection levels and the alarms for the unavailability positioning are both reduced. Therefore, the ODO-MHSS method can effectively detect and eliminate the gross error and improve the protection levels.

The overall availability of the conventional MHSS method was 99.11%, with average HPL and VPL values of 5.56 and 14.46 cm at the 99.0% availability level, respectively. It should be noted that three MI events occurred, and no hazardously misleading information (HMI) was provided. While for the OOD-MHSS method, the overall availability was improved to 99.73%, the average HPL and VPL were up to about 5.21 and 13.95 cm, respectively. Both MI rate and HMI rates were zero. Therefore, the addition of OOD significantly improves availability as well as the HPL/VPL performance.

3.3. BDS/INS CRAIM Performance Between Two Devices Installed in the Same Vehicle

The previous section showed that the inclusion of the OOD-MHSS method can improve the performance of integrity monitoring. This section examines CRAIM performance with INS aiding in a Wuhan suburban scenario in China. This dataset was collected by two devices installed in the same vehicle on 11 September 2023. For the two devices in the same vehicle that run to the end point, there are several moments during the collection where partial or full BDS measurement outages were experienced. It should be noted that during the experience, there were periods when the vehicle passed under elevated highways and traveled beneath them. A high-precision navigation grade position system, POS620, is used as a reference system, which can provide positioning reference accurate to 5 cm. The reference data from the POS620 system were used to validate the performance of CRAIM with INS aiding. Overall, the data collection for the vehicle was about 70 min, and Figure 7 shows the experiment platform, trajectory, and two of the streets traveled.

Although the trajectory in Figure 7 appears to be open-sky at first glance, the overhead traffic signs and overpasses cause partial or full satellite reduction and outages quite frequently, as shown in Figure 8. The typical velocity throughout this experiment was approximately 10 to 15 m/s. This high speed exacerbates the problem of satellite signal outages because the vehicle moves quickly through areas with obstructions. These outages pose significant challenges for the integrity monitoring solution. This environment was far harsher than the experiment between UAV and vehicle because of these features and the frequency due to the velocity of the vehicle. During the data collection, the vehicle experienced multiple periods of partial or full BDS measurement outages. At about 33,925 s, the vehicle began to enter the harsh urban area from the suburban area, and the number of satellites sharply decreased from 14 to 6–7. These outages posed significant challenges for the integrity monitoring solution, as they tested the robustness of the CRAIM algorithm in maintaining accurate positioning during periods of reduced satellite visibility. The proposed CRAIM algorithm is designed to maintain accurate positioning during periods of reduced satellite visibility.

To assess the CRAIM results between the T007 and T013 installed in the same vehicle, the known baseline with a fixed distance of 0.767 m was used as a reference. Therefore, the relevant definition of baseline PE ($BPE$), baseline PL ($BPL$), and baseline AL ($BAL$) can be expressed as follows:

$$\left\{\begin{array}{l}BPE=\left|Dis-0.767\right|\\ BPL=\sqrt{{HPL}^2+{VPL}^2}\\ BAL=\sqrt{{HAL}^2+{VAL}^2}\end{array}\right.$$

(11)

where $Dis$ denotes the length of the baseline solution between T007 and T013.

The blue line in Figure 9 represents BAL thresholds, as we can see that the protection levels jumped several times and had to re-converge since partial or full BDS measurement occurred whenever the vehicle passes under viaducts or next to tall buildings. When the signals reappeared, the ambiguities had to be re-estimated, a process that required time to converge and fix, leading to multiple convergence episodes throughout the data collection. Since BDS satellite geometry determines the performance of kinematic positioning, the positioning errors and PLs of T007-T013 were significantly larger than those of T013-T003, and 99% of their BPL reached 16.63 cm. After the satellite fault was identified and eliminated, the position error was within the normal range, and the position service could continue to be provided, indicating that the new FDE algorithm in this paper could correctly identify and eliminate faults. The BPL values after FDE were also smaller than BAL; hence, the new algorithm improved the continuity of the positioning system. There are sections when the PLs increased somewhat, and this was driven by worsened BDS geometry, which still drove the PLs in the long term. During the period from the suburban area to the harsh urban area, the availability had decreased from about 98.1% to 60.4%.

With the aid of the INS, the jump in protection levels significantly decreased and was maintained throughout the experiment duration. The INS could obtain superior and continuous positioning results over a short time, which was free from the impact of signal deterioration and outage environments. This high precision and high-rate INS over a short time could still be used as the position results of the moving platform, of course, accompanied by an improvement in the availability, which was improved to over 96.67%. Compared with the BDS-only results, the availability had improved from about 60.4% to 73.3% during the period entering the harsh urban area.

BPL could reflect and accommodate BPE variations. Whenever the BPE value exceeded the BAL, the algorithm issued an alarm while maintaining an availability of 91.53%. Two MI events occurred, but no HMI was detected. The BPL values with the INS aiding compared with the BDS-only results were improved to 12.59 cm, with the improvement of about 24.3%, with no MI or HMI events. Thus, the inclusion of an IMU in the high-integrity kinematic positioning solution could significantly improve CRAIM performance when geometries deteriorate.

4. Discussion

This study builds upon previous research on BDS/INS integration and RAIM algorithms, particularly focusing on carrier phase-based RAIM (CRAIM) and its application in challenging environments. The proposed OOD-MHSS method and INS-aided CRAIM algorithm significantly improve availability, protection levels (PLs), and integrity monitoring compared to conventional methods. For instance, the OOD-MHSS method raised availability from 99.11% to 99.73% and reduced average HPL and VPL values, which aligns with the findings of other studies that highlight the benefits of combining outlier detection with RAIM algorithms. Integrating INS into the CRAIM algorithm is shown to enhance the continuity and availability of position solutions, especially in harsh urban environments. This finding is consistent with previous studies that demonstrated the effectiveness of INS in improving the robustness of GNSS positioning systems. The results show that the INS-aided system achieved a 24.3% reduction in PLs and an availability of 96.67%, which underscores the importance of INS integration for high-integrity positioning solutions.

These findings have significant implications for applications requiring high-precision and high-integrity kinematic positioning. By validating the effectiveness of the OOD-MHSS method and INS-aided CRAIM algorithm, we offer a robust framework for ensuring reliable positioning solutions in challenging environments. This is particularly important for the applications where hazardous situations must be avoided.

This research opens up several avenues for future research. One potential direction is the development of more advanced fault detection and exclusion (FDE) algorithms that can handle multiple simultaneous faults more efficiently. Combining OOD with MHSS ARAIM provides a foundation for exploring other hybrid approaches that can further reduce computational burden and improve real-time performance. Another area for future research is the integration of more sophisticated INS systems, such as higher-grade inertial measurement units (IMUs), to enhance the accuracy and reliability of the positioning solutions. The results also highlight the potential for further advancements in multi-sensor integration, such as vision or odometer, to complement GNSS and INS data. This could lead to the development of more comprehensive and robust positioning systems that can operate reliably in a wider range of environments. Future research also could focus on developing algorithms and protocols that enable seamless operation in multi-device moving platform systems, ensuring high integrity and reliability even in the presence of partial or complete sensor outages.

5. Conclusions

Based on the MHSS algorithm, a CRAIM method that uses BDS and a consumer-grade INS to perform kinematic relative positioning between two moving platforms has been developed and described in this paper, which offers a promising approach to provide precise kinematic positioning with integrity information. Additionally, the OOD-MHSS method was proposed. A set of experiments was conducted between two different types of moving platforms, leading to the following conclusions:

The CRAIM algorithm could perform effective FDE and provide centimeter-level relative position solutions together with decimeter-level protection levels. With the aid of the OOD method, the performance of the MHSS algorithm, including the availability and PLs, has been improved. The inclusion of the INS improves solution continuity and availability of position solutions when compared to the BDS-only results. With the aid of the INS, the number of re-convergences and jumps for the PLs is significantly reduced. Particularly, in harsh urban environments, PLs have been significantly reduced by up to 24.3%, and availability improved to 96.67%. These demonstrate that INS can enhance the performance of the CRAIM algorithm for kinematic positioning services. The results are consistent with previous research and highlight the potential for further advancements through multi-sensor integration and more advanced FDE algorithms. These findings have broad implications for various applications and pave the way for future research in this important field.

Finally, we emphasize the importance of integrity monitoring for BDS/INS real-time kinematic relative positioning and its potential for continued advancements in this field. By continuously incorporating multiple sensors, such as odometers or vision, the performance of CRAIM for real-time kinematic relative positioning between two platforms can be further enhanced.